F-Purity of Hypersurfaces

by Daniel Jes´us Hern´andez

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2011

Doctoral Committee: Professor Karen E. Smith, Chair Professor Melvin Hochster Professor Mircea Immanuel Mustat¸˘a Professor Guy A. Meadows Assistant Professor Wenliang Zhang c Daniel Jes´us Hern´andez 2011 All Rights Reserved Dedication: To my friends and family; I love you all

ii ACKNOWLEDGEMENTS

I thank my parents, Luis and Mercedes, and my brothers, Alfonso, Carlos, and

Manuel, for their unconditional love and support. They inspire everything I do, and without them I would never have made it to where I am now. I also thank my grandparents, Maria de Jes´us and Leopoldo Hern´andez, and Elisa and Roberto

Mart´ınez, as well as my many aunts, uncles and cousins. I’m lucky to have such a wonderful family.

The presence of the beautiful Emily Witt in my life further illustrates how lucky

I am. Emily has supported me every day for almost six years, and I surely would never have ﬁnished my Ph.D. without her. The time we spent together has been the best part of my time in Michigan.

Coming to Ann Arbor has been the best decision I’ve made in my life, and Karen

Smith is the person most responsible for making it happen. I can’t imagine having had a better advisor and mentor than Karen, and I thank her for having the patience to take me as a student. Karen is one of my best friends, and I’m very happy to have her, Sanelma, Helena and Tapio in my life. We have been through a lot together, and I thank her for teaching me so much about life and math.

I hold Mel Hochster in the highest esteem, and am very grateful for all that he has taught me. The dedication and generosity he shows towards his students is something that I deeply admire, and his example is one I will strive to follow in my career. I also especially thank him for being so good to Emily, and for introducing us

iii to cryptic crossword puzzles, as well as to the Michigan Math and Science Scholars program, of which I have many fond memories. I am very proud to say that I’d run through a brick wall for Mel.

My decision to pursue a graduate degree in mathematics is due almost entirely to David Fried and Jeehoon Park, both of whom I met in Boston. Then a graduate student, Jeehoon was my first mentor. I thank him for all the time and attention he gave me, and for being such a good friend. I also apologize to him for not studying number theory. I also thank David Fried for giving me my first reading course, for being such a positive influence on me, and for helping me find the confidence to study math. I’ll never forget either of them.

Without a doubt, the best memories I have of Ann Arbor are of time spent with my friends. I’d like to especially thank Gerardo Hern´andez Due˜nas, Yadira

Baeza Rodr´ıguez, Gerardito Hernández Baeza, Luis Núñez Betancourt, Luis Serrano

Herrera, Jos´eLuis Gonzalez Zapata and Erin Emerson, Hyosang Kang, Richard and La´ısVasques, Ben and Alison Weiss, and Greg McNulty for their friendship.

Though many others deserve mention, Rackham requires that my acknowledgements be shorter than the actual thesis, and I apologize for any omissions.

Additionally, I’d like to thank an important group of people, the Math Basketball

All-Stars. I am grateful to the following (math-hoops) friends for coming out and sharing part of their weekend with me: Ryan Kinser (the founder), Kevin Tucker,

Alan Stapledon, Yogesh More, Diane Vavrichek, Jasun Gong, Russell Golman, Kyle

Hofmann, Kacey Walker, Gerardo Hernández Dueñas, Luis Núñez Betancourt, Jae

Kyoung (JK) Kim, Yu-Jui (Jay) Huang, Ming Qin, Chong Hyun Park, David Benson-

Putnins, Greg Simon, and Jonah Blasiak. The time spent playing with them has made every sprained ankle and loose tooth worthwhile.

iv I am especially grateful to Karen Smith, Mel Hochster, Mircea Mustat¸˘a, Guy

Meadows, and Wenliang Zhang for serving on my thesis committee, to the NSF for support through RTG grant number 0502170, and to Rackham for support through a Rackham Merit Fellowship.

Last, but not least, I would like to thank Bennet Fauber for being a cool and helpful guy, Stephen DeBacker for teaching me some number theory and for some awesome MMSS celebrations, Igor Kriz for teaching me some topology and for hang- ing out at social hour, Karen Rhea for being a great friend and teaching mentor,

Yvonne Lai for showing me how to run a course, Bob Cranson for trying to get me into shape, and Tracy Coburn for being the friendliest custodian in East Hall.

v TABLE OF CONTENTS

DEDICATION ...... ii

ACKNOWLEDGEMENTS ...... iii

LIST OF FIGURES ...... viii

CHAPTER

1. Introduction ...... 1

1 Motivation ...... 1 2 Outline ...... 6

2. Singularities of hypersurfaces ...... 8

1 Singularities of hypersurfaces deﬁned via L2 conditions .... 8 2 Some standard constructions in positive characteristic .... 13 3 Singularities of hypersurfaces deﬁned via the Frobenius map . 14

4 F -singularities of hypersurfaces in F -ﬁnite regular rings ... 18 th 1 1 pe Frobenius powers ...... 18 2 Test Ideals and F -jumping numbers ...... 21

3. Some connections ...... 24

1 Preliminaries ...... 25 2 On reduction to positive characteristic ...... 28 3 On F -purity for pairs reduced from C ...... 30 4 A connection between singularities ...... 33

4. On base p expansions ...... 37

5. F -pure thresholds of hypersurfaces ...... 43

1 Truncations of the F -pure threshold ...... 43

vi 2 The set of all F -pure thresholds ...... 46 3 Purity at the F -pure threshold ...... 48

6. Splitting polytopes ...... 51

7. F -purity and log canonical singularities ...... 55

1 Some background on F -purity of monomial ideals ...... 55 2 F -pure thresholds for polynomials with good support ..... 58 3 Log canonical singularities and dense F -pure type ...... 60

8. F -singularities of diagonal hypersurfaces ...... 67

1 A detailed discussion of the main results ...... 67 1 The F -pure threshold of a diagonal hypersurface .. 67 2 Test ideals and higher jumping numbers ...... 69 2 F -pure thresholds of diagonal hypersurfaces ...... 71 3 Test ideals of diagonal hypersurfaces ...... 73 1 ProofofTheorem8.4 ...... 73 2 Some supporting lemmas ...... 75 4 On higher jumping numbers of Fermat hypersurfaces ..... 78 1 More supporting lemmas ...... 79

9. F -pure thresholds of binomial hypersurfaces ...... 85

1 The polytope P associated to a binomial ...... 85 2 Some lemmas on the computation of fptm f ...... 88 3 The F -pure threshold of a binomial hypersurface ...... 91 1 An example ...... 92 2 An algorithm ...... 93 3 ProofofTheorem9.14 ...... 95 4 Proof of the main technical lemmas ...... 96

BIBLIOGRAPHY ...... 104

vii LIST OF FIGURES

Figure

5.14.1 Three intervals that do not intersect FPT2...... 48

3 a b c c

6.7.1 The rational polytope P 0, 1 associated to x ,y , x y . ... 53

3 d1 d2 d3

6.9.1 The rational polytope P 0, 1 associated to x1 , x2 , x3 . .. 53

2 4 7 9 8 4

9.3.1 The rational polytope P 0, 1 associated to xy z , x y z . . 86

9.7.1 The decomposition of P from Example 9.3...... 87

9.15.1 p 43, ℓ L 1, fpt f 16 1 .8 base 43 ...... 93

m 9.15.2 p 37, ℓ L 2, fpt f 16 2 ε .6 34 22 7 14 29 base 37 . 94

viii CHAPTER 1

Introduction

1 Motivation

Let R be a commutative ring of prime characteristic p 0. The e -iterated

F pe Frobenius map R R (deﬁned by r r ) is a ring homomorphism whose image

pe e th

is the subring R R consisting of all p powers of elements of R. We call R

F -finite whenever the Frobenius map is a finite map of rings, and in this thesis we will deal almost exclusively with F -finite rings.

The Frobenius map has been an important tool in commutative algebra since

Kunz characterized regular rings as those for which R is ﬂat over Rp [Kun69]. In general, singular rings exhibit pathological behavior with respect to the Frobenius endomorphism, and by imposing conditions on the structure of R as an Rp-module, new classes of singularities can be deﬁned. For example, we say that R is F -pure

p p (or F -split) if the inclusion R R splits as a map of R -modules [HR76]. This condition is equivalent to the condition that R contain a free Rp-module summand of rank one. Note that a regular ring is F -pure, though the converse need not be

2 2

true: the ring Fp x, y x y is F -pure for p 2, yet always has an isolated singularity at the origin. The notion of F -purity is a critical ingredient in the proof of the well-known Hochster-Roberts Theorem on the Cohen-Macaulay property of

1 2 rings of invariants [HR74].

Recently, more subtle applications of Frobenius to deﬁne singularities has led to

new classes of F -singularities, many of which are motivated by the theory of tight

¦ closure. The tight closure of an ideal I R is an ideal I R which contains I and is “tight” in the sense that it is contained in (and is often much smaller than)

many of the more common closures of I (e.g., the radical and integral closure of

I). The ring R is called F -regular if I I for all ideals I R, and is called ¦

F -rational if I I for every ideal I generated by a system of parameters. Regular rings are F -regular, and in general, F -regular rings satisfy many nice properties.

For example, F -regular rings must be F -pure, Cohen-Macaulay, normal and must also have rational singularities [Smi97a]. For more on tight closure, the reader is referred to the original source [HH90]. For an account of the many applications of tight closure theory, see [Hoc04].

Amazingly, the F -singularities discussed above are closely related to singularity types that appear in the theory of birational geometry for varieties deﬁned over

C, and in particular, to those appearing in the so-called Minimal Model Program

[Fed83, Smi97a, Har98, Smi00, KM98].

It is standard practice in birational geometry to study the singularities of pairs

X,λ Y , where X is a variety over C, and Y X is a hypersurface in X. Via integrability conditions (or alternately, via resolution of singularities), one deﬁnes the notion of KLT and log canonical singularities for such pairs [Kol97]. By varying

the parameter λ, we may deﬁne an important invariant of Y X. The log canonical

threshold of Y X, denoted lct X,Y , is deﬁned to be the supremum over all

parameters λ such that the pair X,λ Y is log canonical. We may replace “log canonical” in the deﬁnition of lctX,Y with “KLT” without aﬀecting the value of

this invariant. It is a fact that lctX,Y 0, 1 , with smaller values corresponding

to worse singularities. This is readily seen in the case that X C , and Y V f

is the hypersurface in X deﬁned a polynomial f C x1, , xm , where we have the

following concrete description of the log canonical threshold:

m 1 m

lct C , V f sup λ R¥0 : 2λ is locally integrable in C . f

By again using resolution of singularities (or integrability considerations), one

may assign to each pair X,λ Y an ideal J X,λ Y called the multiplier ideal

of the pair X,λ g . The ideal J X,λ Y must be trivial for small values of λ,

and get smaller as λ increases. By varying the parameter λ, we may recover the log

canonical threshold of X,Y : lct X,Y sup λ 0 : J X,λ Y is trivial .

For more on the basic properties of multiplier ideals and their jumping numbers, as well as their role in higher dimensional birational geometry, see (for example)

[BL04, Laz09, Laz04].

Motivated by geometric considerations, Hara and Watanabe recently extended

the notion of F -purity to pairs of the form R,λ f , where f is a non-zero, non-unit element of R, and λ is a non-negative real parameter [HW02, Tak04]. For example,

e e

Ô ¡ Õ

p p 1 λ

we say that the pair R,λ f is F -pure if the inclusion R f R splits

pe pe N pe as a map of R -modules for all e 0. Here, R f denotes the R -submodule

of R generated by f N . This generalizes the deﬁnition of F -purity for rings as R is

F -pure if and only if the pair R, 0 f is F -pure. This notion, though technical,

adds great ﬂexibility to the theory, and allows one to deﬁne F -pure thresholds. The

F -pure threshold of an element f R, denoted fpt R, f , is the supremum over all

λ 0 such that the pair R,λ f is F -pure. In the case that f is an element of an F -ﬁnite regular local ring R, m , the F -pure 4

threshold has the following concrete description:

e e

Ö ×

p λ p

m fpt R, f sup λ R¥0 : f for some e 1 ,

e F e × Öp where m is the ideal of R generated by the image of m under R R. Note

e e e ×

Öp p p

m m that if g1, ,ga , then g1 , ,ga . The deﬁnition of the F -pure threshold is analogous to that of the log canonical threshold in complex algebraic

geometry.

By again using the Frobenius morphism, one deﬁnes a family of ideals τ R,λ f

of R indexed by a non-negative real parameter λ. The ideal τ R,λ f is called the

test ideal of the pair R,λ f . Test ideals (deﬁned in the context of tight closure) were originally introduced in [HH90], and generalized to pairs in [HY03]. Test ideals vary with respect to λ in the same way that multiplier ideals do: they must be trivial

for small values of λ, and also get smaller as λ increases. They also may be used to

recover the F -pure threshold: fpt R, f sup λ R¥0 : τ R,λ f R . There is a very close relationship between F -purity and log canonical singularities, which

m m

we will sketch below in the special case of a hypersurface sitting in Q C .

Fix g Q x1, , xm Q x , a non-negative real parameter λ 0, and consider

the ideal J Q x ,λ g Q x . For p 0, we may reduce the coeﬃcients of g, as

well as those of some ﬁxed set of generators of J Q x ,λ g modulo p to obtain a

polynomial gp and ideal J Q x ,λ g p in Fp x1, , xm Fp x . The following

theorem allows us to compare the ideals J Q x ,λ g p and τ Fp x ,λ gp .

Theorem 1.1. [Smi00, HY03] For p 0, τ Fp x ,λ gp J Q x ,λ g p

uniformly in λ. Furthermore, for every λ, there exists a positive integer Nλ such that

τ Fp x ,λ gp J Q x ,λ g p for all p Nλ.

We stress that the assignment λ Nλ is typically an increasing function of λ. We 5

also have the following relationship between log canonical singularities and F -purity.

Theorem 1.2. [HW02, Tak04] The pair Q x ,λ f is log canonical if the pairs

Fp x ,λ fp are F -pure for inﬁnitely many p 0.

The preceding results imply the following relationship between thresholds:

(1.2.1) fptfp lct f and lim fpt fp lct f .

pÑ8

This behavior is illustrated in the following example.

2 3 5

Example 1.3. If f x y Q x, y , then lct f 6 , and

1 2 p 2

23 p 3

fptfp .

56 p 1 mod 6

5 1

6 6p p 5 mod 6

In this example, it follows from Dirichlet’s Theorem on primes in arithmetic progres-

sions that there exist inﬁnitely many primes p such that fptfp lct f .

Theorem 1.2 and Example 1.3 motivate the following two conjectures.

Conjecture 1.4. If the pair Q ,λ V f is log canonical, then Fp x ,λ fp is

F -pure for inﬁnitely many p.

Conjecture 1.5. There exist inﬁnitely many primes such that fptfp lct f .

It turns out that these conjectures are equivalent (see Theorem 5.19), and their veriﬁcation represents a long-standing open problem [Fed83, Smi97b, EM06]. They also serve as motivation for much of the work in this thesis. 6

2 Outline

In Chapter 2, we consider two different notions of singularity. We first consider log canonical singularities, which is defined for polynomials over C via the use of resolution of singularities and L2-methods. This discussion of log canonical singularities is kept light, as it will only serve as motivation for our main results. We next consider F -purity. Since our main results deal with this singularity type, we include a detailed discussion of F -purity of hypersurfaces, and place emphasis on the special case that R is a regular ring. Chapter 2 concludes with a discussion of the deep connection between log canonical singularities and F -purity. To precisely describe this connection, we provide a careful exposition on the process of reduction to positive characteristic.

In Chapter 4, we examine base p expansions of non-negative real numbers. Though the ideas in this chapter are elementary, they prove to be useful in the study of singularities in positive characteristic.

In Chapter 5, we study F -purity of hypersurfaces in the most general settings. In particular, we begin our study of the F -pure threshold. We examine the relationship

between diﬀerent variants of F -purity, and characterize their behavior in terms of

the F -pure threshold. For example, in Theorem 5.19, we show that R, fpt f f is F -pure, and is sharply F -pure if and only if fptf is a rational number with denominator not divisible by the characteristic. In Proposition 5.12, we deduce important restrictions on the set of all F -pure thresholds. Many of the results in

Chapter 5 generalize results previously shown to hold in F -ﬁnite regular local rings.

In Chapter 6, we associate to polynomial a rational, convex polytope P . This

polytope is “small” in the sense that it is contained in 0, 1 for some n 1. This 7 polytope will be key to our study of F -purity of polynomials, and in this chapter we develop some of the important properties of P .

In Chapter 7, we show that log canonical singularities is “equivalent” to F -purity for most polynomials. Precisely stated, Theorem 7.17 veriﬁes Conjectures 1.4 and

1.5 for polynomials whose associated polytopes satisfy a natural non-degeneracy condition. Theorem 7.18 builds on this to show that a natural generalization of these conjectures holds for a very general complex polynomial.

In Chapter 8, we build on the methods of Chapter 7 to study invariants of singularities associated to diagonal hypersurfaces over ﬁelds of prime characteristic. A

d1 dn

diagonal hypersurface is given by a polynomial of the form x1 xn , and we

call a diagonal hypersurface Fermat whenever d1 d2 dn n d for some

d 1. In Theorem 8.1, we present a formula for the F -pure threshold of an arbitrary diagonal hypersurface. Furthermore, we calculate the ﬁrst non-trivial test ideal associated to a diagonal hypersurface. (See Theorem 8.4.) In Theorem 8.6, we examine the existence of, and give formulas for, higher jumping numbers of diagonal hypersurfaces.

In Chapter 9, we examine the F -pure threshold of a binomial hypersurface. By deﬁnition, a binomial is the sum of two distinct monomials. Our main result of this chapter is Algorithm 9.16, an algorithm for computing the F -pure threshold of an arbitrary binomial hypersurface. This algorithm depends on the characteristic of the ambient space and the geometry of the polytope P . CHAPTER 2

Singularities of hypersurfaces

1 Singularities of hypersurfaces deﬁned via L2 conditions

Let f C x1, , xm be a polynomial with complex coeﬃcients, and consider

its zero set Z z C : f z 0 . We also consider its singular set Zsing

m B f f

z C : z z f z 0 . For every non-negative real param- B B x1 xm

2m m Γλ 1

eter λ, consider the function R C R deﬁned by Γλ x1, , xm λ .

Ô Õ| | f x

Note that Γλ has a pole at every point z Z, and understanding how “bad” these poles are provides a measure of the singularities of f. Recall that the function Γλ is

2 m 2 1

called locally L at a point z C if Γλ x 2λ is integrable in a neighborhood

Ô Õ| | f x

of z. Note that Γλ is locally integrable at z for every λ whenever z Z.

Deﬁnition 2.1. We say that the pair C ,λ f is KLT at z if the function Γλ is locally L2 at z. A pair is said to be KLT if it is KLT at every point of Z.

m m

Deﬁnition 2.2. We say that C ,λ f is log canonical at z if C ,ε f is KLT at z for all 0 ε λ. A pair is log canonical if it is log canonical at every point of Z.

m Deﬁnition 2.3. We deﬁne the log canonical threshold of f at z, denoted lctz C , f ,

m m

as follows: lctz C , f sup λ R¥0 : C ,λ f is log canonical at z . We de-

m ﬁne the (global) log canonical threshold of f, denoted lctC , f , as follows:

8 9

m m

lctC , f sup λ 0 : C ,λ f is log canonical . We will often write lct f

m m

and lctz f instead of lct C , f and lctz C , f .

m m

As C ,λ f being log canonical is a local condition, we have that lct C , f

m m

inf lctz C , f : z Z . Furthermore, it follows by deﬁnition that lctz C , f

sup λ R¥0 : C ,λ f is KLT at z , and thus the log canonical threshold may also be called the KLT threshold.

a1 am m

If f x1 xm is a monomial, then C ,λ f is KLT if the integral

dx1 dxm 2

a1 am λ

x1 xm exists in a neighborhood of the origin. By Fubini’s Theorem, it suﬃces to show that

dx (2.3.1) i

2aiλ

xi exists in a neighborhood of the origin for every 1 i m, and by referring to polar

coordinates, we see that the integral in (2.3.1) exists in neighborhood of the origin

1 m 1 1

if and only if 0 λ . It follows that lct C , f min , , . If f is not ai a1 am

m a monomial, is not at all clear how one might compute lctC , f . Fortunately, we can always monomialize f via a log resolution.

Remark 2.4. Let π : X˜ C be a holomorphic map of complex varieties. If

U, x˜1, , x˜m are coordinates on X˜, we may consider JC π , the complex Jacobian

of π on U: if π x˜1, , x˜n g1 x˜ , ,gm x˜ on U, then JC π is deﬁned on U as

B gi x˜ . It is clear that this deﬁnition defends on the choice of coordinate system.

B x˜j

½ ½ ½ ½

On the other hand, if U , x˜1, , x˜m is another set of local coordinates with U U,

then the determinant of the Jacobian with respect to the coordinates U and U ½ diﬀer

by a non-zero constant. Thus, the closed set Eπ : z˜ X : det JC π z˜ 0 is

well deﬁned, and it is a consequence of the Implicit Function Theorem that Eπ

z˜ X˜ : π is not an isomorphism locally at z˜ . 10

Theorem 2.5. [Hir64] There exists a smooth complex variety X˜ and a smooth

proper map π : X˜ C of varieties satisfying the following conditions:

1. π Eπ Zsing.

2. X˜ can be covered by local coordinates U, x˜1, , x˜m such that both f π and

a1 am

det JC π are monomials on U. Precisely stated, f π α x˜1 x˜m and

b1 bm

det JC π β x˜1 x˜m for some functions α and β that do not vanish on U.

Remark 2.6. The set of integers appearing as exponents in some local monomial-

ization of f π is ﬁnite. Similarly, the set integers appearing as an exponent in some

local monomialization of det JC π is ﬁnite. Indeed, they are the coeﬃcients of the

divisors div det JC π and div f π on X˜.

Remark 2.7. There exist local coordinates U, x˜1, , x˜m on X such that f π is

a1 am

a non-vanishing multiple ofx ˜1 x˜m on U, det JC π is a non-vanishing multiple of

b1 bm

x˜1 x˜m on U, and such that bi 0 and ai 0 for some i. If not, then det JC π

¡1

would vanish whenever f π vanishes, so that π Z Eπ. However, this is impossible as π is an isomorphism at every point in Z not in π Eπ .

m Fix a log resolution π : X˜ C of f as in Theorem 2.5. By the change of

variables formula for integration, we see that the function 2λ is locally integrable | | f

in Cm if and only if the functions

b1 bm

det JR π det JC π x˜1 x˜m

(2.7.1)

2λ λ a1λ amλ

f π f π x˜1 x˜m

are integrable in the local coordinate patches U, x˜1, , x˜m given by Theorem 2.5. In (2.7.1), JR π is the real Jacobian of π: it follows from the Cauchy-Riemann

equations that det JR π det JC π . By Fubini’s Theorem, these functions are

¡ Õ

2Ôbi λai integrable on U if and only if each function x˜i is integrable on U, and we

may readily verify that this occurs if and only if

(2.7.2) bi aiλ 1 for 1 i m.

Though Theorem 2.5 guarantees the existence of log resolutions of f, computing such resolutions is often diﬃcult, and it remains highly non-trivial to compute the log canonical threshold of an arbitrary polynomial. However, the use of log resolutions

allows one to deduce the following facts regarding log canonical thresholds.

Lemma 2.8. Let f C x1, , xm be a non-constant polynomial.

1. lctf 0, 1 Q.

2. The pair C , lct f f is log canonical.

Proof. From (2.7.2), it follows that

m bi 1

(2.8.1) lctC , f min :1 i m ,

where we range over all coordinates U, x˜1, , x˜m satisfying the condition in The-

orem 2.5. This is a minimum, and not an inﬁmum, by Remark 2.6. Furthermore, by

Remark 2.7, there exist coordinates U, x˜1, , x˜m with bi 0 and ai 0 for some

i, which shows that lctf 1. To see that the pair C , lct f f is log canonical,

note that the equations in (2.7.2) are satisﬁed (by deﬁnition) if 0 λ lct f .

We now consider another important invariant of singularities of pairs.

Deﬁnition 2.9. We deﬁne the multiplier ideal of the pair C ,λ f as follows:

m g m

J C ,λ f g C x1, , xm : f λ is locally integrable on C . 12

m m

Remark 2.10. Note that C ,λ f is KLT if and only if 1 J C ,λ f , and it

follows that lct f sup λ R¥0 : J C ,λ f C x1, , xm . Furthermore,

if z C and m C x1, , xm is the corresponding maximal ideal, we also

have that lctz f sup λ R¥0 : J C ,λ f m C x1, , xm m . Indeed, this

follows from the fact that the ideal J C ,λ f m is equal to

g C x1, , xm m : is locally integrable at z . f λ

m m

Note that J C ,λ1 f J C ,λ0 f whenever λ0 λ1. By deﬁnition,

m m

J C , 0 f C x1, , xm , and we have just observed that J C ,ε f is also

trivial for every 0 ε lct f . Thus, the multiplier ideal is locally constant to the

right of λ 0, and this behavior is typical: for every λ0 0, there exists λ1 λ0 such that

m m

(2.10.1) J C ,ε f J C ,λ0 f whenever λ0 ε λ1.

The statement in (2.10.1) motivates the following deﬁnition.

Deﬁnition 2.11. We say that λ 0 is an jumping number of f if λ 0, or if λ 0

m m

and J C ,λ f is strictly contained in J C , λ ε f for all 0 ε λ.

The following proposition shows that to study the jumping numbers of f, it suﬃces to study the jumping numbers that are contained in the unit interval.

Proposition 2.12. [Laz04] Every natural number is a jumping number of f, and

a positive real number λ N is a jumping number of f if and only if the fractional

part of λ, which is contained in 0, 1 , is a jumping number of f.

By deﬁnition, lctf is the ﬁrst non-zero jumping number of f, and in this way we may consider the jumping numbers of f to be generalizations of the lctf . Though 13 they will not be explicitly considered in this thesis, they will be compared compared to their positive characteristic analogs in Chapter 8.

Remark 2.13. Fix a log resolution π : X˜ C . Then, as in (2.7.1), the condition

g ˜ that f λ be locally integrable can be translated to an integrability condition on X via π. As before, the advantage in doing so is that f becomes a monomial in the local

coordinates of X˜, which allows one to explicitly compute J C ,λ f in terms of the exponents ai and bi appearing in Theorem 2.5. This way of thinking about multiplier ideals is very powerful, and the statement in (2.10.1) is obvious from this perspective. This characterization also allows one to give a deﬁnition of multiplier ideals and log canonical singularities in every setting in which log resolutions exist.

This generalization will not be discussed further in this thesis, and the reader is referred to [BL04, EM06] for an introduction, and to [Laz04] for a more rigorous development.

2 Some standard constructions in positive characteristic

F e

Let R be a reduced ring of prime characteristic p 0. For e 1, let R R

th pe e

e r r F R

denote the iterated Frobenius morphism deﬁned by . We will use ¦

e pe pe to denote R when considered as an R-algebra via F . If R r : r R is the

e th e

p R R F R subring consisting of powers of , then the -algebra structure of ¦ and

the Rpe -algebra structure of R are isomorphic.

e e ß

1ßp 1 p For e 1, let R be the set consisting of the formal symbols f : f R .

e e e e

ß ß ß

1ßp 1 p 1 p 1 p

We deﬁne a ring structure on R by setting f g : f g and

e e e e

ß ß ß

1ßp 1 p 1 p 1 p

f g : fg . If R is a domain, then R admits a more concrete

e description: Let L be a ﬁxed algebraic closure of the fraction ﬁeld of R, and let f 1ßp

denote the unique root of the equation T f L T in L. We may then describe 14

1ßp e th R as the subring of L consisting of all p -roots of elements of R. For example,

e e ß e 1ßp 1 p

1ßp

if R Fp x1, , xm , then R Fp x1 , , xm .

e e e ß

1ßp p 1 p

As R is reduced, we have an inclusion R R given by r r . This e

pe 1ßp pe

is an inclusion because r 0 r 0 r 0, and it is a ring

e 1 p e e e

ß ß

p pe pe 1ßp pe 1 p pe 1 p

map since r s r s r s r s . We also

e e ß

e 1ßp 1 p

F R R R r r note that ¦ as -modules via the isomorphism . This map and

e th e th

its inverse are often referred to as taking “p roots / raising to p powers.”

d e d d e e d

ß ß ß

1ßp 1 p 1 p p 1 p

We have an inclusion (of rings) R R given by r r , and

d e e d d

ß ß ß

1ßp 1 p 1 p 1 p

in this way, we identify R and R as R -modules. Also note that

e ß for any multiplicative set of W of R, we may canonically identify W ¡1 R1 p and

1ßp ¡

¡1 1 W R as modules over the localization W R.

1ßp

We say that R is F -ﬁnite if R (equivalently, F¦ R) is a ﬁnitely generated R-

R F R1ßp F eR

module. One can show that is -ﬁnite if and only if (equivalently, ¦ ) is a ﬁnite R-module for every e 1, or even for one single e 1.

Example 2.14. By definition, a field K of characteristic p 0 is F -finite if and

only if K : K , and a finitely generated algebra over K is F -finite if and only if K is F -finite. [BMS08, Example 2.1]

3 Singularities of hypersurfaces deﬁned via the Frobenius map

The Frobenius map has been an important tool in commutative algebra since

e Kunz characterized regular rings as those for which R1ßp is ﬂat over R [Kun69].

1ßp Recall that that R is said to be F -pure (or F -split) if the inclusion R R splits

as a map of R-modules for some e 1. An F -pure ring is necessarily reduced, and

1ßp R is F -pure if and only if the inclusion R R splits as a map of R-modules for

15 every e 1, or even for one single e 1 [HR76].

We would like to generalize of the notion of F -purity to pairs. By deﬁnition,

a pair consists of the ambient ring R, an element f R, and a non-negative real

parameter λ, and is denoted by R,λ f .

e e ß

N ßp 1 p

We say that the inclusion R f R splits over R (or splits as a map of

e e ß

1ßp N p

R-modules) if there exists an map θ HomR R , R with θ f 1. Note that

e e ß

N ßp 1 p if f is a unit, then R f R splits over R if and only if R is F -pure. For the

remainder of this section, R will denote an F -pure ring of characteristic p 0, and f will denote a non-zero, non-unit element of R.

λ 1ßp

We would like to say that the pair R,λ f is F -pure if R f R splits for

λ some (or all) e 1. Of course, the problem in doing so is that f need not represent

1ßp an element of R for any e 1. On way of getting arounds this is to approximate

λ by a sequence of rational numbers λe with λe pe N and lim λe λ, and then eÑ8 e

λe 1ßp require that some (or all) of the inclusions R f R split over R. Of course,

e e e

Ô ¡ Õ Ô ¡ Õ p λ p 1 λ p 1 λ there are numerous ways to approximate λ (for example, pe , pe , and pe

would all work) and each choice may lead to a distinct notion of F -purity for pairs.

Deﬁnition 2.15. [Tak04, TW04, Sch08] The pair R,λ f is said to be

e e e

Ô ¡ Õ ß ß

p 1 λ p 1 p

1. F -pure if R f R splits over R for all e 1,

e e e

ß ß

p λ p 1 p

2. strongly F -pure if R f R splits overR for some e 1, and

e e e

Ô ¡ Õ ß ß

p 1 λ p 1 p

3. sharply F -pure if R f R splits over R for some e 1.

If ℘ R is a prime ideal, we say that R,λ f is (strongly/sharply) F -pure at ℘ if

the pair R℘,λ f is (strongly/sharply) F -pure.

The following lemma shows that F -purity may be detected locally.

Lemma 2.16. R,λ f is (strongly/sharply) F -pure if and only if R℘,λ f is

(strongly/sharply) F -pure at every prime ideal ℘ R.

Proof. We will only prove the statement regarding F -purity. By deﬁnition, R,λ f

e e e

Ô ¡ Õ ß

1ßp p 1 λ p

is F -pure if the map Θ : HomR R , R R given by evaluation at f is

1ßp

surjective for all e 1. However, Θ is surjective if and only if Θ℘ : HomR R , R ℘ e

1ßp HomR℘ R℘ , R℘ R℘ is surjective for every ℘ Spec R, and under this identiﬁ-

e e

Ô ¡ Õ ß p 1 λ p cation is is easy to verify that Θ℘ is also given by evaluation at f .

Of course, Lemma 2.16 also holds with “prime ideal” replaced by “maximal ideal.”

We now examine how F -purity for a pair is aﬀected by varying the parameter λ.

e e ß

N ßp 1 p Lemma 2.17. If the inclusion R f R splits as a map of R-modules, so

e e ß

aßp 1 p

must the inclusion R f R for all 0 a N.

e e ß

1ßp N p

Proof. By hypothesis, there exists a map θ HomR R , R with θ f 1. If

N ¡a pe ¤f

e e e θ

ß ß

1ßp 1 p 1 p

we let φ HomR R , R denote the composition R R R, then e

aßp φ f 1.

Lemma 2.18. Let λ be a positive real number.

1. If R,λ f is F -pure, then so is R,ε f for every 0 ε λ.

2. R,λ f is not F -pure if λ 1.

e e e

Ù Ø Ù Ø Ù

Proof. As Ø p 1 0 0, the pair R, 0 f is F -pure. As p 1 ε p 1 λ ,

Lemma 2.17 shows that R,ε f is F -pure whenever R,λ f is. For the second

point, suppose that R, 1 ε f is F -pure for some ε 0, so that

e e e

Ô ¡ ÕÔ Õ ß ß

p 1 1 ε p 1 p

(2.18.1) R f R splits over R for every e 1. 17

e e e e

Note that p 1 1 ε p 1 p 1 ε p for e 0. By (2.18.1) and

1ßp

Lemma 2.17, we have that R f R splits over R for e 0. We conclude that

1ßp

there exists a map θ HomR R , R with 1 θ f f θ 1 , which is impossible

as f is not a unit.

Deﬁnition 2.19. [TW04] We deﬁne the F -pure threshold of f R, denoted fpt R, f ,

as follows: fpt R, f sup λ R¥0 : R,λ f is F -pure . Note that fpt R,u

whenever u is a unit in R. We deﬁne the F -pure threshold of f at ℘ Spec R, de-

noted fpt℘ R, f , as follows: fpt℘ R, f sup λ R¥0 : R,λ f is F -pure at ℘ .

When there is possibility for confusion, we write fptf and fpt℘ f rather than

fptR, f and fpt℘ R, f . Note that, by deﬁnition, fpt℘ R, f fpt R℘, f .

Remark 2.20. By Lemma 2.18, we see that fptR, f 0, 1 .

Remark 2.21. Though not at all obvious from the deﬁnition, fpt f Q whenever

R is F -ﬁnite and regular [BMS08, Theorem 3.1]. For rationality results in a more general setting, see [KLZ09, BSTZ09].

Proposition 2.22. The F -pure and strongly (respectively, sharply) F -pure thresh-

old agree: fptR, f supλ R,λ f is strongly (respectively, sharply) F -pure .

Proof. The equality of these thresholds is shown in [TW04, Proposition 2.2] and

[Sch08, Proposition 5.3].

Remark 2.23. Let W R be any multiplicative set. If R,λ f is F -pure, it ¡

¡1 1

follows that W R,λ f is also F -pure, and thus fpt R, f fpt W R, f .

Proposition 2.24. The F -pure threshold may be computed locally. Precisely stated,

we have that fptf inf fpt℘ f : ℘ Spec R .

Proof. Note that fptf inf fpt℘ f : ℘ Spec R by Remark 2.23. We may

now assume that λ : inf fpt℘ f : ℘ Spec R 0. For 0 ε λ, we have

that R, λ ε f is F -pure at every ℘, so that R, λ ε f is also F -pure by

Lemma 2.16. Thus, fptf λ ε, and the claim follows by letting ε 0.

4 F -singularities of hypersurfaces in F -ﬁnite regular rings

We will next focus on the case of a smooth ambient ring, and will closely follow

[BMS08]. Throughout this section, R will be assumed to be an F -ﬁnite regular ring

of characteristic p 0, and f will denote a non-zero, non-unit in R. In this setting,

e R1ßp F eR R

we know that (equivalently, ¦ ) is a ﬁnitely generated ﬂat -module [Kun69]. th 1 1 pe Frobenius powers

In this subsection, we deﬁne (see Deﬁnition 2.26) and study the properties of th 1 pe Frobenius powers of principal ideals. These ideals will be used in the next

subsection to deﬁne the test ideal associated to a pair.

e e ×

Öp p For every ideal I R, let I denote the ideal generated by the set g : g I .

e e

× Ö × Öp e p th

I IF R I e I

It is clear that ¦ , and we call the Frobenius power of . Note

Ö ×

e p e d

× Ö × Öp p e

I I F R R that by deﬁnition. Furthermore, as ¦ is a ﬂat -module, we

e e ×

p Öp

have that g I if and only if g I.

e e

× Ö ×

Öp p

Lemma 2.25. Let In n be a family of ideals of R. Then n In n In .

e ×

e Öp

S F R I IS IS I S Proof. Let : ¦ . As , it suﬃces to show that n n . Furthermore, it suﬃces to check this equality after localizing at every prime ideal

of S. However, S is a ﬁnitely generated ﬂat R-algebra (and hence locally free), in

which case it is easy to verify that n IS℘ n I S℘ for all ℘ Spec S.

19 th 1

Deﬁnition 2.26. [BMS08] We deﬁne the pe Frobenius power of f to be the th

e × Öp 1 minimal ideal J, with respect to inclusion, such that f J . We denote the pe

pe Frobenius power of f by f . This is well deﬁned by Lemma 2.25.

Lemma 2.27. Let f R.

pe Öp

1. If J R is an ideal, then f J if and only if f J .

d ×

1 1 Öp

pe pe d

2. f f .

N pe n pe

3. If N n are integers, then f f .

1 1

d pe d

p pe

4. f f .

Öp pe

Proof. By deﬁnition, we have that f J f J. On the other hand, if

e × 1 1 Öp 1

Ö ×

pe pe p pe d

f J, then f f J . For the second point, let J : f .

e 1 Ö

d p × d

× Ö ×

Öp pe p

Then f J , which implies that f J . The third point follows from

e e

× Ö ×

n Öp N p the fact that if f I , then f I , while the last follows from the fact that

d e d e

× Ö ×

p Öp p

f I f I . th 1 Lemma 2.28. [BMS08, Lemma 2.7] The formation of pe powers commutes with

1 1

¡1 pe pe

localization: If W R is a multiplicative set, then W f fW , where

¡1 fW denotes the image of f in W R.

Proposition 2.29. [BMS08, Proposition 2.5] Suppose that R is a ﬁnitely-generated

pe pe

free module over the subring R R. If β1, , βN is a basis for R over R , and

pe pe pe

f a1 β1 an βN is the unique representation of f as an R -linear combination

of this basis, then f a1, , an R.

e e × Ö ×

p Öp

Proof. It is obvious that f a1, , an . Next, suppose that f J for some

r pe N pe

ideal J. Express f g h with h J, and express g s β as an i1 i i i i j 1 ij j 20

R -linear combination of the basis elements β1, , βN . Equating the coeﬃcients of

r pe pe N pe pe

the β in the expression f g h will show that a s h , and so we i i1 i i i j 1 ij i

N may conclude that a s h J. i j 1 ij i

pe 1ßp

Corollary 2.30. The ideal f is the image of the map HomR R , R R

e given by evaluation at f 1ßp .

Proof. The image of the evaluation map obviously commutes with localization, and

pe Lemma 2.28 shows that the same is true for f , which allows us to assume that

R is local. In this case, R is a ﬁnitely-generated, ﬂat (and hence free) module over the

subring Rpe , and the claim is then an immediate corollary of Proposition 2.29. th 1

Corollary 2.30 allows us to relate pe Frobenius powers with F -purity. Corollary 2.31. Let S, n be an F -ﬁnite regular local ring, and let f be a non-unit

e e e

ß Ö ×

1ßp 1 p p

of S. Then the inclusion S f S splits over S if and only if f n .

e e

1ßp 1 p pe

Proof. By Corollary 2.30, S f S splits if and only if f n, which by

e × Öp Lemma 2.27 occurs if and only if f n .

Ô ¡ Õ

p 1 λ

Corollary 2.32. R,λ f is F -pure at a maximal ideal m if and only if f

e ×

Öp

m for all e 1. Similarly, R,λ f is strongly F -pure at m if and only if

e e e e

Ö × Ô ¡ Õ Ö ×

p λ p p 1 λ p

f m for some e 1, and is sharply F -pure at m if and only if f m

for some e 1.

e e 1ßp

Ô ¡ Õ ß

p 1 λ p

Proof. R,λ f is F -pure at m if and only if the inclusion Rm f Rm

splits for every e 1. However, Corollary 2.31 shows that this map splits if and

e e e e Ô ¡ Õ Ö × Ö × ×

p 1 λ p Öp p

only if f m Rm. As m m, m is a primary ideal, and hence

e e e e

× Ö × Ô ¡ Õ Ö ×

Öp m p p 1 λ p

m R R m . From this, we can conclude that f m Rm if and

e e

Ô ¡ Õ Ö × p 1 λ p only if f m . 21

The next statement follows from Corollary 2.32 and Proposition 2.22.

e e

Ö ×

p λ p

m Corollary 2.33. fptm f sup λ R¥0 : f for some e 1 .

2 Test Ideals and F -jumping numbers

In this subsection, we deﬁne and present some of the basic properties of the test

ideal corresponding to a pair R,λ f . In order to deﬁne the test ideal, we will need

the following lemma.

e 1 e 1

e e p ×

p λ p Öp λ

Lemma 2.34. For every λ 0, we have that f f .

e 1 pe 1 e

× Öp λ p λ Proof. Let J : f . By Deﬁnition 2.26, it suﬃces to show that f is

Öp e 1 e e e 1 e

Ö × Ö × Ö ×

contained in J . Note that p λ p p λ p p λ , so that p λ p p λ as

e e 1 e p e 1

× Ö × Ö ×

Öp λ p p λ p well. By deﬁnition, we have that f J , and so f J as well.

e e

Ö × p λ p

As R is regular, we may conclude that f J .

Definition 2.35. [BMS08, Definition 2.9] We define the test ideal of R,λ f as

e e

p λ p

follows: τ R,λ f : f . This is an ideal of R by Lemma 2.34. e¥1

e e

p λ p

Remark 2.36. As R is Noetherian, τ R,λ f f for e 0.

Remark 2.37. It follows immediately from Remark 2.36 and Lemma 2.28 that ¡

¡1 1

W τ R,λ f τ W R,λ f for every multiplicative set W R.

The theory of test ideals is a key component in the theory of tight closure [HR76,

HH90, HY03]. The following result, which is an immediate corollary of the last point of Lemma 2.27, allows us to determine when this ascending chain stabilizes in an important special case.

1 p λ pe

Lemma 2.38. [BMS09, Lemma 2.1] If λ pe N, then τ R,λ f f . 22

Proposition 2.39. [BMS08, Proposition 2.11, Corollary 2.16]

1. τ R,λ1 f τ R,λ0 f if λ1 λ0.

2. τ R,λ f R for 0 λ 1.

3. For every λ 0, there exists ε 0 so that τ R,λ f τ R, λ ε f .

1 1

e e e e

p λ1 p p λ0 p

Proof. Fix e 0 so that τ R,λ1 f f and τ R,λ0 f f .

The assertion that τ R,λ1 f τ R,λ0 f then follows from Lemma 2.27.

To prove the second point, it suﬃces to show that τ R,λ f is trivial for some

λ 0. By Lemma 2.27, the ideals f form an increasing chain, so we may

1 1 1

pe pd pd

ﬁx a d such that f f for all e d. If f R, there exists a

1 1

pe pd

maximal ideal m with f f m for all e d. Lemma 2.27 then shows

e ×

Öp

that f m for all e d, contradicting the fact that f 0. We conclude that

1 pd

τ R, pd f f R.

For the last point, it suﬃces to show that τ R, λ ε f τ R,λ f for

d pd ×

Öp λ d

some ε 0. Fix d such that τ R,λ f f . If p λ N, there exists

d d

0 ε 1 such that p λ p λ ε . Then

1 1

d d d d

Ô p p

× × Ö

Öp λ ε p λ

τ R, λ ε f f f τ R,λ f .

We now assume that p λ N. After possibly increasing d, we may also assume

1 d pd

1 pd p λ

that τ R, pd f f R. We now show that τ R,λ f f is

1 1

2d p2d 2d p2d

1 p λ 1 p λ 1

contained in τ R, λ p2d f f . Let J : f . By

2d 2d

Ö × p λ 1 p

deﬁnition, we have that f J , so that

d ×

2d 2d d d Öp

× Ö ×

Öp p λ p p λ

(2.39.1) f J : f J : f .

d d ×

pd Öp p λ

However, (2.39.1) and the fact that f R imply that J : f R as

d d d pd ×

p λ Öp p λ well. From this, we conclude that f J , so that f J.

Deﬁnition 2.40. We say that λ 0 is an F -jumping number of f if λ 0, or if

λ 0 and τ R,λ f is strictly contained in τ R, λ ε f for all 0 ε λ.

The following result, whose proof we omit, shows that the jumping numbers of f are completely determined by those contained in the unit interval.

Proposition 2.41. [BMS08, Proposition 2.25] Every natural number is an F -

jumping number of f, and a positive real number λ N is an F -jumping number if and only if the fractional part of λ, which is contained in 0, 1 , is an F -jumping

number.

We have seen in Proposition 2.39 that τ R,λ f R for small values of λ,

and so the ﬁrst F -jumping corresponds to the ﬁrst value at which τ R,λ f is

non-trivial. We see below that this ﬁrst jumping number coincides with fptR, f .

Proposition 2.42. fpt R, f sup λ R¥0 : τ R,λ f R

e e

p λ p

Proof. τ R,λ f R if and only if f R for some e 1, which by

1ßp

Corollary 2.30 happens if and only if there exists a map θ HomR R , R such

e e

p λ p

that θ f 1. This is precisely the condition that R,λ f be strongly

F -pure, and the assertion follows from Proposition 2.22.

Remark 2.43. As fptm R, f fpt Rm, f , it follows from Proposition 2.42 and

Remark 2.37 that fptm f sup λ 0 : τ R,λ f m τ Rm,λ f Rm . CHAPTER 3

Some connections

Suppose that f Q x1, , xm . By reducing each of the coeﬃcients of f modulo

p 0, we obtain a family of models fp Fp x1, , xm . In general, given a polyno-

mial f C x1, , xm , we may produce a family of positive characteristic models of f via the method of reduction to positive characteristic.

2 11

We begin with an example. Consider the polynomial f 5x πwy e 3z

in Cx,y,z,w . If we let A : Z π, e, 3 C, then A is a ﬁnitely-generated ¤

π ¤e 3

Z-subalgebra of C, and f A x,y,z,w . Note that from A x,y,z,w , we can recover

Cx,y,z,w via base change: C A A x,y,z,w C x,y,z,w . If µ A is a maximal

ideal, then Aµ is a ﬁnite ﬁeld by Corollary 3.2, which also shows that all but

ﬁnitely many p appear in the set char A µ : µ A is maximal . By construction,

each coeﬃcient of f is a unit in A, and hence will have non-zero image under the

map A A µ. Thus, if fµ denotes the image of f under the map A x,y,z,w

Aµ A A x,y,z,w A µ x,y,z,w , fµ is a polynomial over a finite field of positive characteristic whose supporting monomials are the same as the supporting monomials of f. Furthermore, by varying µ we obtain models fµ over fields of all but finitely many characteristics.

We will discuss generalizations of this process in what follows. Rather than focus

24 25 on polynomial rings, we will discuss the process of creating positive characteristic models for for any element (and ideal) in a ﬁnitely generated C-algebra.

1 Preliminaries

In this section, we gather a collection of results related to the method of reduction to positive characteristic. These results are consequences of the following variant of

Noether Normalization.

Lemma 3.1 (Noether Normalization for Domains). Let A be a ﬁnitely-generated al-

gebra over a domain D with D A. There exists 0 N D such that the extension

DN AN can be factored as DN DN z1, , zd AN , where z1, , zd AN are

algebraically independent over DN , and DN z1, , zd AN is module-ﬁnite.

If L Frac D, and R is the localization of A at the non-zero elements of D, then

by Noether Normalization for fields, the inclusion L R can be factored as a purely transcendental extension followed by a module-finite extension. As only finitely many

“denominators” are involved at this process, one can localize at a single element of

D and preserve this factorization. For a proof of Lemma 3.1 that is independent of

Noether Normalization for ﬁelds, we refer the reader to [Hoc].

Corollary 3.2. If Z A is a ﬁnitely‘generated Z-algebra, then

1. every maximal ideal of A contains a prime p, and

2. all but ﬁnitely many primes p are contained in a maximal ideal of A.

3. Furthermore, Aµ is a ﬁnite ﬁeld for every maximal ideal µ A.

Proof. Let µ A be a maximal ideal, and suppose that µ Z 0. It follows

that Aµ is also a ﬁnitely generated Z-algebra, so by Lemma 3.1, there exists an

integer N 0 such that A µ A µ N is module ﬁnite over a polynomial ring with

coeﬃcients in ZN . This implies that 0 dim A µ dim ZN 1, a contradiction.

For the second point, let 0 N Z and z1, , zd AN be as in Lemma 3.1,

so that the extension ZN AN factors as ZN ZN z1, , zd AN . Every p

not dividing N generates a prime ideal in both ZN and ZN z1, , zd , and so by the Lying Over Theorem, there exists a prime (and hence maximal) ideal of A not containing N, and lying over p.

For the last point, we have that Aµ is ﬁnitely generated algebra over Fp for some

p, and thus is module ﬁnite over a polynomial ring Fp z1, , xd by Lemma 3.1. As

Aµ is a ﬁeld, dimension considerations force that d 0, and thus A µ is a ﬁnite

extension of Fp.

Lemma 3.3. Let A B be an inclusion of ﬁnitely-generated Z-algebras. If µB B

is maximal, then so is µA : µ A A.

Proof. By Corollary 3.2, we know that B µB is a ﬁnite ﬁeld, and the inclusion

AµA B µB then shows that A µA must also be a ﬁnite ﬁeld.

Corollary 3.4. Let A be a domain, A B be an inclusion of ﬁnitely-generated

A-algebras, and let π : Spec B Spec A be the induced map of schemes. Then the

(inverse) image of a dense set under π is also dense. Furthermore, the image under

π of a non-empty open set contains a non-empty open subset of A.

Proof. We will ﬁrst show that if W Spec B is dense in Spec B, then π W is dense

in Spec A. By means of contradiction, suppose that π W is not dense in Spec A, so

that π W U for some non-empty open set U Spec A. It follows that

¡ ¡

¡1 1 1

(3.4.1) W π U π π W π U . 27

¡1

However, π, being induced by the inclusion A B, is dominant, and so π U .

¡1

As W is dense in Spec B, we must have that π U W , contradicting (3.4.1).

¡1

We will now show that if Λ Spec A is dense, then π Λ is dense in Spec B.

¡1

It suﬃces to show that Spec Bf π Λ for every non-zero element f B. By

assumption, B is a ﬁnitely-generated A-algebra, and thus so is Bf B T 1 T f .

By Lemma 3.1, there exists N A non-zero such that AN BfN factors as

(3.4.2) AN AN z1, , zd BfN ,

where z1, , zd are elements of BfN that are algebraically independent over AN , and the second inclusion in (3.4.2) is module ﬁnite. By the Lying Over Theorem,

it follows that the map Spec BfN Spec AN induced by (3.4.2) is surjective. As Λ

is dense in Spec A, we have that Λ Spec AN . The previous surjection shows ¡

¡1 1

that Spec BfN π Λ , and thus that Spec Bf π Λ .

Let U Spec B be an open set, and suppose that U contains the basic open set

Spec Bf . We have seen in the preceding paragraph that there exists N 0 in

A such that Spec AN π Spec BfN π Spec Bf π U .

The following result shows an explicit way to reduce a polynomial over C to

positive characteristic, and will be important in Chapter 7.

Corollary 3.5. Let f C x1, , xs with f 0 0. Then there exists a ﬁnitely-

generated Z-subalgebra A C with f A x1, , xs satisfying the following prop-

¡1

erty: for p 0, π p and Support fµp Support f for every maximal ideal

¡1

µp π p . Here π : Spec A Spec Z denotes the map induced by Z A.

a1 an ¦

Proof. Let f u1x unx for some coeﬃcients ui C , and let B ¨

1 ¨1

Zu1 , ,um ; note that B is a ﬁnitely-generated Z-algebra. Choose an integer N so the inclusion ZN BN factors as in Lemma 3.1. We may take A : BN . 28

2 On reduction to positive characteristic

If Z A C is a ﬁnitely-generated Z-algebra and g C x1, , xs , we say

that g is deﬁned over A if g A x1, , xs C x1, , xs . Similarly, we say

that an ideal J C x1, , xs is deﬁned over A if some ﬁxed set of generators

of J are deﬁned over A. This condition is equivalent to the existence of an ideal

JA A x1, , xs with the property that JA C x1, , xs J. For example,

J πx ,y 2z and f ln 5 x y S are both deﬁned over Z π, 2, ln 5 . In

fact, if J and f are defined over A, then they are defined over any finitely-generated Z-algebra B C with A B. If Σ is any finite set of ideals (or elements) of S,

there exists a ﬁnitely generated Z-algebra A C such that every member of Σ is

deﬁned over A.

Let S C x1, , xs I be a ﬁnitely-generated C-algebra. Suppose that I is

deﬁned over A, and ﬁx an ideal IA A x1, , xs that expands to I. We say that

S is deﬁned over A, and set SA : A x1, , xs IA. We now point out some of the subtleties of these deﬁnitions and constructions.

Remark 3.6. First, note that if I is deﬁned over A, then I does not canonically de-

termine an ideal of Ax1, , xs . Indeed, if IA and IA are both ideals of A x1, , xs

that expand to I, then IA and IA need not be equal as ideals of Ax1, , xs . For ex-

ample, choose complex numbers α and β that are pairwise algebraically independent

over Q (i.e., π and e is very likely an example of such a pair). Let A Z α, β , and

½ ½

consider the ideals IA x, α y and IA x, β y in A x, y . Note that IA IA as

ideals of Ax, y , though IA C x, y IA C x, y x, y . In this speciﬁc case, we

½ ½

see that though IA IA, we still have that A x, y IA A x, y IA. This suggests that distinct choices for IA might yield isomorphic quotients.

However, SA A x1, , xm IA may depend on the choice of IA. For example,

let A Z π , and consider the ideals IA x and IA πx of A x . Then

½ ½

IA C x IA C x x , yet A x IA A x IA as rings, for

Ax IA A Z t Z t, x tx A x IA,

where t is an indeterminate over Z. That SA depends on the choice of A is clear.

Many of the ambiguities described above can be resolved by expanding our “co-

eﬃcient” base A. Indeed, if IA A x1, , xs and IB B x1, , xs are ideals

whose expansions to Cx1, , xs are equal, then there exists a ﬁnitely generated

Z-algebra C containing both A and B such that the expansions of IA and IB are

equal in C x1, , xs . Indeed, one need only attach the ﬁnitely many coeﬃcients involved in expressing generators of IA in terms of those for IB over C, and vice versa.

Another unifying point is that, regardless of the choice of A or IA, we can always recover A from SA, as C A SA S. This justiﬁes the point of view that SA, though

not canonically determined, is a good approximation of S.

Consider the map Spec SA Spec A induced by the map A SA. The ﬁber over

the generic point 0 Spec A is the variety corresponding to Frac A A SA, while

the ﬁber over a closed point (i.e., maximal ideal) µ A is the variety corresponding

to Aµ A SA. By Corollary 3.2, this is a variety over the ﬁnite ﬁeld A µ.

By extending the scalars of the closed ﬁber from Frac A to C, we obtain the

variety corresponding to S. This follows from the fact that C A Frac A A SA

C A SA S. As the extension Frac A C is nice (i.e., faithfully ﬂat), we see that

much of the information carried by S is carried by the fiber of Spec SA Spec A over the generic point of Spec A, regardless of whatever choices were made. On the 30 other hand, it is a general principle that properties of the generic fiber are closely related to those of the closed fibers. However, each closed fiber is a variety over a

ﬁnite ﬁeld, and can be studied via the Frobenius map on the ambient ring.

Suppose that we want to study a ﬁnitely-generated C-algebra S. If S is deﬁned

over A, we may then hope to use properties satisﬁed by the ﬁbers of Spec SA

Spec A over “most” closed points of Spec A to say something about S. In doing so, we would like to know that this is independent of the choices being made.

3 On F -purity for pairs reduced from C

The aim of this section is to examine the F -purity property for pairs that are

reduced from positive characteristic. Let f denote a polynomial with non-zero image

in S : C x1, , xs I. Suppose that A C is a ﬁnitely-generated Z-algebra

such that f and I are both deﬁned over A. Fix an ideal IA A x1, , xm with

IA C x1, , xm I, and let SA A x1, , xm IA be as deﬁned above, so

that C A SA S. For every maximal ideal µA A, fµA will denote the image

of f under the map Ax1, , xs SA SA µA : A µ A SA. Let B C

be another ﬁnitely-generated Z-algebra with f and I both deﬁned over B, and let

IB B x1, , xm ,SB B x1, , xm IB, and fµB SB µB be as above.

Though the construction of SA depends on the choice of A and IA, we will show

that the issue of whether or not the positive characteristic pairs SA µA ,λ fµA

are F -pure for “most” maximal ideals µA A is independent of these choices. Here,

“most” maximal ideals will mean all maximal ideals in some dense (or dense open)

subset of Spec A. In doing so, we will rely on the following result.

Lemma 3.7. Let K L be perfect ﬁelds of characteristic p 0. Let R be a

K-algebra, and consider the inclusion R T : L K R. Then there exists a

e e ß

1ßp 1 p

map θR HomR R , R with θR g 1 if and only if there exists a map

e e ß

1ßp 1 p θT HomT T , T with θT g 1.

Proof. As T is a free R-module (and hence faithfully ﬂat), the map of R-modules e

1ßp HomR R , R R

given by evaluation at g1ßp is surjective if and only if the map of T -modules e

1ßp (3.7.1) HomT T R R , T T

e e e

ß ß

1ßp 1 p 1 p given by evaluation at 1 g is surjective. We identify T and T R R as

T -algebras via the canonical isomorphisms

e e e e e

ß ß ß ß 1ßp 1 p 1 p 1 p 1 p

e T R R L K R R R L K R L K1ßp R

e e ß

1 p 1ßp

L K R T .

Here, we have crucially used that L and K are perfect. It is easy to check that

e e ß

1ßp 1 p

the isomorphism T R R T induces an isomorphism between the map

e e e e

ß ß ß

1ßp 1 p 1 p 1 p

HomT T , T T given by evaluation at g R T with the map

appearing in (3.7.1).

Lemma 3.8. Suppose that A B, and that IB B x1, , xm is equal to the

expansion of IA A x1, , xm to B x1, , xm . Let µB be a maximal ideal of

Spec B, and let mA : µB A denote the corresponding maximal ideal of A. Then

SA µA ,λ fµA is F -pure if and only if SB µB ,λ fµB is F -pure. In particular,

fptfµA fpt fµB .

Proof. Let p : char A µA char B µB. Note that A µA B µB is an extension of

ﬁnite (hence, perfect) ﬁelds. That IA B x1, , xm IB implies that SB B ASA, 32 and furthermore we have

(3.8.1)

SB µB SB B B µB SA A B B B µB SA A B µB

SA A A µA AßµA B µB

SA µA AßµA B µB

Under this identiﬁcation, it is easy to see that fµA fµB under the inclusion

SA µA SA µA AßµA B µB SB µB . Finally, it follows from Lemma 3.7 that e

e N ßp

1ßp

there exists a map in HomSA ÔµA SA µA ,SA µA sending fA to 1 if and only e

e N ßp

1ßp

Õ if there exists a map in HomSB ÔµB SB µB ,SB µB sending fB to 1, and the

claim follows.

Corollary 3.9. The pair SA µA ,λ fµA is F -pure for all maximal ideals µA

in some dense (respectively, dense open) subset of Spec SA if and only if the pair

SB µB ,λ fµB is F -pure for all maximal ideals µB in some dense (respectively,

dense open) subset of Spec SB.

Proof. We will ﬁrst assume that A B and that IB IA B x1, , xm , so that

Lemma 3.8 applies. Suppose there exists a dense (respectively, dense open) set

WA Spec A such that SA µA ,λ fµA is F -pure for every maximal ideal µA WA,

¡1

and set WB : π WA . By Corollary 3.4, WB is a dense (respectively, dense open)

subset of Spec B. Let µB denote a maximal ideal in WB, and let µA µB A be the

image of µB under π. It follows from Lemma 3.8 that SB µB ,λ fµB is F -pure.

Instead, suppose there exists a dense (respectively, dense open) set WB Spec B

such that SB µB ,λ fµB is F -pure for every maximal ideal µB WB. Let WA denote any dense (or dense open) subset of π WB (such a set exists by Corollary 3.4), and let µA denote a maximal ideal in WA. By deﬁnition, we may choose a maximal

ideal µB in B such that µA π µB µB A. Once again, Lemma 3.8 shows that

SA µA ,λ fµA must also be F -pure.

We now address the general case. As IA C x1, , xs IB C x1, , xs , there

is a ﬁnitely-generated Z-algebra D C such that A, B D, and IA D x1, , xs

IB D x1, , xs : ID. By construction, our initial argument applies to the inclusions A D and B D, and thus the claim comparing A and B follows.

4 A connection between singularities

In this subsection, f S C x1, , xm will denote a polynomial with f 0 0.

If A C is a ﬁnitely-generated Z-algebra, then SA will denote the polynomial ring

Ax1, , xm S. If µ A is a maximal ideal, we will use SA µ to denote

the polynomial ring Aµ A SA A µ x1, , xm . By Corollary 3.2, SA µ is a

polynomial ring over a ﬁnite ﬁeld. By abuse of notation, we will use m to denote the ideal generated by the variables in the polynomial rings S, SA, and SA µ .

Deﬁnition 3.10. We say that C ,λ f has dense F -pure type at 0 if for every

ﬁnitely generated Z-algebra A C with f SA, the positive characteristic pairs

SA µ ,λ fµ are F -pure at m for all maximal ideals µ in some dense (though not neccesarily open) subset of Spec A.

This deﬁnition of dense F -pure type can easily be extended to study the singu-

larities of f at any point z with f z 0. One way to do so is as follows: Fix an

φ m m

automorphism S S such that z 0 under the induced isomorphism C C ,

and let g φ f . Note that g 0 0. Then C ,λ f has dense F -pure type at z

if C ,λ g has dense F -pure type at 0.

Remark 3.11. If follows from Corollary 3.9 that it suﬃces to check the condition

in Deﬁnition 3.10 for one single A with f SA. 34

The following theorem allows us to compare log canonical singularities with F - pure singularities.

Theorem 3.12. [HW02, Tak04] If the pair C ,λ f has dense F -pure type at 0, then it is also log canonical at 0.

The following result further illustrates the close relationship between singularities

in characteristic zero and those in characteristic p 0.

Theorem 3.13. [Smi00, HY03] There exists a ﬁnitely generated Z-algebra A C

λ λ m

with f SA, and ideals H SA m for every λ 0 with H Sm J C ,λ f m

satisfying the following conditions: 1. There exists a dense open set U Spec A such for every maximal ideal µ U

and for every λ 0, we have τ SA µ ,λ fµ m H SA µ m. 2. For every λ 0, there exists a dense open set Uλ Spec A such that for every

maximal ideal µ Uλ, we have τ SA µ ,λ fµ m H SA µ m.

We emphasize that the set U Spec A does not depend on λ, while the sets

Uλ do, and typically get smaller as λ increases. As a corollary of Theorem 3.13, we

obtain the following relationship between thresholds.

Theorem 3.14. For every A C with f SA, the following hold:

1. There exists a dense open set W Spec A such that fptm fµ lct0 f for

every maximal ideal µ U.

2. For every 0 λ lct0 f , there exists a dense open set Wλ Spec A such

that λ fptm fµ lct0 f for every maximal ideal µ Wλ.

We again emphasize that the set Uλ varies with λ, and typically shrinks as λ increases. 35

Proof. Corollary 3.9 can be used to show that the conclusions of the theorem hold if

and only if they hold for a specific A with f SA. Let A be as in Theorem 3.13. For the first point, let W U be as in the first statement in Theorem 3.13. For µ U,

the inclusion τ SA µ , fµ λ m H SA µ and Remark 2.43 imply that

(3.14.1)

fptm fµ sup λ : τ SA µ , fµ λ m SA µ m sup λ : H SA µ m SA µ m .

However, as SA m Sm and SA m SA µ m are local maps of local rings, we have

λ λ λ m

that H SA µ m is trivial H is trivial H Sm J C ,λ f m

is trivial. This observation, combined with (3.14.1) and Remark 2.10 show that

fptm fµ lct0 f .

For the second point, let 0 λ lct0 f , let Wλ Uλ U, where U and Uλ

are as in Theorem 3.13, and ﬁx a maximal ideal µ Wλ. That fptm fµ lct0 f follows from the preceding paragraph. By our choice of λ, Remark 2.10 shows that

m λ m

J C ,λ f m is trivial. This shows that H Sm J C ,λ f m is also trivial,

which we have already seen holds if and only if H SA µ m τ SA µ ,λ fµ m is

trivial as well. It then follows from Remark 2.43 that λ fptm fµ .

Conjecture 3.15. For every A C with f SA, there exists a dense set U in

Spec A such that fpt fµ lct f for all maximal ideals µ U.

Whenever f has rational coeﬃcients, this conjecture is equivalent to the statement

that fptm fp lct0 f for inﬁnitely many p 0. For f Q x1, , xm , the set

of all primes such that fptm fp lct0 f also appears to encode subtle arithmetic

information of f, as illustrated by the following example.

Example 3.16. Let f Q x, y, z be a homogeneous polynomial of degree 3 with

isolated singuarity at 0, so that f deﬁnes an elliptic curve E P . Then, lct0 f 1,

and fptm fp 1 if and only if Ep V fp is not supersingular. It is known that there are inﬁnitely many primes such that Ep is not supersingular, though the

collection of such primes may be small (i.e., have density zero in the set of all primes)

[Ser72]. It is also not possible that fptm fp lct0 f for all p 0, as it is known that there are inﬁnitely many primes for which Ep is supersingular [Elk87]. See

[MTW05, Example 4.6] for more details on this example. CHAPTER 4

On base p expansions

Much of this thesis is dedicated to studying the properties of F -pure thresholds, and by Remark 2.20, we know that these invariants are always contained in the unit interval. In this section, we introduce some notation regarding base p (or p-adic) expansions of numbers contained in the unit interval, and derive some easy results.

Though elementary, the idea of studying a number via its base p expansion will be

useful when applied to F -pure thresholds.

Deﬁnition 4.1. Let α 0, 1 . A non-terminating base p expansion of α is an

expression α i , with 0 ai p 1, such that N 0, e N with ae 0. i¥1 p th The number ae is called the e digit of the non-terminating base p expansion of α.

Remark 4.2. Though base p expansions are not unique in general, we note that

every α 0, 1 possesses a unique non-terminating base p expansion.

1 1 0 p¡1 Example 4.3. The non-terminating base p expansion of is e . p p p e¥2 p

Example 4.4. Let α b be a rational number in 0, 1 , and ﬁx a prime p 1 mod b,

bω 1

so that p bω 1 for some ω 1. Dividing both sides by p shows that 1 p p ,

a and multiplying both sides by α b shows that

aω 1

(4.4.1) α α. p p

As a b, we also have that aω bω p 1. This, along with (4.4.1), shows that

aω the non-terminating base p expansion of α is constant, and is given by α e . e¥1 p

We refer the reader to [HW08, Chapter 9] for the standard algorithm on comput-

ing base p expansions.

Deﬁnition 4.5. Let α 0, 1 , and let p be a prime number, and ﬁx e 1.

1. 0 ce α p 1 will denote the e digit in the non-terminating base p

expansion of α. By convention, cα 0 0.

2. For e 1, we deﬁne the e truncation of the non-terminating base p expansion

Õ Ô Õ

c1 Ôα ce α

of α by α e : p pe . By convention, 0 e 0.

3. For e 0, we deﬁne the e tail of the non-terminating base p expansion of α

cd Ôα

by α : d . By convention, 0 0.

e d¥e 1 p e

4. For α α1, ,αn 0, 1 , we set α e : α1 e , , αn e .

Lemma 4.6. Let α 0, 1 . Then the following hold:

1. α e pe N.

1 1

2. 0 α e pe , with equality if and only if α pe N.

3. α α e α e.

4. α e α and α e 0 for all e.

1 p α

5. If α pe N, then α e pe .

e e

× 6. Öp α p α e 1.

e e e

Ø Ù 7. p α e 1 p 1 α p α e. 39

e e e

Ö × 8. p α e p 1 α p α e 1.

9. If β 0, 1 pe N and α β, then α e β.

e e e

Ù Ø Ù Proof. Points 1 - 4 follow by deﬁnition. For 5, note that Øp α p α e p α e

e e 1 1 e

Ø Ù Ø Ù p α e p α e . As α pe N, we have that 0 α e pe , and so p α e 0.

e e e e

For 6, we have that p α p α e p α e, with 0 p α e 1 by 2. Thus, we see

e e e

× that Öp α p α e 1. As p α e α 1, points 7 and 8 follow from rounding

e e e e

the equation p 1 α p α α p α e p α e α. We now prove the last

1 e

point. By 2, we have that α e pe α β, and multiplying by p shows that

e e

p α e 1 p β. As both sides of this inequality are integers, we can conclude that

e e

p α e p β.

d 1

Lemma 4.7. If p 1 α N, then α α ed α for all e 1. ed d ed p d

Proof. Left to the reader.

e Ý

Üα e 1 p

Deﬁnition 4.8. If α 0, 1 , let α en α en α e .

¥ ¥

e n 0 p e n 0 p e p ¡1 We observe that α e is the rational number whose base p expansion is obtained by

“repeating” the ﬁrst e digits of the non-terminating base p expansion of α.

Lemma 4.9. If α 0, 1 , then the following hold:

1. α e α e. e

e e

2. p α e p 1 α e.

Proof. Left to the reader.

Lemma 4.10. Let α 0, 1 . For e 1, the following conditions are equivalent:

e e

Ù 1. Ø p 1 α p α e. 40

2. α p α e.

3. α α e.

Proof. We may assume that α 0. By Lemma 4.6, α α e α e, and so

e e e

(4.10.1) p 1 α p α e p α e α.

e e e

Ù Ø Ù From this, we gather that Ø p 1 α p α e p α e α , which shows that 1

e e

holds if and only if 2 holds. By Lemma 4.9, p α e p 1 α e. Substituting

this into 4.10.1 and gathering the p 1 terms yields the equation

e e

(4.10.2) p 1 α α e p α e α.

Thus, (4.10.2) shows that 2 holds if and only if 3 holds.

Deﬁnition 4.11. Let α1, ,αn 0, 1 , and let p be a prime number.

1. We say that the non-terminating base p expansions of α1, ,αn add without

carrying if ce α1 ce αn p 1 for every e 1.

2. For k k1, ,kn N , we say that the base p expansions of k1, ,kn add

without carrying if the obvious analogous condition holds.

Remark 4.12. The non-terminating base p expansions of α1, ,αn add without

e e

carrying if and only if the base p expansions of the integers p α1 e , ,p αn e

add without carrying for all e 1.

Remark 4.13. If the non-terminating base p expansions of α1, ,αn add without

carrying and α : α1 αn, then ce α ce α1 ce αn for all e 1.

The following classical results will play a key role in this thesis. 41

Lemma 4.14 ([Dic02, Luc78]). Let k k1, kn N , and set N : k ki.

N N! Then : 0 mod p if and only if the base p expansions of the entries k k1!¤¤¤km!

of k add without carrying.

Theorem 4.15 (Dirichlet). For any collection α1, ,αn of rational numbers, there

exist inﬁnitely many primes p such that p 1 αi N for 1 i n.

n n

Lemma 4.16. Let α1, ,αn Q 0, 1 .

1. If α1 αn 1, then there exist inﬁnitely many primes p such that the

non-terminating base p expansions of α1, ,αn add without carrying.

2. If α1 αn 1, then there exist inﬁnitely many primes p such that

c1 α1 c1 αn p.

Proof. By Theorem 4.15, there exist inﬁnitely many primes p congruent to 1 modulo

the denominator of each αi. As in Example 4.4, we have that the non-terminating

base p expansion of each αi is constant. In the case that α1 αn 1, one must

have that i ce αi p 1, and so 1 follows. Similarly, if α1 αn 1, then

one must have that i ce αi p, and so 2 follows.

The following lemma will be very useful in Chapter 9 when computing the F -pure threshold of binomial hypersurfaces.

Lemma 4.17. Let α, β 0, 1 . If ce 1 α ce 1 β p, then

α β α β . e e pe e

1 Proof. If both α and β are in pe N, then so is α β. By Lemma 4.6, we have

1 1 1

that α e pe β e pe α β α β e pe , and the result follows. We

1 1

may now assume that α pe N, so that α e pe . By our choice of e, we also

Ô Õ Ô Õ

ce α ce 1 β 1 1 2

have that α e β e pe 1 pe 1 pe . Thus, pe α e β e pe and

1 1

α β α e β e pe α e β e pe , which shows that

1 1

(4.17.1) α β α β δ for some 0 δ . e e pe pe

1 e e e

Ø Ù From (4.17.1), we see that α β pe N and p α β p α e p β e 1. The result then follows from point 5 from Lemma 4.6. CHAPTER 5

F -pure thresholds of hypersurfaces

The main results of this chapter clarify the relationship between the various types of F -purity appearing in the literature, namely F -purity and (sharp/strong)

F -purity. Theorem 5.19 tells us that (strong) F -purity is always a stronger condition

than sharp F -purity, which is (possibly) stronger than the condition of F -purity. We

also see that F -purity and sharp F -purity are the same if and only if fptR, f is a

rational number whose denominator is not divisible by p, and that R, fpt f f is always F -pure. In Proposition 5.12, we derive some interesting restrictions on the set of all F -pure thresholds in a ﬁxed characteristic. We emphasize that these results are valid assuming only that R is F -pure, and thus generalize facts which are known to hold whenever R is an F -ﬁnite, (complete) regular local ring.

1 Truncations of the F -pure threshold

Throughout this chapter, R will denote an F -pure ring of characteristic p 0, and f will denote a non-unit in R. In this section, we will prove Key Lemma

5.2 (which shows that the truncations of the F -pure threshold encode important

“splitting data”), and deduce some important consequences. In doing so, we will

1ßp use the fact that an R-linear map θ : R R gives rise, in a natural way, to an

ℓ ℓ e ℓ ℓ ℓ e ℓ e ℓ

ß ß ß ß ß ß ß

1ßp 1 p 1 p 1 p 1 p 1 p 1 p 1 p

R -linear map θ : R R deﬁned by θ r θ r . We

ℓ

1ßp ℓ th call θ the p -root of the map θ.

e e e

ß ß

1 p α p 1 p

Lemma 5.1. Let α 0, 1 pd N. If the inclusion R f R splits as a

α 1ßp

map of R-modules for some e 1, so must the inclusion R f R .

e e e e e

ß ß ß

p α p 1 p p α p

Proof. Suppose R f R splits, so that θ f 1 for some map

e d e

ß ß

1ßp 1 1 α 1 p 1 p

θ HomR R , R . If e d, then α pd N pe N, and so f R R . It is

e e d e θ

ß ß ß

p α p α 1 p 1 p e

then clear that f f maps to 1 under the composition R R R.

e d ß

1ßp 1 p e e

Ö × Instead, suppose that d e, so that R R . Note that p α p α ,

e e

Ö × so α p α p . By Lemma 2.17, it suﬃces to show that there exists a map

d e e e

ß ß 1ßp p α p 1 p R R sending f to 1. As R is F -pure, there exists an R -linear map

d e d φ e θ

ß ß ß

1ßp 1 p 1 p 1 p e

φ : R R with φ 1 1. If σ denotes the composition R R R,

e e

ß p α p

then σ f 1.

Key Lemma 5.2. Let R be an F -pure ring of characteristic p 0, and ﬁx d 1.

d d ß

d aßp 1 p

Then p fpt f d max a N : R f R splits as a map of R-modules .

d d ß

d aßp 1 p

Proof. Set λ : fpt f and νf p : max a : R f R splits . Note that

d d d

0 νf p p 1. If λ 0, then νf p 0, and we will now assume that

λ 0, 1 . By Lemma 4.6, λ d λ, it follows from Proposition 2.22 that R, λ d f

e e e

ß Ü Ý

× ß

Ö p λ d p 1 p

is strongly F -pure, and hence there exists e 1 such that R f R splits.

d d

By Lemma 5.1, we may assume that e d, and it follows that p λ d νf p .

Ý Üλ d

d pd 1ßp

To prove equality, we must show that the inclusion R f R never

d d

ß ×

d d Öp λ p

splits. By Lemma 4.6, p λ d 1 p λ , so it suﬃces to show that θ f 1

1ßp 1 d d

for every θ HomR R , R . If λ pd N, it follows that p λ p λ ε for

d d d d

Ô Õ ß ß

× × Ö

Öp λ p p λ ε p

0 ε 1. By Deﬁnition 2.19, it follows that θ f θ f 1

1ßp

for every θ HomR R , R .

1 d d

Now, suppose that λ pd N, so that p λ p λ. By way of contradiction, d

λ 1ßp

suppose that θ f 1 for some θ HomR R , R . As0 λ, it follows that

d d ß

1 1ßp 1 p λ pd . By Lemma 2.17, there exists an R-linear map R R sending f to 1.

d 2d d

ß ß

d th 1ßp 1 p 1 p

Taking p roots of this map produces an R -linear map φ : R R with

2d 2d φ d θ

ß ß

1ßp 1 p 1 p

the property that φf 1. Under the composition R R R, it

d follows from the R1ßp -linearity of φ that

1 2 2

λ d d ß

p2d λ 1ßp λ 1 p λ λ

f f f f φ f f 1 θ f 1.

1 2 λ d

p2d 1ßp

We see that R f R splits, contradicting the deﬁnition of λ fpt f .

e e ß

N ßp 1 p Remark 5.3. It follows from Key Lemma 5.2 and Lemma 2.17 that R f R

splits as a map of R-modules if and only if pe fpt f e.

Ý ß

d Ü α d 1 p

Proposition 5.4. Let α 0, 1 with p 1 α N . If R f R splits as

Ý ß

Üα ed 1 p

a map of R-modules, then so does R f R for every e 1.

Ý ß

Üα d 1 p Proof. We induce on e, the base case being our hypothesis. Suppose R f R

Ý ß

Üα ed 1 p and R f R split as maps of R-modules, so that there exists

Ü Ý

1ßp α d 1. an R-linear map R R with f 1, and

Ü Ý

1ßp α ed

2. an R-linear map θ : R R with θ f 1.

ed d

Ü Ý α 1ßp

ed d We now show that R f R splits as a map of R-modules. By taking

ed th

p -roots of the map in 1, we obtain

Ý Üα d

ed ed d ed

ß ß

1ßp 1 p 1 p ped

(3) an R -linear map φ : R R with φ f 1. Ý

Üα d

By Lemma 4.7, we have that α α ed for all e 1, and it follows that

ed d ed p Ý

Üα d

Ý Ü Ý Üα ed α

ed d p ed (5.4.1) f f f . 46

ed d φ ed θ ß

1ßp 1 p

Under the composition R R R, it follows from (5.4.1), and the

Ý Ü Ý Üα α

ed d d

Ü Ý

Ü Ý Ü Ý Ü Ý 1ßp α ed α α ed α

ed d p ed ed p ed

R -linearity of φ, that f f f f φ f f 1 Ý

Üα ed

θ f 1.

Ý ß

d Üα d 1 p

Corollary 5.5. Let α 0, 1 with p 1 α N. If R f R splits, then

α fpt f .

Ý ß

Üα ed 1 p

Proof. By Proposition 5.4, R f R splits for all e 1. By Key Lemma

4.6, we have that α ed fpt f ed for every e 1, and taking the limit as e gives the desired inequality.

We arrive at the following generalization of [MTW05, Proposition 2.16].

¡ Õß ß

Ôp 1 p 1 p

Corollary 5.6. fptf 1 if and only if R f R splits over R.

p¡1

Proof. If fptf 1, then fpt f 1 p , and it follows from Key Lemma 5.2

¡ Õß ß Ô ¡ Õß ß

Ôp 1 p 1 p p 1 p 1 p

that R f R splits. On the other hand, if R f R splits, then

Corollary 5.5 shows that fpt f 1. The claim then follows, as fpt f 1 by

Remark 2.20.

Corollary 5.7. If R, m is an F -ﬁnite regular local ring, then R f is F -pure if

Ö ×

p¡1 p

and only if f m if and only if fpt f 1.

Proof. The ﬁrst assertion is just Feder’s Criteria [Fed83, Proposition 2.1]. One the

Ö × Ô ¡ Õß ß

p¡1 p p 1 p 1 p

other hand, Corollary 2.31 shows that f m if and only if R f R

splits, which by Corollary 5.6 holds if and only if fptf 1.

2 The set of all F -pure thresholds

Deﬁnition 5.8. FPTp will denote the set of all characteristic p F -pure thresholds:

FPTp : fpt R, f :0 f, a non-unit in an F -pure ring R of characteristic p . 47

Remark 5.9. We stress that the ring R is allowed to vary in Deﬁnition 5.8. By

Remark 2.20, we have that FPTp 0, 1 .

p ¡1

Recall from Deﬁnition 4.8 that if λ 0, 1 , then λ e pe λ e 0, 1 is the rational number whose base p expansion is obtained by repeating the ﬁrst e digits of

the non-terminating base p expansion of λ.

Proposition 5.10. For any λ FPTp, and for any e 1, we have that λ e λ.

Proof. There exists an F -pure ring R and an element f R such that λ fpt R, f .

Set α : λ e. By Lemma 4.9, it follows that

1. α e λ e λ e, and e

e e

2. p 1 α p λ e N.

Ý Ü Ý ß

Üα e λ e 1 p

By Key Lemma 5.2 and 1, the inclusion R f R f R splits as a map

of R-modules. Then, Corollary 5.5 and 2 imply that λ α. Corollary 5.11. For any e 1 and λ FPTp, the following hold:

e e

Ù 1. Ø p 1 λ p λ e.

2. λ p λ e.

3. λ λ e.

Proof. The three assertions are equivalent by Lemma 4.10, and the third point follows from Proposition 5.10.

Corollary 5.11 places severe restrictions on the set FPTp.

Proposition 5.12. Fix e 1. For any β 0, 1 pe N we have that

FPT p β, e β . p 1

Proof. Let λ FPTp. If λ β, then λ e β by Lemma 4.6. Combining this with

pe pe

Corollary 5.11 and Deﬁnition 4.8 shows that λ λ e λ e β. ¡ e p ¡1 e p 1

Remark 5.13. Proposition 5.12 is a generalization of [BMS09, Proposition 4.3], in

which it is assumed that R is an F -ﬁnite regular ring.

Example 5.14. Proposition 5.12 states that for every e 1, there exist p 1 disjoint open subintervals of 0, 1 that do not intersect FPTp. Figure 5.14.1 shows

the intervals corresponding to e 1, 2 and 3 that cannot intersect FPT2.

e 1

e 2

e 3

1 1 3 1 5 3 7 0 8 4 8 2 8 4 8 1

Figure 5.14.1: Three intervals that do not intersect FPT2.

Remark 5.15. As illustrated by Example 5.14, many of the intervals from Proposi-

pe 1 tion 5.12 overlap. However, as length β, e β , Proposition 5.14 p ¡1 2

×X ¤ β ÈÖ0,1 pe N 1 shows that for every e 1, there is a set of Lebesgue measure 2 that does not

intersect FPTp. This was ﬁrst observed in [BMS09].

Remark 5.16. Proposition 5.12 may be used to show that FPTp is a set of Lebesgue

measure zero. This is not surprising, as very often FPTp is contained in 0, 1 Q

[BMS08, BMS09, KLZ09, BSTZ09]. We stress, however, that in the generality dealt

with in this chapter, the issue of whether FPTp Q is open.

3 Purity at the F -pure threshold

In this section, we prove Theorem 5.19. We now recall the various singularity

types and invariants associated to pairs R,λ f via the Frobenius morphism.

Deﬁnition 5.17. The pair R,λ f is said to be

e e e

Ô ¡ Õ ß ß

p 1 λ p 1 p

1. F -pure if R f R splits over R for all e 1,

e e e

ß ß

p λ p 1 p

2. strongly F -pure if R f R splits over R for some e 1,

e e e

Ô ¡ Õ ß ß

p 1 λ p 1 p

3. sharply F -pure if R f R splits over R for some e 1.

The F -pure threshold f is deﬁned as fptf supλ R,λ f is F -pure , and we may deﬁne the (strongly/sharply) F -pure thresholds similarly. By Proposition

2.22, all of these thresholds agree, and consequently, F -purity, strong F -purity, and

sharp F -purity are equivalent conditions on pairs R,λ f with 0 λ fpt f .

The following example shows that these conditions need not be equivalent at the

parameter fptf .

1 p

Example 5.18. Consider the pair Fp x , p x , and let x Fp x . It

℄ ℄ ¡

p 1 e 1 e¡1 1 e 1

is easy to see that fptx p . Note that p 1 p p p p 1, 1

e e e ¡ Õ Ô

Ö × ¡

p p 1 p p ¡p p e 1 e 1

m and so x x . Similarly, p 1 p p , and consequently 1

e e e Ô ¡ Õ¤

× Ö ×

p p 1 p p Öp

x x m for every e 1. Corollary 2.32 allows us to conclude that

1 p

Fp x , p x is F -pure, but not sharply F -pure.

In the main result of this chapter, we see that the structure of the base p expansion

of fptf completely determines whether these are equivalent conditions in general.

Theorem 5.19. The pair R, fpt f f is F -pure, not strongly F -pure, and sharply

F -pure p 1 fpt f N for some e 1.

Remark 5.20. The ﬁrst assertion above generalizes [Har06, Proposition 2.6], in which it is assumed that R is a complete, F -ﬁnite regular local ring. The third is a generalization of [Sch08, Corollary 5.4 and Remark 5.5], in which it is assumed that

R is an F -ﬁnite regular local ring. 50

Remark 5.21. The condition that p 1 fpt f N for some e 1 is equivalent

to the condition that fptf Q, and that the denominator of fpt f is not divisible by p. We will notice in Chapters 8 and 9 that the denominator of fptf is often a power of p, and more often is divisible by p. Thus, there are many instances in which F -purity is not equal to sharp F -purity.

e e

Ù Proof of Theorem 5.19. By Corollary 5.11, Ø p 1 fpt f p fpt f e, and so

e e e

Ô ¡ Õ Ô Õ ß ß

p 1 fpt f p 1 p

Key Lemma 5.2 implies that the inclusion R f R splits as a map

of R-modules. We see that R, fpt f f is F -pure.

e e

× By Lemma 4.6, Öp fpt f p fpt f e 1, and so Key Lemma 5.2 shows that

e e e

Ô Õ ß ß

p fpt f p 1 p

the inclusion R f R never splits over R. We see that R, fpt f f is not strongly F -pure.

e e

Ô ¡ Õ Ô Õ ß

p 1 fpt f p

By Deﬁnition 5.17, R, fpt f f is sharply F -pure if and only if R f

1ßp splits oﬀ from R over R for some e 1, and Remark 5.3 show that this inclusion splits if and only if

e e

× (5.21.1) Ö p 1 fpt f p fpt f e .

However, Lemma 4.6 shows that (5.21.1) holds if and only if

e e e

× Ø Ù (5.21.2) Ö p 1 fpt f p fpt f e p 1 fpt f , where the last equality follows from Corollary 5.11. Finally, we observe that (5.21.2)

holds if and only if p 1 fpt f N. CHAPTER 6

Splitting polytopes

In this chapter, we associate to any collection of n distinct monomials a rational

polytope P 0, 1 , and derive some basic properties of P . The polytopes P have appeared previously in [MTW05, ST09, LM10, BMS06], and the geometry of

P will be used crucially in Chapter 7, 8, and 9 in giving bounds for and explicitly

computing F -pure threshold of certain special classes of polynomials.

Throughout this chapter, let R : K x1, , xm denote a polynomial ring over a

a1 an

ﬁeld K of arbitrary characteristic, and C : x , , x will denote a collection

of distinct monomials in R. Furthermore, if s s1, ,sn R , we will use s

to denote the coordinate sum s1 sn. We stress that is not the usual

Euclidean norm on Rn.

Deﬁnition 6.1. We call the matrix M M C a1 an Mm¢n N the splitting matrix associated to C.

Remark 6.2. Deﬁnition 6.1 is motivated by the following fact: For every n-tuple of

× Ö ×

n a1 k1 an kn ÖMk 1 Mk m

integers k k1, ,kn N , we have that x x x1 xm

Mk th m

x , where Mk i denotes the i entry of Mk N . It follows from this that

a1 an N N k Mk

u1x u x u x , where the u are elements of K, and |

n |k N k i

k N! k1! kn! .

51 52

Deﬁnition 6.3. We deﬁne the splitting polytope associated to C as follows:

P P s R Ms i , , m , C ¥0 : i 1 1

th m

where Ms i denotes the i entry of the element Ms R . By Deﬁnition 6.3, we

have that P 0, 1 .

Deﬁnition 6.4. A point η P is called maximal if η max s : s P , and

P max will denote the face consisting of all maximal points of P .

# Deﬁnition 6.5. We say that P contains a unique maximal point if P max 1.

Lemma 6.6. If P has a unique maximal point η P , then η must be a vertex of P .

Furthermore, if the set of vertices of P contains a unique element η with maximal

coordinate sum, then η must also be the unique maximal point of P .

Proof. If P max η , it follows from the fact that P max is a face of P that η must be a vertex of P [Bar02]. On the other hand, suppose that η is the unique vertex of

P with maximal coordinate sum, so that η P max. If η is not the unique maximal point of P , then the convex polytope P max contains inﬁnitely many points, and so

must contain a vertex ν η. As vertices of P max are vertices of P , this contradicts

the uniqueness of η.

a 0 c Æ

a b c c

Example 6.7. Let C x ,y , x y . Then, M , and consequently

0 b c

as1 cs3 1

3 P s R P

¥0 : . Note that is the convex hull of the points

bs2 cs3 1

1 1 1 1 1

0, v1 a , 0, 0 , v2 0, b , 0 , v3 a , b , 0 , and v4 0, 0, c . See Figure 6.7 for a

# 1 1 1

picture of P . By Lemma 6.6, P 1 if and only if v3 a b c v4 , in which

1 1 1

case P max v3 or v4 . In the case that a b c , P max is equal to the edge determined by the vertices v3 and v4. 53

s3 v b 4

b v b 2 v1 s b 2

b s1 v3

3 a b c c

Figure 6.7.1: The rational polytope P 0, 1 associated to x ,y , x y .

a1 am b1 bm 2

Example 6.8. If C x1 xm , x1 xm , P 0, 1 contains a unique max-

imal point if ai bi for all 1 i m. (see Corollary 9.5.)

d1 dn n 1

Example 6.9. C x , , x P s R s i n

If , then ¥ : i for 1 ,

1 n 0 di

1 1

and P , , . max d1 dn

P max

b b

b b s b 2

3 d1 d2 d3

Figure 6.9.1: The rational polytope P 0, 1 associated to x1 , x2 , x3 .

Remark 6.10. It follows from the proof of Lemma 6.6 that if P contains a unique maximal point η, then η must be a vertex of P . As P is deﬁned by hyperplanes with coeﬃcients in N, it follows that η must have rational coordinates. In fact, if

H s R : L s 1 is a halfspace deﬁned by a linear equation L with rational

coeﬃcients, then every vertex of the polytope P H also has rational coordinates.

Lemma 6.11. Fix e 1. If s is in the boundary of P , then s e s1 e , , sn e is in the interior of P . 54

Proof. As the deﬁning inequalities of P have coeﬃcients in N, the assertion follows

from the fact that si e si.

a1 an N N k Mk

Recall from Remark 6.2 that u1x u x u x . We | n k |N k will soon be interested in knowing, after gathering of terms, what the coeﬃcient of a

given monomial in this expression is. Speciﬁcally, we are interested in knowing when

½ ½

½ n

there exist indices k k in N such that k k N and Mk Mk . The following lemma allows us to address this issue whenever the geometry of P is nice.

Lemma 6.12. Suppose that P has a unique maximal point η P .

1 n

1. If s η e and Ms M η e for some s pe N , then s η e.

1 n 1 n

2. If s ν , Ms Mν, and η e ν pe N for some ν, s pe N , then

s ν. ½

½ n

η s η η η R

Proof. For the ﬁrst statement, let : e. By deﬁnition, ¥0.

½ ½

By assumption, we also have Mη Ms Mη M η e Mη, and η

½ ½

s η η e η , which shows that η is a maximal point of P . Thus η η,

½ 1 n

and s η e. For the second statement, let s : s η e ν pe N . By

½ ½

½ 1 n

hypothesis, s pe N , s η e , and Ms M η e. The ﬁrst statement,

½ ½

applied to s , shows that s η e, and thus s ν.

Corollary 6.13. Suppose that P has a unique maximal point η P .

e e e e

|Ü Ý |

Ý |Ü Ý | Ü Ý Ü p η

p M η e a1 an p η e e p M η e

1. The coeﬃcient of x in u1x unx is e u . Ý p Üη e

1 n 1 n

2. Let ν pe N be an index such that η e ν pe N . Then the coeﬃcient

e e e e

| | |

p Mν a1 an p |ν p ν p Mν

of x in u1x unx is peν u . CHAPTER 7

F -purity and log canonical singularities

The main result of this chapter, Theorem 7.6, gives bounds for, and in an important special case, allows us to explicitly compute the F -pure threshold of a polynomial f whenever the polytope P associated to the supporting monomials of f contains a unique maximal point. These bounds will be crucial when computing the F -pure threshold of diagonal and binomial hypersurfaces in Chapters 8 and 9.

Let f denote a polynomial over the complex numbers. In Theorem 7.17, we invoke Theorem 7.6 to show that Conjecture 3.15, which states that log canonical singularities is equivalent to dense F -pure type, holds for f whenever the associated polytope P has a unique maximal point. Finally, in Theorem 7.18, we apply these methods to show that Conjecture 3.15 holds for very general complex polynomials.

1 Some background on F -purity of monomial ideals

Let R K x1, , xm denote a polynomial ring over a ﬁeld of characteristic

p 0 with K : K K. By Example 2.14, R is an F -ﬁnite regular ring. We

will let m R denote the ideal generated by the variables, and f will denote a

non-zero polynomial in m.

In Chapter 2, we introduced the notion of F -purity for pairs of the form R,λ f ,

55 56

and to this point we have dealt exclusively with pairs of this form. However, the no-

tion of (strong/sharp) F -purity can be extended to pairs of the form R, a f , where

a is any ideal of R [Tak04]. For example, we say that the pair R,λ f is F -pure at m

e e e e

Ô ¡ Õ Ö × Ö ×

p 1 λ p p λ p

if a m for all e 0, and is strongly F -pure at m if a m for some

e 1 (c.f, Corollary 2.32). We also deﬁne the F -pure threshold of a at m as in the

a a m

principal case: fptm : sup λ R¥0 : R,λ is F -pure at . As in Proposi-

a a m

tion 2.22, we have that fptm sup λ R¥0 : R,λ is F -strongly pure at . Below, we gather these facts to give our working deﬁnition for fptm a for this thesis.

Deﬁnition 7.1. Let a m be an ideal of R. Then

e e

Ö ×

p λ p

1. fptm f sup λ R : f m for some e 1 , and

e e

Ö ×

p λ p

2. fptm a sup λ R : a m for some e 1 .

The ﬁrst point in Deﬁnition 7.1 above is a restatement of Corollary 2.33. Our

goal in this Chapter is to compare the value of fptm f with that of fptm a , where here a is the monomial ideal generated by the supporting monomials of f.

a a

Deﬁnition 7.2. If f a uax R, Support f : x : ua 0 . We call

Supportf the set of supporting monomials of f.

Notation 7.3. For the rest of this section, we will assume that Supportf

a1 an

x , , x . We will let a Support f R denote the monomial ideal

generated by Supportf . Following the notation of Chapter 6, M a1 an will denote the splitting matrix associated to Supportf , and P will denote the polytope

associated to Supportf . Recall that P 0, 1 . We now gather some facts regarding F -pure thresholds, and relate fptm a to the geometry of P . 57

Proposition 7.4.

1. fptm f Q 0, 1 .

2. R,λ f is F -pure at 0 if and only if 0 λ fptm f .

3. fptm f min 1, fptm a .

4. fptm a max s : s P .

Proof. The ﬁrst point follows from Remarks 2.20 and 2.21, while the second is a

restatement of Theorem 5.19. The third point follows from the fact that f a.

e e

Ö × cp p It remains to prove the last point. If a m , then at least one of the

e e

Ö ×

cp p n

generators of a is not in m , and so there exists k k1, ,kn N such

1 e ×

e a1 k an kn Mk Öp

Ö × that k1 kn cp and x x x m , i.e., every entry of

e 1

Mk is less than p . Thus, pe k P , and consequently,

cp × k 1

(7.4.1) c k max s : s P .

pe pe pe

It follows from Deﬁnition 7.1 and (7.4.1) that fptm a max s : s P .

Next, choose η P max, and ﬁx e 1. By Lemma 6.11, every entry of M η e

e e e e Ý Ü Ý

Ü 1

Ý |Ü Ý | p η p ηn Ü

a1 e an e p M η e p η e

is less than 1. Thus, x x x is contained a , but

e ×

Öp

m a

not in . By Deﬁnition 7.1, we have that fptm η e . As limeÑ8 η e η,

this shows that fptm a η max s : s P .

Remark 7.5. The inequality in the third point of Proposition 7.4 above may be

p p 1

strict. Indeed, it is easily veriﬁed from Deﬁnition 7.1 that fptm x y p , while

p p 2

fptm x ,y p . 58

2 F -pure thresholds for polynomials with good support

Recall that if η 0, 1 , then ce η denotes the e digit in the non-terminating base p expansion of η. (See Chapter 4.)

Theorem 7.6. Suppose that P 0, 1 contains a unique maximal point η

η1, , ηn , and let L sup N : ce η1 ce ηn p 1 for all 0 e N .

1. If L , then fptm f fptm a η1 ηn.

2. If L , then η1 L ηn L pL fptm f .

The following lemmas will be used often in the proof of Theorem 7.6.

e ×

e a Öp

m Lemma 7.7. For every e 1, we have that p fptm f e max a N : f .

Proof. This is a restatement of [MTW05, Proposition 1.9], and a special case of

Lemma 5.2

Lemma 7.8. Let α 0, 1 be a rational number such that p 1 α N for some

e e

¡ Õ Ö ×

Ôp 1 α p

e 1. If f m , then α fptm f .

Proof. This is an immediate consequence of Corollary 5.5 and Corollary 2.31.

a1 an

Proof of Theorem 7.6. We may write f u1x unx , where each ui is

non-zero. We ﬁrst assume that the non-terminating base p expansions of η1, , ηn

add without carrying. By Proposition 7.4, it suﬃces to show that fptm f η . If

η e η1 e , , ηn e , then Corollary 6.13 shows that

p η e e

Ý Ü Ý p Üη p M η e u e x e

e p η e

Ý | p |Ü η appears as a summand of f e . 59

Ü Ý

¦ p η e 1. By deﬁnition, each ui K , so u 0.

e e

2. By Remark 4.12, the integers p η1 e , ,p ηn e add without carrying for all

e 1. This, combined with Lemma 4.14, shows that

p η e

e 0 mod p.

p η e 3. By Lemma 6.11, every entry of M η e is less than 1, and consequentially every

e e

entry of p M η e is less than p .

e e

Ý |Ü Ý |

p M Üη e p η e

The ﬁrst two points above show that x Support f , while the third

e e e e

Ý Ö × |Ü Ý | Ö ×

p M Üη e p p η e p

point shows that x m . Thus, f m , and Lemma 7.7 allows us to

conclude that fptm f e η e η1 e ηn e. Taking the limit as e

shows that fptm f η1 ηn fptm a , while the opposite inequality holds by

Proposition 7.4.

We now assume that the non-terminating base p expansions of η1, , ηn add

with carrying. Let ce ti and L be as in the statement of the Theorem. Note that

4. 0 L (as c0 ti 0 by convention), and that

5. cL 1 η1 cL 1 ηn p.

By replacing each cL 1 ti with an integer less than or equal to cL 1 ti , we see that

there exist integers δ1, , δn such that δ1 δn p 1 and 0 δi cL 1 ti ,

with the second inequality being strict for at least one index. We will assume without

loss of generality that δ1 cL 1 η1 . For e L 2, set

δ1 p 1 p 1 δ2 δn

(7.8.1) ν e η , , , .

L pL 1 pL 2 pe pL 1 pL 1

The following summarizes the important properties of ν e ν1 e , , νn e . 60

1 n

6. ν e pe N .

7. As δ c 1 t for 2 i n, it follows that ν e t , while the

i L i i i L 1

fact that δ1 c 1 η1 shows that ν1 e η1 . Thus, we have that L L 1

1 n

0 η e ν e pe N for e L 2.

8. As δ1 δn p 1, it follows from the deﬁnition of L that the base p

e e

expansions of the integers p ν1 e , ,p νn e add without carrying.

9. Finally, we have that

1 p 1 p 1

ν e η δi

L pL 1 pL 2 pe

p 1 p 1 p 1

η .

L pL 1 pL 2 pe

Points 6 and 7 above, along with Remark 6.2 and Corollary 6.13, show that, after

gathering terms, the monomial

p ν e e e

Õ Ô Õ up ν Ôe xp Mν e

e p ν e

Ô Õ|

appears as a summand of f p |ν e . Point 8 above, along with Lemma 4.14, gives that

e e e

| Ô Õ|

Õ | Ô Õ|

p ν e p Mν Ôe p ν e

e 0 mod p. We can conclude from this that x Support f Õ p ν Ôe

e e e e

Õ Ö × | Ô Õ| Ö ×

p ν Ôe p p ν e p

and x m . Thus, f m , and by Lemma 7.7, we have that

p 1 p 1

(7.8.2) fptm f ν e η ,

e L pL 1 pe where the last inequality follows from point 9 above. Finally, letting e in (7.8.2)

gives that fptm f η L pL .

3 Log canonical singularities and dense F -pure type

In this section, we prove Conjecture 3.15, which states that log canonical singularities is equivalent to dense F -pure type, for a large class of polynomials over the 61

complex numbers. Let S denote a polynomial ring over C, and let f S be a polyno-

mial with f 0 0. Recall that the pair S,λ f is said to be KLT at 0 if the func-

tion 2λ is locally integrable at 0, and is said to be log canonical at 0 if it is KLT for |

every 0 ε λ. As with singularities deﬁned via Frobenius, these singularity types

can be extended to pairs of the form S,λ a , where now a is allowed to be any ideal

of S [Kol97]. For example, if a f1, , fa is a set of generators for f, we say that λ

S,λ a is KLT at 0 if the function a 2 is integrable in some neighborhood of | | fi i1

the origin. By again referring to a log resolution, this can be shown to be independent

of the set of generators. We say that S,λ a is log canonical at 0 if it is KLT at 0 for

every 0 ε λ. We may also deﬁne the log canonical threshold of a at 0, which we

denote lct0 , to be the common value of sup λ R¥0 : S, a λ is KLT at 0

sup λ R¥0 : S, a λ is log canonical at 0 [BL04, EM06, Laz09].

Below, we gather some important facts regarding log canonical thresholds.

Proposition 7.9. If a is the monomial ideal of S generated by Supportf , then

1. lct0 f Q 0, 1 ,

2. S,λ f is log canonical at 0 if and only if 0 λ lct0 f , and

3. lct0 f min 1, lct0 a .

Proof. The ﬁrst two points above are a restatement of Lemma 2.8, while the third

point follows from the fact that f a.

Remark 7.10. By Proposition 7.4, each of the points in Proposition 7.9 holds whenever “log canonical at 0” is replaced by “F -pure at m” and “lct0” is replaced by

“fptm.” 62

The following well-known lemma shows that the log canonical threshold and the

F -pure threshold of a monomial ideal agree in all characteristics.

Lemma 7.11. Let K be a ﬁnite ﬁeld of characteristic p 0 with K : K ,

and let C be a collection of monomials in the variables x1, , xm. By abuse of

notation, let a denote the monomial ideal generated by C in both Cx1, , xm and

K x1, , xm . Then, lct0 aC fptm aK .

N R a P Proof. Let ¥0 denote the Newton polytope associated to , and let denote

the polytope associated to the set C. Then

a a (7.11.1) lct0 max λ R¡0 : 1 λ N max s : s P fptm ,

where 1 1, , 1 R . The ﬁrst equality in (7.11.1) follows from [How01,

Example 5], the second by setting w 1 in [BMS06, Lemma 4.3], and the last by

Proposition 7.4.

Notation 7.12. For the rest of this chapter, f will denote a non-zero element of

S : C x1, , xm C x with f 0 0. Furthermore, we set Support f

a1 an

x , , x , and a Support f S. If A is a ﬁnitely generated Z-subalgebra

of C, SA will denote the subring Ax1, , xm A x S. Note that C A SA S.

For a maximal ideal µ A, SA µ will denote the polynomial ring SA A A µ

SA µSA A µ x . By Corollary 3.2, char A µ 0. For g SA, gµ will denote

the image of g in SA µ . Similarly, aµ will denote the ideal in SA µ generated by

Supportfµ . Finally, the symbol m will always be used to denote the ideal generated

by the variables x1, , xm in the polynomial rings S,SA, and SA µ .

We recall the deﬁnition of dense F -pure type, and its relationship with log canonical singularities.

Deﬁnition 7.13. The pair S,λ f is of dense F -pure type at m if there exists a

ﬁnitely generated Z-algebra A C with f SA, and a dense subset W Spec A

such that SA µ ,λ fµ is F -pure at m for all maximal ideals µ W .

Theorem 7.14. [HW02, Tak04] If S,λ f is of dense F -pure type at m, then

S,λ f is log canonical at 0.

The converse to Theorem 7.14 is Conjecture 3.15, and verifying this conjecture remains a long-standing open problem [Fed83, Smi97b, EM06].

Deﬁnition 7.15. If Conjecture 3.15 holds, we say that log canonical singularities at

0 implies dense F -pure type at m for f.

Below, we give a slightly diﬀerent characterization of dense F -pure type.

Lemma 7.16. Log canonical singularities at 0 implies dense F -pure type at m for f

if there exists a ﬁnitely generated Z-algebra A C with Z A and f SA, and an

inﬁnite set of primes Λ satisfying the following property: For every p Λ, there exists ¡

¡1 1

a non-empty set Wp π p , dense in the ﬁber π p , such that for every maximal

ideal µp Wp, Support fµp Support f , and fptm fµp min 1, fptm µp .

Here, π : Spec A Spec Z denotes the map induced Z A. ¡

¡1 1

Proof. As Wp is dense in π p , it follows that Wp π p , and thus

¡1 1

(7.16.1) Wp Wp π p π Λ .

È È

pÈΛ p Λ p Λ

¡1 By Lemma 3.4, π Λ is dense in Spec A, and so (7.16.1) shows that W is pÈΛ p

¡1

also dense in Spec A. For p Λ and µp π p , it follows from Theorem 3.14 that

(7.16.2) fptm fµp lct0 f for p 0. 64

# As Λ , we may replace Λ with a slightly smaller set (7.16.1) remains

true after this replacement), and thus assume that (8.10.1) holds for all p Λ and

¡1

µp π p . Note that

a a (7.16.3) lct0 f min 1, lct0 min 1, fptm µp fptm fµp lct0 f .

Indeed, the ﬁrst inequality in (7.16.3) above holds by Proposition 7.9, the second by

Lemma 7.11, the third by assumption, and the last by (8.10.1).

We know from Proposition 7.9 that S,λ f is log canonical at 0 if and only if

0 λ lct0 f . If µ A is a maximal ideal, we know that the pair SA µ ,λ fµ

is F -pure at m if and only if 0 λ fptm fµ by Proposition 7.4. Examining

Deﬁnitions 7.15 and 7.13, we see that to demonstrate that log canonical singularities

at 0 implies dense F -pure type at for f, it suﬃces to show that fptm fµp lct0 f for every p Λ and maximal ideal µp Wp. However, this is precisely the content of

(7.16.3).

Theorem 7.17. If P contains a unique maximal point, then log canonical singularities at 0 implies dense F -pure type at m for f.

Proof. Let A be a ﬁnitely-generated Z-algebra satisfying the conclusion of Corollary 3.5, and let π : Spec A Spec Z denote the map induced by the inclusion Z A.

¡1

Let µp denote an arbitrary element of π p . As Support f Support fµp for

p 0, fptm fµp can be computed via P , as in Theorem 7.6. Let η be the unique

maximal point of P .

If fptm µp η 1, let Λ consist of all primes p 0 such that the non-

terminating base p expansions of η1, , ηn add without carrying. By Lemma 4.16,

a Λ , and it follows from Theorem 7.6 that fptm fµp fptm µp for all p Λ.

If fptm µp η 1, instead let Λ denote the set of all primes p 0 such that

(7.17.1) c1 η1 c1 ηn p.

By Lemma 4.16, Λ . Theorem 7.6 and (7.17.1) above show that fptm fµp 1,

¡1

and Proposition 7.9 shows that equality must hold. If we set Wp : π p , we see that A, Λ, and Wp satisfy the hypotheses of Lemma 7.16, and so we are done.

Theorem 7.18. Suppose that the coeﬃcients of the supporting monomials of f are algebraically independent over Q. Then log canonical singularities at 0 implies dense

F -pure type at m for f.

¦ n Remark 7.19. Let Z denote a countable union of Zariski-closed subsets of C ,

and let g be a polynomial over C such that g 0 0 and Support g n. If

u1, ,un denote the coeﬃcients of the supporting monomials of g, we say that g is

very general (with respect to Z) if u1, ,un Z. Note that the coeﬃcients of the supporting monomials of g are algebraically independent over Q if and only if g

is very general with respect to ZQ, where

¦ n

(7.19.1) ZQ : V h C .

Ö ¤¤¤ × hÈQ η1, ,ηn

¦ n In (7.19.1) the ti denote the coordinates of C . In this language, Theorem 7.18 shows that log canonical singularities at 0 implies dense F -pure type at m for a very general complex polynomial.

a1 an n

Proof of Theorem 7.18. Let Support f x , , x , and let P 0, 1 de-

note the polytope associated to Supportf . Set α min 1, max s : s P ,

and ﬁx a point η P with η α. By Remark 6.10 , we may assume that η Q .

n #

If Λ denotes the set of primes p such that p 1 η N , then Λ by Theo-

a1 an ¦

rem 4.15. We may write f u1x unx , where the ui are elements of C .

If A : Z u1, ,un ui C, then f SA, and Support fµ Support f for all

maximal ideals µ A. Consequently, the F -pure threshold of the ideal generated

by Support fµ in SA µ is equal to max s : s P . Fix a prime p Λ.

¡ Õ Ô ¡ Õ

Remark 6.2 shows that the monomial xÔp 1 Mη appears in f p 1 α with coeﬃcient

p 1 α k

(7.19.2) 0 Θ u u Z u , ,u A.

η,p k 1 n

| ¡ Õ

k |Ôp 1 α

¡ Õ¤

MkÔp 1 Mη

Ô Õ¤

p¡1 α

As α 1, we have that k 0 mod p for each k in (8.13.3). By hypothesis,

Zu is a polynomial ring, and it follows that Θη,p u induces a non-zero element of

the polynomial ring ZpZ u A pA. Let π : Spec A Spec Z be the map induced

¡1

by the inclusion Z A. If we deﬁne Wp : D Θη,p u π p , we have just shown

¡1

that Wp , and so Wp is a dense (open) subset of the ﬁber π p .

Let µp denote an arbitrary maximal ideal in Wp. By construction, Θη,p u has

¡ Õ

Ôp 1 α

¡ Õ

Ôp 1 Mη

non-zero image in Aµp, and so (8.13.3) shows that x Support fµp .

¡ Õ Ö ×

Ôp 1 Mη p

As η P , every entry of p 1 Mη is less than p, and so x m . Thus,

¡ Õ

Ôp 1 α ×

Öp

(7.19.3) fµp A µp x SA µp .

By (7.19.3), Lemma 7.8, applied to fµp SA µp , shows that

(7.19.4) fptm fµp α min 1, max s : s P , and Proposition 7.4 shows that equality must hold in (7.19.4). We have just shown

¡1

that A, Λ, and Wp π p satisfy the hypotheses of Lemma 7.16. We conclude that log canonical singularities at 0 implies dense F -pure type at m for f.

Remark 7.20. An important diﬀerence between Theorem 7.17 and Theorem 7.18 is ¡

¡1 1

that Wp π p in the former, while we only know that Wp π p in the latter.

It would be interesting to investigate under what conditions F -pure thresholds remain constant over the ﬁbers of certain distinguished primes in Spec Z. CHAPTER 8

F -singularities of diagonal hypersurfaces

In this chapter we compute various F -invariants of diagonal hypersurfaces. A

d1 dn

diagonal hypersurface is a polynomial of the form u1x1 unxn , and a diagonal

d d

hypersurface u1x1 unxd is called a Fermat.

1 A detailed discussion of the main results

Throughout, R K x1, , xn will denote a polynomial ring over an F -ﬁnite

ﬁeld K of characteristic p 0, and m x1, , xn will denote the ideal generated by the variables of R.

1 The F -pure threshold of a diagonal hypersurface

Our main result, Theorem 8.1 below, gives a formula for the F -pure threshold of any diagonal hypersurface as a function of the characteristic p. Recall that ce α denotes the eth digit in the non-terminating base p expansion of α. (See Chapter 4.)

n d1 dn

Theorem 8.1. Let d1, ,dn N , and let f u1x1 unxn .

1 1

If L : sup N : c c p 1 for all 0 e N , then

e d1 e dn

1 1

if L

d1 dn

fptm f

1 1 1

d1 d pL if L L n L

67 68

2 3 2 7 Formulas for the F -pure threshold of the hypersurfaces x y and x y are given in [MTW05, Example 4.3/4.4]. At ﬁrst glance, these formulas appear to be quite diﬀerent from those appearing in Theorem 8.1 above. Below, we give an example of how Theorem 8.1 may be used to recover the formulas already in the literature.

Example 8.2. We adopt decimal notation for base p expansions. For example, if a

and b are integers with 0 a, b p 1, then the expression .a b base p will denote

the unique number α with the property that ce α a for e odd and ce α b for

2 3

e even. Let f u1x u2y . We now use Theorem 8.1 to compute fptm f as a

function of the congruency class of p modulo 6. If p 3, then

1 1

(8.2.1) .1 base 3 and .1 .0 2 base 3 . 2 3

Carrying is required to add the expansions in (8.2.1), and the ﬁrst carry occurs at

1 1 1 1 1 2

m the second spot. We conclude that fpt f 2 1 3 1 3 0 3 3 3 .

Similarly, one can show that fptm f 2 if p 2. If p 6ω 1 for some ω 1 , then

1 1

(8.2.2) .3ω base p and .2ω base p . 2 3

We notice that the expansions in (8.2.2) add without carrying, and consequently

1 1 5

fptm f 2 3 6 . Finally, if p 6ω 5 for some ω 0, then

1 1

(8.2.3) .3ω 2 base p and .2ω 1 4ω 3 base p . 2 3

In adding the expansions in (8.2.3), the ﬁrst carry occurs at the second spot, and so

1 1 1 3ω 2 2ω 1 1 5ω 4

(8.2.4) fptm f . 2 1 3 1 p p p p p

5ω 4 1 5 5 1

The reader may verify that p 6p 6 , and so (8.2.4) becomes fptm f 6 6p . 69

As a special case, we recover the following formula from [MTW05, Example 4.3]:

1 2 p 2

23 p 3

2 3

fptm x y .

56 p 1 mod 6

5 1

6 6p p 5 mod 6

By using methods similar to those in the Example 8.2, we are able to give a formula for the F -pure threshold of the degree d Fermat hypersurface in terms of the lease positive residue of p modulo d.

d d

Corollary 8.3. Let f u1x1 udxd K x1, , xd . Then

1 ℓ ℓ 1

pℓ p d p for some ℓ 1

fptm f

a 1

1 p 0 d p and p a mod d with 1 a d

2 Test ideals and higher jumping numbers

By Proposition 2.42, τ R,λ f R for 0 λ fpt f , while necessarily we

have that τ R, fpt f f R. We now focus on understanding τ R, fptm f f

for diagonal hypersurfaces f. By Lemma 8.10, the ideals τ R, fptm f f and

τ R, fpt f f are equal whenever p does not divide any of the exponents in f, and so Theorem 8.4 below often allows to to compute the ﬁrst non-trivial test ideal.

d1 dn

Theorem 8.4. Let f u1x1 unxn . Then

f fptm f 1

τ R, fptm f f 1 1

m fptm f

d1 dn

m fptm f min 1, and p max d1, ,dn di

Remark 8.5. Note that τ R, fptm f f need not equal m if fptm f is less than

1 1

min 1, and p is less than or equal to some exponent of f. For exam- d1 dn ples of this pathological behavior, see [MY09, Proposition 4.2].

Our ﬁnal result computes higher jumping numbers for the degree d Fermat hy-

persurface. By Proposition 2.41, it suﬃces to only consider those jumping numbers contained in 0, 1 . As we have seen in Corollary 8.3, the F -pure threshold of such a hypersurface depends strongly on the congruence class of p modulo d. In Theorem

8.6 below, we will see that the existence (and value) of a second jumping number strictly between fptf and 1 also depends strongly on information encoded by the congruence class of p modulo d.

d d

Theorem 8.6. Let f u1x1 udxd. Suppose that p d and write p d ω a

Ö ×

for some ω 1 and 1 a d. Let σp : ω 2a d .

1. If a 1, then fpt f 1 is the only F -jumping number of f in 0, 1 .

We now assume that a 2.

p¡a σp

2. If p a d 1 , then fpt f p 1 are F -jumping numbers of f in 0, 1 .

3. If p a d 1 , then fpt f 1 are the only F -jumping numbers of f in 0, 1 .

Remark 8.7. As a is strictly less than d, the condition that p a d 1 in the

last point above is automatically satisﬁed whenever p is larger than d 1 . Thus,

Theorem 8.6 says that if p d 1 , then fpt f and 1 are the only jumping

numbers of f in 0, 1 .

Example 8.8. Suppose that d 4, and p 7. Then ω 1, a 3, and p a d 1 .

p¡a σp

Ö × Ö × Note that σp ω 2a d 1 6 4 3 a, and thus p 1. In this case, Theorem 8.6 provides no new information.

Example 8.9. Let d 6 and p 11, so that ω 1, a 5, and p a d 1 . We

2a 10

have that σp ω d 1 6 3, and Theorem 8.6 shows that

p a 1 7 p a σp 9

fptf , , and 1

p 11 p 11

are F -jumping numbers of f contained in 0, 1 . In this case, the reader may verify that these are all of the F -jumping numbers of f in 0, 1

2 F -pure thresholds of diagonal hypersurfaces

Throughout this chapter, R K x1, , xn will denote a polynomial ring over

an F -ﬁnite ﬁeld K of characteristic p 0, and m x1, , xn will denote the ideal

pe pe pe pe pe

generated by the variables of R. R r : r R K x1 , , xn will denote

e th p

the subring of R consisting of p powers of elements of R. As K : K , we

pe e pe

have that K : K for all e 1. If Σ is a ﬁnite basis for K over K , the

reader may verify that

e ×

Öp e

(8.9.1) σ µ : µ monomial ,µ m , σ Σ

pe pe is a free basis for R as an R -module. If K is perfect, so that K K, the basis in

(8.9.1) is the unique free basis for R as an Rpe -module consisting of monomials.

The following lemma says that to compute the F -pure threshold of a diagonal polynomial f, it often suﬃces to compute the F -pure threshold of f at m.

d1 dn

Lemma 8.10. Let f u1x1 unxn be a diagonal polynomial in R. If p di

for every 1 i n, then fpt f fptm f .

Proof. It follows from the Jacobian criterion for regularity that R℘ f is regular (and

hence F -pure) for every ℘ m. It follows that fpt℘ f 1 by Corollary 5.7, and the claim then follows from Proposition 2.24. 72

We will now prove Theorem 8.1, and in doing so, we will rely heavily on (the methods of) Theorem 7.6.

n d1 dn

Theorem 8.1. Let d1, ,dn N , and let f u1x1 unxn .

1 1

If L : sup N : c c p 1 for all 0 e N , then

e d1 e dn

1 1

if L

d1 dn

fptm f

1 1 1

d1 d pL if L

L n L

1 n

Proof. Let P denote the rational polytope s :0 si 0, 1 . Then P is

1 1

also the polytope associated to Supportf , and , , is the unique maximal d1 dn

1 1

point of P . (See Chapter 6.) It follows from Theorem 7.6 that fptm f d1 dn

1 1

whenever the non-terminating base p expansions of , , add without carrying.

d1 dn

Suppose now that L . It follows from Theorem 7.6 that

1 1 1

fptm (8.10.1) f L . d1 L dn L p

By way of contradiction, suppose the inequality in (8.10.1) is strict. By Lemma 4.6,

1 1 1

fptm we have that f L d1 d pL , and Lemma 7.7 implies that

L n L

A E

L 1 L 1

¤¤¤ L

p p 1 k × k d1k1 d k Öp

d1 L dn L n n (8.10.2) f u x x m . k 1 n k

It follows from (8.10.2) that there exists an index k k1, ,kn N with

L 1 L 1

(8.10.3) k p p 1 d1 L dn L

L 1 1

such that diki p for all i. Restated, L ki, and applying Lemma 4.6 to

di p

1 1 these inequalities yields the inequalities d pL ki. These inequalities imply

i L

L 1 the bounds ki p d for all i, and summing these yields

i L

L 1 L 1

k k1 kn p p , d1 L dn L which contradicts (8.10.3). Thus, equality holds in (8.10.1), and we are done. 73

d d

Corollary 8.3. Let f u1x1 udxd K x1, , xd . Then

1 ℓ ℓ 1

pℓ if p d p for some ℓ 1

fptm f

a 1

1 p if 0 d p and p a mod d with 1 a d

ℓ ℓ 1 1 1 1

Proof. If p d p for some ℓ 1, then pℓ 1 d pℓ . Consequently, the

1 1 ωe

non-terminating base p expansion of is of the form e , with ω 1 0.

d d e¥ℓ 1 p ℓ

1 1 1

Thus, ce d 0 for 1 e ℓ, cℓ 1 d ωℓ 1 0, and adding d copies of cℓ 1 d

1 1 ℓ

yields cℓ 1 d cℓ 1 d dωℓ 1 d p p. In the notation of Theorem

1 1 1 1

8.1, we have that L ℓ, and as d 0, we have fptm f d d pℓ pℓ .

ℓ ℓ For the remainder, we will assume that p d. Fix an integer ω 1 such that

1 ω a 1

p d ω a with 1 a d. From this equation, we see that d p d p , from

1 1 a

which we may conclude that c1 d ω and ce 1 d ce d for e 1. If a 1,

1 1

it follows that ce d ω for all e 1. As d ce d d ω p 1 for all e 1,

it follows that the non-terminating base p expansions of d copies of d add without

carrying, so that fptm f 1 by Theorem 8.1.

1 1 1

Next, suppose that a 2. Adding d copies of c1 d gives c1 d c1 d

d ω p a p 1, while d c2 d p by Lemma 8.19. This shows that, in

adding the expansions of d copies of d , the ﬁrst carry occurs at the second digit, and

1 1 d¤ω 1 p a 1

Theorem 8.1 implies that fptm f d d p p p p . 1

3 Test ideals of diagonal hypersurfaces

1 Proof of Theorem 8.4 Having computed fptm f for diagonal hypersurfaces, we now change our focus

to computing the corresponding test ideals at these parameters. We will follow the

d1 dn 1 1 n

previous notation. In particular, f u x u x and η , , Q . 1 1 n n d1 dn 74

We begin with Proposition 8.11 , a technical result that will be key to the proof of

Theorem 8.4. We will prove Proposition 8.11 in the next subsection.

e r

Proposition 8.11. Suppose that di p , and that di p for any 1 r e 1.

Ü Ý |

1 1 |p η e

Furthermore, assume that ce ce p 2 and e 0 mod p. Ý d1 dn p Ü η e

A E

e 1 e 1 pe

¤¤¤ p p 1 d1 e dn e Then xi f .

d1 dn

Theorem 8.4. Let f u1x1 unxn R : K x1, , xn be a diagonal

hypersurface over a perfect ﬁeld of characteristic p 0. Then

f fptm f 1

τ R, fptm f f 1 1

m fptm f

d1 dn

m fptm f min 1, and p max d1, ,dn . di

Proof. It follows immediately from Deﬁnition 2.26 and Lemma 2.38 that

p p 1 1

f f τ R, 1 f . We will now assume that fptm f 1.

d1 dn

1 1 n

Let η : , , Q . In this case, it follows from Theorem 8.1 that the d1 dn non-terminating base p expansions of the entries of η add without carrying, so that

1. di p for all r 1 and 1 i n (for, otherwise, carrying would be neces-

sary),

Ü Ý |

|p η e

2. and e 0 mod p for all e 1 (by Lemma 4.14). Ý

p Üη e

As fptm f 1, ce fptm f p 1 for some e 1. Choose e 0 such that

3. di p for 1 i n,

1 1

4. c fptm f c c p 2 (see Remark 4.13), and e e d1 e dn

A E

1 1 1 e

e e p

¤¤¤

e e p p 1

Ô Õ m

p fpt f p d1 e dn e 5. τ R, fptm f f f f .

e 1 e 1 e

fptm In point 5, we have used that p d1 p d p f e (as the entries e n e

e e

× of η add without carrying) and that Öp fptm f p fptm f e 1. (See Lemma

4.6.) Points 1 5 above allow us to apply Proposition 8.11, which says that

A E e

e 1 e 1 p

¤¤¤ p p 1

d1 e dn e

x1, , xn f τ R, fptm f f .

1 1 1

fptm f For the remaining case, we assume that d1 d pL 1,

L n L

1 1

where L max e : c c p 1 . By Remark 4.13, we have that

1 de n de

1 1

c fptm f c c for 1 e L, and that c fptm f p 1

e 1 de n de e

for e L 1. As fptm f 1, we must have that ce fptm f p 2 for some

1 e L.

1 1

6. Choose 1 ℓ L so that cℓ fptm f c1 cn p 2 and

dℓ dℓ

ce fptm f p 1 for e ℓ 1.

1 1 1

ℓ fptm f 7. By our choice of , we have that d1 d pℓ .

ℓ n ℓ

ℓ

Ü Ý | |p η ℓ

8. We also see that ℓ 0 mod p by Lemma 4.14. Ý

p Ü η ℓ

9. As p max d1, ,dn , each di is strictly less than any power of p.

As before, points 6 9 and Proposition 8.11 allow us to conclude that

A E

ℓ 1 ℓ 1 pℓ

¤¤¤ p p 1

d1 dn ℓ

x1, , xn f ℓ τ R, fptm f f .

2 Some supporting lemmas

In this section, we prove Proposition 8.11, and we do so via a series of lemmas.

e 1 e

Lemma 8.12. Let d and e be positive integers. Then d 1 p d p N.

e e 1 e e

Furthermore, if d p , then d 1 p d p p 1. e

e 1 e 1 1 1

Proof. It is clear that d 1 p d p Z. By deﬁnition, d d d ,

e e e

e 1 e e 1 1 e 1

and consequently d 1 p d p d 1 p d d d 1 p d .

e e e

1 1 e 1

Finally, by Lemma 4.6, we have that 0 d pe ,so0 d 1 p d d.

e e

Lemma 8.13. Fix e 1, let i be an integer with 1 i n, and let vi denote the

n th e

element of N with 1 in the i spot and zeroes elsewhere. Suppose that di p ,

| Ý |

r p Ü η e 1

di p for any 1 r e 1, and e 0 mod p. Let

p Ü η e vi

e 1 e e 1 e 1 e 1 ¡

d 1 p p d1p d p i i dnp

di e d1 e di e dn e

µi : x x x xn .

i 1 i omitted

By Lemma 8.12,

p η 1 e

e p Ü η e vi

βi : e u µi

p η e vi

A E

e 1 e 1

¤¤¤ pe p p 1 is part of a free basis for R over R as described in (8.9.1). When f d1 e dn e

pe pe is written as an R -linear combination of this basis, the coeﬃcient of βi is equal to xi .

Proof. After relabeling the variables, we may assume that i 1. We have that

A E

e 1 e 1

¤¤¤ p p 1 k

d1 dn e k d1k1 dnkn

(8.13.1) f e u x x , k 1 n

n e 1 e 1

k N k p p where the sum ranges over all such that d1 d 1. e n e

Note that the monomial given by the index k p η e v1 is

d1 p 1 d2p dnp e

d1 e d2 e dn e p

(8.13.2) x1 x2 xn x1 µ1.

Monomials in (8.13.1) that correspond to distinct indices are themselves distinct.

Thus, it suﬃces to show that the monomial given by p η e v1 is the only mono-

pe mial in (8.13.1) that is an R -multiple of µ1. However, a monomial in (8.13.1)

corresponding to the index k κ1, , κn is an R -multiple of µ1 whenever

e e

d1κ1 dnκn a1p anp

(8.13.3) x1 xn x1 xn µ1 77

for some a1, , an N . Equating the exponents in (8.13.3) shows that

e e 1 e 1

d1κ1 p a1 1 d1 1 p and diκi p ai di for i 2. d e di e

Dividing each equation by the appropriate di and adding the resulting equalities

e a1 ¡1 ai e 1 e 1

shows that k p p p 1, and comparing d1 i¥2 d d1 d

i e n e

e 1 1 a1 ¡1 ai

this with the equality k p 1 shows that 0. d1 d d1 i¥2 d e n e i

As a1, , an N , we conclude that a1 1 and ai 0 for i 2. Substituting these values into (8.13.3) shows that

d1k1 dnkn p

(8.13.4) x1 xn x1 µ1,

and comparing (8.13.4) with (8.13.2) show that k p η e v1.

Ü Ý |

1 1 |p η e

Lemma 8.14. If ce ce p 2 and e 0 mod p, then Ý

d1 dn p Ü η e

p η e 1

e 0 mod p for 1 i n.

p η e vi

1 1 1

Proof. The assumption that ce ce ce p 2 implies that

di d1 dn

e 1 1

p d 1 ce d 1 0 mod p for 1 i n. Similarly, we also have that

i e i

e 1 1

p η 1 c c 1 0 mod p. e e d1 e dn

The result then follows by reducing the equality

e e e

p η e 1 p η e p η e 1

e e

p η vi p η e 1 e e p d 1 i e (over Q) modulo the prime p.

Proof of Proposition 8.11. This is an immediate consequence of Proposition 2.29 and

Lemmas 8.13 and 8.14. 78

4 On higher jumping numbers of Fermat hypersurfaces

d d

Throughout this section, f : u1x1 udxd will now denote the degree d

Fermat hypersurface in R. Our goal is to prove Theorem 8.6, which is concerned

with the higher jumping numbers of f. Note that, by Proposition 2.41, it suﬃces to only consider those jumping numbers contained in 0, 1 . Theorem 8.6 shows that

whenever p is greater than d, yet still “small”, there exist exotic jumping numbers.

Theorem 8.6 also shows that for p 0, the only jumping numbers in 0, 1 are the expected ones (namely, fptf and 1).

Notation 8.15. In what follows, we will always assume that p d. Accordingly,

we ﬁx integers ω 1 and 1 a d such that p d ω a.

Ö ×

Theorem 8.6. Let σp : ω 2a d N.

1. If a 1, then fpt f 1 is the only F -jumping number of f in 0, 1 .

We now assume that a 2.

p¡a σp

2. If p a d 1 , then fpt f p 1 are F -jumping numbers of f in 0, 1 .

3. If p a d 1 , then fpt f 1 are the only F -jumping numbers of f in 0, 1 .

The proof of Theorem 8.6 will depend heavily on the following two lemmas, whose

proofs we postpone until the next subsection.

Lemma 8.16. Suppose that a 2, and p a d 1 . Then

¡ ¡ ¡

p a σp 1 p¡1 p 1

1. τ R, p p2 pe f for all e 1.

p¡a σp m 2. τ R, p f .

p¡1 p 1

m Lemma 8.17. If a 2 and p a d 1 , then τ R, p pe f

for all e 1.

Proof of Theorem 8.6. By Corollary 8.3, fptm f 1 whenever a 1. As the F - pure threshold is the smallest jumping number, the ﬁrst assertion of Theorem 8.6

follows. We now assume that a 2.

As p d, it follows from Theorem 8.4 that τ R, fptm f f m. Lemma 8.16

shows that if p a d 1 , then τ R,λ f continues being equal to m for all λ

¡ ¡

p¡a 1 p a σp p a σp

with fptm f p λ p , and that τ R, p f . It follows

p¡a σp

from Deﬁnition 2.40 that p is an F -jumping number of f. The assertion that p¡a σp

p 1 follows from Lemma 8.20.

Similarly Lemma 8.17 shows that τ R,λ f m for every λ fptm f , 1 . As

we have already seen that τ R, 1 f f m, this shows that fptm f 1 are the only F -jumping numbers of f contained in 0, 1 .

1 More supporting lemmas

We now prove the two technical lemmas cited in the proof of Theorem 8.6.

1 ω a 1

Remark 8.18. From the equation p d ω a, we have that d p d p , from

1 1 a

which we may conclude that c1 d ω while ce 1 d ce d for e 1.

Lemma 8.19. If a 2, then d 1 c2 d p 1.

Proof. Let ω1 : c1 d . By Remark 8.18, it suﬃces to show that d 1 ω1 p.

a ω1 a

We may write d p d 1, from which it follows that

d 1 a d 1 ω1 a d 1 ω1 d 1

(8.19.1) d 1 .

d p d 1 p p

¡ Õ¤

Ôd 1 a a

First, suppose that a 3, so that d a d a 1 2. Substituting this into 80

(8.19.1) shows that

d 1 ω1 d 1 d 1 ω1 p 1 d 1 ω1 1

2 1.

p p p p p

¡ Õ ¤ ¡

Ôd 1 ω1 1

From this, we may conclude that 1 p , so that d 1 ω1 p 1.

We now deal with the case that a 2. By way of contradiction, suppose that

d 1 ω1 p. As p is prime, the equality d 1 ω1 p implies that d 1 p

or ω1 p. The later is impossible as ω1 is a digit in a base p expansion, while the

former is impossible as p d. Thus, we may assume that d 1 ω1 p 1, and

substituting this into (8.19.1) shows that

2d 2 p 1 d 1 d 2

1 .

d p p p ¡

d¡2 d 2 1 1

Consequently, d p , so d p , which implies that p d, a contradiction.

Lemma 8.20. Let σp : ω d .

1. p d ω σp 1 2p.

2. If p a d 1 , then σp a.

3. If p a d 1 , then p d ω a 1 2p.

Proof. For 1, substituting the identity dω p a into d ω σp 1 yields

(8.20.1) dω σ 1 2p 2a d d.

p d

2a 2a 2a

Substituting the inequalities d d d 1 into (8.20.1) shows that

2p d d ω σp 1 2p.

For 2, we have that dω a p a d 1 by assumption. It follows that

2a 2a

dω a a d 1 ω a σ ω a. d p d

The third point follows along the same lines, and is left to the reader. 81

e ×

N Öp

Lemma 8.21. Let µ,µ1,µ2 be monomials with µ1,µ2 Support f and µ m .

Suppose that d 2 is not a power of p. If µ1 xi µ for some i and µ2 is another

pe R -multiple of µ, then µ1 µ2.

Proof. By assumption, we have that

pe pea n

(8.21.1) µ1 xi µ and µ2 x µ for some a N .

The assumption that all d1 dn d implies that f is homogeneous, so that

deg µ1 deg µ2. Applying this to (8.21.1) shows that a 1, so that µ2 xj µ

for some j. If i j, then degxi µ1 p degxi µ, while degxi µ2 degxi µ, and so

e (8.21.2) degxi µ1 degxi µ2 p .

We have already observed that the left-hand side of (8.21.2) is divisible by d, con-

tradicting the fact that d is not a power of p. Thus, i j, and so µ1 µ2.

Lemma 8.16. Suppose that a 2 and p a d 1 . Then

¡ ¡ ¡

p a σp 1 p¡1 p 1

1. τ R, p p2 pe f for all e 1, and

p¡a σp m

2. τ R, p f .

Proof. By Lemma 8.19, we know what d 1 c2 d p 1. Thus, there exist non-

negative integers δ1, , δd¡1 such that δi p 1 and δi c2 d for 1 i d 1,

with the preceeding inequality being strict for at least one index i (which we are free

to choose). In what follows, we will assume that δd¡1 c2 d . Fix e 1, and let

(8.21.3)

ω δ1 ω δd¡2 ω δd 1 p 1 p 1 ω σp 1

s : , , , , . p p2 p p2 p p2 p3 pe p 82

Note that s pe N. Furthermore, we claim that the ﬁrst d 1 entries of s are less

1 ω c d 1

than or equal to d 2 p p2 . Indeed, this follows from the fact that δi c2 d

for 1 i d 1, with the inequality being strict for i d 1. Let k : p s N.

We summarize some important properties of k below.

1. By the preceding remarks, the ﬁrst d 1 entries of k are less than or equal to

e 1 p d e, and it follows that the ﬁrst d 1 entries of d k are less than or equal

e to p 1.

e e

2. By Lemma 8.20, the last entry of d k is strictly between p and 2p .

e dω σp 1 δi p 1 p 1

k p

3. p p2 p3 pe

e p a σp 1 p 1 p 1 p 1

p .

p p2 p3 pe

4. By construction, the integers k1, ,kd add without carrying. By Lemma 4.14,

| |k

we conclude that k 0 mod p. |

As there is no gathering of monomials when expanding f |k using the multinomial

| |

d¤k k

theorem, the last point above shows that x Support f , while the ﬁrst two points show that

d¤k p (8.21.4) x xd µ,

e ×

Öp

where µ is some monomial with µ m . By point 4 above, the element β :

e | k | k p

k u µ is part of a free basis for R over R as described in (8.9.1), and Lemma | 8.21 (combined with (8.21.4)) show that, when writing f |k in terms of this basis, the

coeﬃcient β is given by xd . We see that

e p a σ 1 p 1 p 1 |

|k p p

x f τ R, f , d p p2 pe 83 where the last equality holds by Lemma 2.38. As this argument is symmetric in the

variables, the ﬁrst claim follows.

p¡a σp m

It remains to show that τ R, p f . By way of contradiction, suppose

p a σp p¡a σp p d

that x1 τ R, p f f . It follows that there exists k N with

d¤k p a σp

k p a σp such that x Support f , p dk1 2p, and dki p 1 for 2 i d. Note that

a 1

(8.21.5) dk p 1 dω a 1 k ω k ω,

i i d i

where here we have used that ki N and that 1 a d. Thus, from the conclusion

of (8.21.5) we obtain the stronger bound dki dm1 p a for 2 i d. It follows

that

d p a σp d k dk1 d 1 p a ,

from which we can conclude that

(8.21.6) p a dσp dk1 2p.

However, substituting the deﬁnition of σp into (8.21.6) shows that

2a 2a

p a dσ p a d ω p a dω d

p d d

p a p a d

2p 2a 2a 2p.

p¡a σp

This contradicts (8.21.6), and it follows that x1 is not contained in τ R, p f .

p¡a σp m

Hence, τ R, p f .

p¡1 p 1

m Lemma 8.17. If a 2 and p a d 1 , then τ R, p pe f

for all e 1. 84

Proof. As in the proof of Lemma 8.16, Lemma 8.19 guarantees that there exists non-

d¡1

negative integers δ1, , δ ¡1 such that δ p 1 and δ ω2 for 1 i d 1,

d i1 i i with at least one inequality being strict. Again, we will assume that δd¡1 ω2. Fix

e 1, and let

(8.21.7)

ω1 δ1 ω1 δd¡2 ω1 δd 1 p 1 p 1 ω1 a 1

s : , , , , . p p2 p p2 p p2 p3 pe p

Note that s pe N. As in the proof of Lemma 8.16, we have that the ﬁrst d 1

1 e

entries of s are less than or equal to p . Let k : p s N. e

1. The ﬁrst d 1 entries of d k are less than or equal to p 1.

e e

2. By Lemma 8.20, the last entry of d k is strictly between p and 2p

d¡1

e dm1 a 1 δi p 1 p 1

k p 3. p p2 p3 pe

e p 1 p 1 p 1 p 1

p p2 p3 pe

4. The integers k1, ,kd add without carrying, so k 0 mod p by Lemma 4.14.

As in the proof of Lemma 8.16, these points imply that

e p a σ 1 p 1 p 1 |

|k p p

x f τ R, f .

d p p2 pe

p¡1 p 1

m By the symmetry of the argument, we conclude that τ R, p pe f . CHAPTER 9

F -pure thresholds of binomial hypersurfaces

This chapter is dedicated to the computation of F -pure threshold for binomial hypersurfaces. Recall that a binomial, by deﬁnition, a K-linear combination of two

distinct monomials. For most binomials f, Theorem 9.14 provides a formula for fptm f in terms of the the characteristic and certain quantities determined by the geometry of P . This formula is also the key step in Algorithm 9.16, an algorithm for computing the F -pure threshold of an arbitrary binomial hypersurface.

1 The polytope P associated to a binomial

Given any binomial f, we may use the methods of Chapter 6 to construct the poly-

tope P associated to Supportf . As f is a binomial, we have that Support f 2,

and thus P 0, 1 . In this case, the polytope P is more easily studied. For the con- venience of the reader, we restate Deﬁnition 6.3 in this simpliﬁed setting. We will use

“” to denote the standard inner product (or dot product) on R : s σ s1σ1 s1σ2.

ν ω

Deﬁnition 9.1. Let x x denote distinct monomials in K x1, , xm such

that every variable appears in either xν or xω. We deﬁne the polytope associated to

ν ω 2

x , x P s R ν ,ω s i , , m as follows: ¥0 : i i 1 for all 1 .

m 2

Remark 9.2. As ν, ω N , we have that P 0, 1 .

Example 9.3. In Figure 9.3, we consider the polytope P for ν 1, 4, 7 and

ω 9, 8, 4 . Observe that adding the condition 2s1 3s2 1 would not change P ,

which motivates the Deﬁnition 9.4 below.

1 3

1 b , b 28 28

0, 9

b 1 3 s1 9s2 1

10 , 40 2

P s ,s R s s

1 2 ¥0 : 4 1 8 2 1

7s1 4s2 1

1 , 0 bb b 7 s1

2 4 7 9 8 4

Figure 9.3.1: The rational polytope P 0, 1 associated to xy z , x y z .

Deﬁnition 9.4. We say that νℓ,ωℓ is active on P if P s : νi,ωi s 1 .

P s R ν ,ω s ν ,ω It follows that ¥0 : i i 1 for all active i i . We say that

ν ω

x , x minimally deﬁne P if νi,ωi is active on P for all i.

Recall that a point η P is called a maximal point of P if η max s : s P .

Corollary 9.5. P has a unique maximal point νi ωi for all active νi,ωi .

Proof. By Lemma 6.6, it suﬃces to show that the set of vertices of P contains a

unique with maximal coordinate sum if and only if νi ωi for all active νi,ωi .

However, it is fairly straightforward to verify that there exist two distinct vertices

with maximal coordinate sum if and only if some bounding line segment of P has

slope equal to 1. As the equations for the bounding line segments of P are given by the active νi,ωi , the claim follows.

Corollary 9.5 shows that P will have a unique maximal point for most choices

of ν and ω. Furthermore, if P does not have a unique maximal point, we may 87 eliminate some variables from the monomials xν and xω to obtain monomials whose associated polytope does have a unique maximal point, as shown in the following remark.

a b

Remark 9.6. Let g be a binomial over K with Supportg x , x for some

m ai

a, b N . Let µ x . Then g µ h for some binomial h satisfying the ai bi i

condition that no variable appears with the same exponent in both of its supporting monomials. By Corollary 9.5, the polytope associated to Supportg must contain a

unique maximal point.

Deﬁnition 9.7. Suppose that P has a unique maximal point η η1, η2 P . We Æ

deﬁne the subpolytopes P , R, and P Æ of P as follows:

Æ 2

1. P : P s R : s2 η2 .

2. R : s R : s1 η1 and s2 η2 .

3. P Æ : P s R : s1 η1 .

Note that this gives a decomposition P P R P Æ .

Æ 1 3

P η , b 10 40

s1 s2 η 40 R

P Æ

bb b s1

Figure 9.7.1: The decomposition of P from Example 9.3. 88

Figure 9.7.1 illustrates some important properties of the polytope decomposition

P P R P Æ which we now summarize. As P is deﬁned by equations with non-negative integer coeﬃcients, the bounding line segments of P that are not on the axes must have negative slope. Furthermore, the convexity of P shows that the slope of these bounding line segments must strictly decrease as s1 increases.

If P has a unique maximal point η P , then the ﬁrst time a line of the form

s1 s2 α intersects P occurs when α η , and this intersection consists precisely of the point η. It follows that none of the bounding line segments have slope equal to negative one (for such a line segment would consist entirely of maximal points).

Thus, the bounding line segments of P not on the axes have negative slope, and none of these slopes are equal to negative one. Furthermore, it is easy to verify that the line segments with slopes greater than negative one give the non-trivial bounding line segments for Æ P . Similarly, the segments with slopes less than negative one

give the non-trivial bounding line segments for P Æ . We record these observations in

Lemma 9.8 below.

Lemma 9.8. Suppose that P has a unique maximal point η P . Then:

Æ 2

P s R s η ν ,ω s ν ,ω ν ω

1. : ¥0 : 2 2 and i i 1 for active i i with i i .

P Æ s R s η ν ,ω s ν ,ω ν ω

2. : ¥0 : 1 1 and i i 1 for active i i with i i . Õ 2 Some lemmas on the computation of fptm Ôf

In this subsection, we derive some lemmas which will simplify our computation of the F -pure threshold of a binomial ideal.

ν ω Lemma 9.9. Let f u1x u2x be a binomial over K. Then there is no gathering of monomial terms in the expansion of f N given by the binomial theorem: if k and 89

κ N with k κ N, then k1ν k2ω κ1ν κ2ω if and only if k κ.

2 k1ν k2ω N

In particular, if k N with k N, then x Support f if and only if N

k 0 mod p.

ν ω

Proof. By hypothesis, we have that k κ is in the kernel of the matrix M .

1 1

However, ker M 0 ν ω, which is impossible as f is a binomial over K.

ν ω

Lemma 9.10. Let f K x1, , xm be a binomial with Support f x , x ,

and let P denote the polytope associated to Supportf . Reorder the variables and

ﬁx ℓ 1 so that νi,ωi is active on P precisely when 1 i ℓ. Let g K y1, ,yℓ

denote the image of f under the map K x1, , xm K y1, ,yℓ given by xi yi

e ×

N Öp

for 1 i ℓ and xi 1 otherwise. Then f x1, , xm if and only if

e Ö

N p ×

g y1, ,yℓ . ½

ν ω ¦

Proof. Write f u1x u2x for some u1 and u2 K . Let ν ν1, , νℓ and

½ ½

½ ν ω

ω ω1, ,ωℓ , so that g u1y u2y . We may write

N N ½

N k k1ν k2ω N k k1ν k2ω (9.10.1) f u x and g u y .

k k

| | | k |N k N By Lemma 9.9, (9.10.1) gives the unique expression for f N and gN as a K-linear

combination of distinct monomials in their respective polynomial rings. Thus, f

e Ö

p × 2 N 1

x1, , xm if and only if there exists k N with 0 mod p and e k k p ¡1

e ×

N Öp

νi,ωi 1 for all 1 i m. Similarly, g y1, ,yℓ if and only if there

2 N 1

exists k N with 0 mod p and e k νi,ωi 1 for all 1 i ℓ. However, k p ¡1

if s : e k, then s νi,ωi 1 for all 1 i m if and only if s P , which (by

p ¡1

our choice of ℓ) holds if and only if s νi,ωi 1 for all 1 i ℓ.

Corollary 9.11. Let f K x1, , xm and g K y1, ,yℓ be as in Lemma 9.10,

and let m x1, , xm and n y1, ,yℓ . Then, the polytopes associated

to Support f and Support g are equal and is minimally deﬁned by Support g .

Furthermore, fptm f fptn g .

As P 0, 1 , it follows that the maximal points of P should have small coor-

dinate sum. Whenever P contains a unique maximal point η P and is minimally

deﬁned by Supportf , the following lemma will allow us to focus on the typical case

in which η 1 when computing the F -pure threshold of f.

ν ω

Lemma 9.12. Suppose P is minimally deﬁned by Supportf x , x and has

a unique maximal point η P . If η 1, then after renaming variables, we have

that f y z for some n 1.

Proof. As P 0, 1 , it follows that both η1 and η2 are non-zero. Thus, there exist Æ

bounding line segments of P and P Æ that contain η, so we may ﬁx i such that

νi ωi and j i with νj ωj such that

(9.12.1) νi,ωi η νj,ωj η 1.

As νi ωi, it follows that νi νi η νi, νi η νi,ωi η 1, and as νi N,

it follows that νi 0. Similarly, the equation νj,ωj η 1 implies that ωj 0,

1 1

and substituting the values νi ωj 0 into (9.12.1) shows that η , . We

νj ωi Æ

also see that the bounding line segments of P and P Æ that intersect η are given by

νj, 0 (which corresponds to a horizontal line) and 0,ωi (which corresponds to a

2 1 1

P s R s s vertical line). This shows that ¥ : 1 and 2 , and as we are 0 νj ωi

νj ωi

assuming that Support f minimally deﬁnes P , we have that f xj xi . Finally,

1 1

the equality η 1 shows that either νj or ωi must equal 1. νj ωi

Lemma 9.13. Let K be an F -ﬁnite ﬁeld, and let m and n denote the homogeneous

maximal ideals of the polynomial rings K x and K y , respectively. If f m and

n m n

g , then fptm n fg min fptm f , fptn g , where is the homogeneous

maximal ideal of the polynomial ring K x, y .

Proof. Suppose λ1 : fptm f λ2 : fptn g . We now show that fptm n fg λ1.

e e

Ý Ö ×

p Üλ1 e p

m By Lemma 4.6, λ1 e λ2 e for all e, and so Lemma 7.7 implies that f

e e e e

Ý Ö × Ü Ý Ü Ý

p Üλ2 e p p λ1 e p λ2 e

and g n . Fix monomials µ1 Support f and µ2 Support g

e e

× Ö ×

Öp p such that µ1 m and µ2 n . As µ1 and µ2 are in diﬀerent sets of vari-

e e e e Ö ×

Ý Ö × Ö × Ü p

p λ1 e p p

ables, µ1µ2 Support fg but not in m n m n , and so

e e e e Ö ×

Ý Ü Ý Ö × Ü p

p λ1 e p λ1 e 1 p

fg m n . By Lemma 7.7, f m , and it follows that

e e e e × × Ö Ö

Ý Ö ×

Ü p p

p λ1 e 1 p e a

m m n m n fg . Thus, p λ1 e max a : fg

p fptm n fg e (see Lemma 7.7) for all e 1.

3 The F -pure threshold of a binomial hypersurface

Theorem 9.14. Suppose P is minimally deﬁned by Supportf with unique maxi-

mal point η satisfying η 1. Let L sup N : ce η1 ce η2 p 1 0 e N .

1. If L , then fptm f η .

For the remainder, we will assume that L and both η1 and η2 are non-zero. Let

ℓ : max e L : ce η1 ce η2 p 2 . (We will see that 1 ℓ L.)

1 1

2. If neither η ℓ pℓ , 0 nor η ℓ 0, pℓ is contained in the interior of P ,

then fptm f η L.

1 1

3. Otherwise, let ε max r : η ℓ pℓ ,r or η ℓ r, pℓ is in P . Then

(a) 0 ε η L, with ε η L either η1 or η2 is in pℓ N, and

(b) fptm f η L ε η , with fptm f η either η1 or η2 is in

1 pℓ N. 92

1 An example

We present an example which we hope clariﬁes the statement of Theorem 9.14.

7 2 5 6 ¦

Example 9.15. Let f u1x y u2x y K x, y with u1,u2 K , so that

2s1 6s2 1

P s ,s R .

1 2 ¥0 :

7s1 5s2 1

Note that P is minimally deﬁned by Supportf and contains a unique maximal

1 5

point η 32 , 32 (see Figures 9.15.1 and 9.15.2), so that fptm f is computable via Theorem 9.14.

Theorem 9.14 states that for primes p with the property that the non-terminating

1 5 1 5 3

base p expansions of 32 and 32 add without carrying, then fptm f 32 32 16 .

As in Example 4.4, if p 1 mod 32, (for example, if p 97, 193, 257, 353 or 449) then the expansions of 132 and 5 32 are constant, and thus add without carrying.

However, there exist primes p such that fptm f 16 with p 1mod32. For

1 5

example, if p 47, then 32 . 1 22 base 47 and 32 . 7 16 base 47 .

We will now compute fptm f when p 43. As

(9.15.1)

1 5

. 1 14 33 25 22 36 12 4 base 43 and . 6 30 38 41 28 9 17 20 base 43 ,

32 32

we see that carrying occurs at the second spot, and that ℓ L 1. By (9.15.1), we

1 6

see that η 1 43 , 43 , and Figure 9.15.1 shows that

1 1

η , 0 , η 0, P . 1 43 1 43

By Theorem 9.14, we conclude that

3 8

fptm f η 1 when p 43. 16 1 43

We now compute fptm f when p 37. As

(9.15.2)

1 5

. 1 5 28 33 19 24 10 15 base 37 and . 5 28 33 19 24 10 15 1 base 37 ,

32 32

we see that the ﬁrst carry occurs at the third spot, and that ℓ L 2. We also see

1 5 5 28

from (9.15.2) that η 2 37 372 , 37 372 , and Figure 9.15.2 shows that

1 1

η , 0 , η 0, Interior P .

2 372 2 372

From Figure 9.15.2, we also see that ε max r : η 2 372 ,r P . More-

over, Figure 9.15.2 shows that the point η 2 372 ,ε lies on the hyperplane

7s1 5s2 1, and an easy calculation shows that ε 6845 .00 22 7 14 29 base 37 .

Thus, Theorem 9.14 shows that

fptm f η 2 ε ε .6 34 22 7 14 29 base 37 when p 37. 16 2

b η b

b b 1 5

b η ,

b 32 32 η 1 1 43

b b

1 η 1 43

m Figure 9.15.1: p 43, ℓ L 1, fpt f 16 1 .8 base 43 .

2 An algorithm

We now show how Theorem 9.14 may be used to construct an algorithm for computing the F -pure threshold at m of any binomial over K. 94

s2 η

bb 1 5

b η 32 , 32 η 2 b

1 372 ε

b b

1 η 2 372

Figure 9.15.2: p 37, ℓ L 2, fpt f 16 2 ε .6 34 22 7 14 29 base 37 .

Algorithm 9.16. Let g K x1, , xm be a binomial. Step 1: Factor g as g µ h for some monomial µ and binomial h so that no variable

appearing in µ appears in h, and so that no variable appears with the same exponent

in both supporting monomials of h. As in Remark 9.6, the polytope P associated

to Supporth will contain a unique maximal point η P .

a1 ad

Step 2: Reorder the variables so that µ x1 xd . By Lemma 9.13,

¤¤¤ Õ Ô ¤¤¤ Õ fptm g fptm µ h min fpt Ô 1 µ , fpt h ,

x , ,xd xd 1, ,xm

1 1

¤¤¤ Õ and it is an easy exercise to verify that fptÔx1, ,x µ min , , . d a1 ad

Step 3: We must now compute the F -pure threshold of h. As in Corollary 9.11,

eliminate inactive variables from h to obtain a polynomial f with the property that Supportf minimally deﬁnes P .

Step 4: If η 1, it follows from Lemma 9.12 that f y z for some n 1. In

this case, it is an easy consequence of Theorem 8.1 that fptÔy,z f 1.

Step 5: If η 1, we may compute the F -pure threshold of f using Theorem 9.14. 95

3 Proof of Theorem 9.14

This subsection is dedicated to the proof of Theorem 9.14. We will rely heavily on the following two technical lemmas whose proofs will be postponed until the following

subsection.

Lemma 9.17. Let ℓ L be as in Theorem 9.14. Then 1. 1 ℓ L and

1 1

2. η1 ℓ η2 ℓ pℓ η1 L η2 L pL η1 η2 L.

3. Furthermore,

L L ℓ ℓ ℓ

Ý Ö × Ü Ý Ü Ý Ö ×

p Ü η1 η2 L p p η1 ℓ p η2 ℓ 1 p

f m f m

1 1

η , 0 , η 0, Interior P .

ℓ pℓ ℓ pℓ

1 1

Lemma 9.18. If ε : sup r : η ℓ r, pℓ , η ℓ pℓ ,r P 0, then

1. 0 ε η L, with ε η L either η1 or η2 is in pℓ N,

e e

Ý Ü Ý Ü

Ö ×

p η1 η2 L ε e p 2. f m for all e L, and

e 1 e

Ü Ý Ü Ý

1 2 ×

p η η L ε e pe Öp

3. f m for all e L.

Proof of Theorem 9.14. If L , it follows from Theorem 7.6 that fptm f η . For the remainder of this proof, we will assume that L and that both η1 and

η2 are non-zero (since if one were zero, they would add without carrying). The assertion that 1 ℓ L is the content of the ﬁrst point of Lemma 9.17. It follows from Theorem 7.6 that

(9.18.1) fptm f η η η η , 1 L 2 L pL 1 2 L 96

where the last equation in (9.18.1) follows from Lemma 9.17.

1 1

We will ﬁrst assume that neither η ℓ pℓ , 0 nor η ℓ 0, pℓ is in Interior P .

By means of contradiction, suppose that the inequality in (9.18.1) is strict. It then

follows from Lemma 4.6 that fptm f L η1 η2 L, and Lemma 7.7 then shows that

L L

Ý Ö × p Üη1 η2 L p f m .

However, this contradicts Lemma 9.17, and thus we have equality in (9.18.1).

It remains to prove the third point. The assertions regarding ε 0 is the content of the ﬁrst point of Lemma 9.18, which also shows that

(9.18.2)

e ×

e N Öp e

m p η1 η2 L ε e max N : f p fptm f e for all e L,

where the last equality holds by Lemma 4.6. Finally, dividing (9.18.2) by pe and

taking the limit as e shows that fptm f η1 η2 L ε.

4 Proof of the main technical lemmas

In this section we prove the two lemmas used in the proof of Theorem 9.14.

Lemma 9.17. Let ℓ L be as in Theorem 9.14. Then 1. 1 ℓ L and

1 1

2. η1 ℓ η2 ℓ pℓ η1 L η2 L pL η1 η2 L.

3. Furthermore,

L L ℓ ℓ ℓ

Ý Ö × Ü Ý Ü Ý Ö ×

p Ü η1 η2 L p p η1 ℓ p η2 ℓ 1 p

f m f m

1 1

η , 0 , η 0, Interior P . ℓ pℓ ℓ pℓ

Proof. We will ﬁrst show that L 1. If L 0, then by deﬁnition we have that

Õ Ô Õ

c1 Ôη1 c1 η2

c1 η1 c1 η2 p. This implies that η1 η2 η1 1 η2 1 p 1,

contradicting the assumption η1 η2 1. We have that ℓ L by deﬁnition, so

it remains to show that that ℓ 0. If ℓ 0, then by deﬁnition we have that

ce η1 ce η2 p 1 for 1 e L, from which it follows that η1 L η2 L

L ¡

L p¡1 p 1

L . By deﬁnition of L, we have that c 1 η1 c 1 η2 p, and so e1 p p L L

cL 1 η1 cL 1 η2 p 1 1

η1 η2 η1 η2 η1 η2 1,

L 1 L 1 L L pL 1 pL pL which again contradicts the assumption that η1 η2 1.

To prove the second point, we may assume that ℓ L. By deﬁnition of ℓ and L,

L p¡1

η1 η2 η1 η2 e . From this, we may conclude that

L L ℓ ℓ eℓ 1 p

1 p 1 1 1

η η η η η η .

1 L 2 L pL 1 ℓ 2 ℓ pe pL 1 ℓ 2 ℓ pℓ

eℓ 1

The assertion that η1 L η2 L η1 η2 L follows from Lemma 4.17.

We now prove the last point. As η1 ℓ η2 ℓ pℓ η1 η2 L, we have that

¡ ©

ℓ 1

Ý Ü Ý

ℓ ℓ p Ü η η ℓ

Ý Ü Ý Ü Ý

p Üη1 ℓ p η2 ℓ 1 ℓ ℓ pℓ p η1 η2 L f f f .

From this, we see that

¡ L ℓ

ℓ ℓ ℓ ℓ ℓ p L L

Ý Ü Ý Ü Ý Ö × Ü Ý Ü Ý Ö ×

p Üη1 ℓ p η2 ℓ 1 p η1 η2 L p p η1 η2 L p η1 η2 L p

f f m f f m .

We now prove the remaining equivalence. We will begin by supposing that

1 1

η , 0 , η 0, Interior P .

ℓ pℓ ℓ pℓ

1 1

Without loss of generality, we assume that σ : η ℓ pℓ , 0 η1 ℓ pℓ , η2 ℓ is in the interior of P . We gather some important properties of σ below:

1 2 1

1. We have that σ pℓ N , and that σ η1 ℓ η2 ℓ pℓ .

2. By deﬁnition of ℓ L, we have that ce η1 ce η2 p 1 for e ℓ, with

ℓ 2 this inequality strict for e ℓ. This shows that the coordinates of p σ N

add without carrying, and by Lemma 4.14, we have that

ℓ p σ

0 mod p. pℓσ

ℓ ℓ ℓ ℓ

3. Finally, as σ Interior P , it follows that p σ1νi p σ2ωi p νi,ωi σ p

ℓ ℓ e

Ö ×

p σ1ν p σ2ω p

for all 1 i m, so that x m .

ℓ ℓ

The ﬁrst two points above, combined with Lemma 9.9, show that xp σ1ν p σ2ω is

ℓ ℓ ℓ ℓ

Ý Ü Ý p Ü ν1 ℓ p ν2 ℓ 1 p σ1ν p σ2ω contained in Support f , while the last point shows that x

ℓ ℓ ℓ ℓ

× Ü Ý Ü Ý Ö × Öp p ν1 ℓ p ν2 ℓ 1 p m . From this we may conclude that f m .

ℓ ℓ ℓ

Ý Ü Ý Ö × p Üη1 ℓ p η2 ℓ 1 p

To conclude the proof, suppose that f m . Our aim now is to

1 1

show that either η ℓ pℓ , 0 or η ℓ 0, pℓ is in the interior of P . By assumption,

2 ℓ ℓ k1ν k2ω

there exists an element k N with k p η1 ℓ p η2 ℓ 1 such that x is

ℓ ×

Öp 1

m contained in Support f but not in . Let α α1,α2 : pℓ k. We summarize some important properties of α:

1 1

4. α pℓ N and α η1 ℓ η2 ℓ pℓ .

ℓ

Ö ×

k1ν k2ω p ℓ 5. As x m , every entry of k1ν k2ω is strictly less than p , which

shows that α Interior P .

By 4 above, it is not possible that both α1 η1 ℓ and α2 η2 ℓ. Without

loss of generality, we will assume that α2 η2 ℓ pℓ , and substituting this into 4

forces the inequality α1 η1 ℓ. By 4 again, we see there exists N N such that

N N 1

6. α pℓ , pℓ η ℓ 0, pℓ .

1 1

We now show that η ℓ 0, pℓ Interior P . By Lemma 4.6, η2 ℓ pℓ η2, so that

1 1

(9.18.3) η 0, P η 0, P .

ℓ pℓ ℓ pℓ

By (9.18.3), to show that η ℓ 0, pℓ Interior P , it suﬃces to show that

(9.18.4)

1 Æ

η 0, Interior P P

ℓ pℓ

s R s η ν ,ω s ν ,ω ν ω .

¥0 : 2 2 and i i 1 i i with i i

In (9.18.4), we have used the assumption that Supportf minimally deﬁnes P , along

with the description of P given in Lemma 9.8. By 6 above,

1 N N

(9.18.5) ν ,ω η 0, ν ,ω α ,

i i ℓ pℓ i i pℓ pℓ

N N

ν ,ω α ν ,ω , i i i i pℓ pℓ

ν ,ω α ν ω . i i pℓ i i

Finally, as α Interior P , it follows from (9.18.5) that

ν ,ω η 0, ν ,ω α 1 for all ν ω , i i ℓ pℓ i i i i

which shows that the desired containment from (9.18.4) holds.

1 1

Lemma 9.18. If ε : sup r : η ℓ r, pℓ , η ℓ pℓ ,r P 0, then

1. 0 ε η L, with ε η L either η1 or η2 is in pℓ N,

e e

Ý Ü Ý Ü

Ö ×

p η1 η2 L ε e p 2. f m for all e L, and

e 1 e

Ü Ý Ü Ý

× 1 2 Ö

p η η L ε e pe p 3. f m for all e L.

100

Proof. We prove the ﬁrst assertion. Suppose that η ℓ r, pℓ is in P . It follows

from the fact that η is a maximal point of P and Lemma 9.17 that

1 1

η r η1 η2 r η r, η η η ,

L ℓ ℓ pℓ ℓ pℓ L L

which shows that r, and consequentially ε, is less than or equal to η L.

Suppose that ε η L and that η ℓ η L , pℓ P . Note that

1 1

η η , η1 η2 η1 η2 η1 η2 η1 η2 ,

ℓ L pℓ ℓ ℓ pℓ L L L

which shows that η ℓ η L , pℓ is a maximal point of P . As η is the unique such

1 1

point of P , we have that η η ℓ η L , pℓ . This shows that η2 η2 ℓ pℓ ,

1 which by Lemma 4.6 shows that η2 pℓ N.

1 1

Conversely, without loss of generality, suppose that η1 pℓ N, so that η1 ℓ pℓ

by Lemma 4.6. We have already seen that ε η1 η2 L, and to prove equality it

suﬃces to show that η ℓ pℓ , η1 η2 L P . Observe that

η1 η2 L η1 η2 L η1 η2

η1 ℓ η1 ℓ η2 ℓ η2 ℓ

η η η

1 ℓ pℓ 2 ℓ 2 ℓ

η1 η2 L η2 ℓ .

From this, we conclude that

(9.18.6) η1 η2 L η2 ℓ .

Finally, from (9.18.6), we have that

1 1 1

η , η η η , η η , η η η P . ℓ pℓ 1 2 L ℓ pℓ 2 ℓ 1 ℓ pℓ 2 ℓ 2 L 101

e L e

Ü Ý Ü Ý

Ö × p η1 η2 ε

L e L p

We will now show that f m for all e 1. We know that

1 1

either η ℓ pℓ ,ε or η ℓ ε, pℓ is contained in P , and without loss of generality

we will assume that η ℓ pℓ ,ε P . Fix e L, and consider the element

1 1

(9.18.7) σ : η , η ε N. 1 ℓ pℓ 2 ℓ e pe

We summarize some important properties of σ below:

1. σ η1 ℓ η2 ℓ pℓ ε e η1 η2 L ε e.

ℓ 1 ℓ

2. By deﬁnition of ℓ L, the integers p η1 ℓ pℓ and p η2 ℓ add without

e 1 e

carrying. As e L ℓ, the integers p η1 ℓ pℓ and p η2 ℓ also add

e 1 e

without carrying. Furthermore, the integers p η1 ℓ pℓ and p η2 ℓ are

both greater than or equal to pe¡ℓ, and hence also greater than or equal to

e¡L 1

p . By Lemma 9.17 we know that ε e ε η1 η2 L pL , so that

e e¡L e

p ε e p . It follows that the entries of p σ add without carrying, so that

e | p |σ

peσ 0 mod p by Lemma 4.14.

3. Finally, as ε e ε, it follows that σ Interior P .

e e

¤ By Lemma 9.9, the ﬁrst two points above show that the monomial xp σ1 ¤ν1 p σ2 ω

Ý Ü Ý Ü p η1 η2 ε is contained in the support of f L e , while the third point shows that

e × this same monomial is not contained in mÖp . From this, we may conclude that

e e

Ý Ü Ý Ü

Ö ×

p η1 η2 L ε e p f m for all e L.

e 1 e

Ü Ý Ü Ý

× 1 2 Ö

p η η L ε e pe p

We will now show that f m for all e L. As

¡ e L

e 1 L p e

Ü Ý Ü Ý

Ü Ý Ü Ý 1 2

p η η L ε e pe p η1 η2 L p ε e 1 f f f , it suﬃces to show that

(9.18.8)

e¡L e L e

× Ü Ý Ü Ý

p Öp p η1 η2 L p ε e 1

m µ1 µ2 for all µ1 Support f and µ2 Support f . 102

L e

× Ö ×

Öp p

We may assume that µ1 m and µ2 m , for otherwise the containment in

a1ν a2ω b1ν b2ω (9.18.8) holds trivially. Thus, we may write µ1 x and µ2 x for

2 L e

some a a1, a2 and b b1, b1 in N with a p η1 η2 L and b p ε e 1

L such that every entry of a1ν a2ω is less than p and every entry of b1ν b2ω is

e 1 1

less than p . We set α : pL a and β : pe b, and summarize some important properties of α and β below:

1 2 1

4. α pL N Interior P , and α η1 η2 L η1 ℓ η2 ℓ pℓ .

1 2 1

5. β pe N Interior P , and β ε e pe .

Under this new notation, the condition appearing in (9.18.8) may be restated as ¡

e¡L e L e e

Ô Õ Ö ×

p a1 b1 ν p a2 b2 ω p α β p x x m .

From this, we see that to conclude the proof, it suﬃces to show that

(9.18.9) α β Interior P .

By 4 above, it is not possible that both α1 η1 ℓ and α2 η2 ℓ. Without

loss of generality, we will assume that α2 η2 ℓ pℓ , and 4 above again shows

that α1 η1 ℓ. Thus, there exists N N such that

N N 1

α , η 0, . pℓ pℓ ℓ pℓ

For indices i with νi ωi, we have that

1 N N

(9.18.10) ν ,ω η 0, β ν ,ω α , β i i ℓ pℓ i i pℓ pℓ

ν ,ω α β ν ω

i i pℓ i i

νi,ωi α β . 103

By (9.18.10), to demonstrate (9.18.9), it suﬃces to show that

(9.18.11) ν ,ω η 0, β 1 for some i with ν ω . i i ℓ pℓ i i

There exists an i with νi ωi by our assumption that η1 and η2 are non-zero.

We will now conclude this proof by showing that the desired statement in (9.18.11)

1 2

holds. As β pe N , it follows that β must be contained the line segment determined

by 0, β and β , 0 , and thus must be of the form

(9.18.12) β β , 0 t β , β for some t 0, 1 .

If νi ωi, then (9.18.12) shows that

(9.18.13)

1 1

ν ,ω η 0, β ν ,ω η β , t β , β

i i ℓ pℓ i i ℓ pℓ

ν ,ω η β , t β ω ν

i i ℓ pℓ i i

ν ,ω η β , i i ℓ pℓ

Comparing (9.18.11) with (9.18.13), we see that it suﬃces to show that

ν ,ω η β , 1 for some i with ν ω . i i ℓ pℓ i i

However, as β ε e pℓ ε, it suﬃces to show that

ν ,ω η ε, 1 for some i with ν ω .

i i ℓ pℓ i i

1 Æ

By deﬁnition, we have that η ℓ 0, pℓ P , and it follows from the deﬁnition

1 Æ

of ε that η ℓ ε, pℓ is not contained in Interior P . By Lemma 9.8, this shows

that

ν ,ω η ε, 1 for some i with ν ω , i i ℓ pℓ i i which allows us to conclude the proof of the Lemma. BIBLIOGRAPHY

104 105

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