Moduli of Surfaces and Applications to Curves

Home , Abelian surface, Kodaira dimension, Moduli scheme, Surface of general type

Monica Marinescu

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

This thesis has two parts. In the first part, we construct a moduli scheme F rns that parametrizes tuples pS1,S2,...,Sn 1, p1, p2, . . . , pnq where S1 is a fixed smooth surface over Spec R and Si 1 is the blowup of Si at a point pi, @1 ¤ i ¤ n. We show this moduli scheme is smooth and projective. We prove that F rns has smooth pnq @ ¤ ¤ ÞÑ divisors Di,j , 1 i j n, which correspond to tuples that map pj pi under the projection morphism Sj Ñ Si. When R k is an algebraically closed field, we ¦p r sq ¦p nq demonstrate that the Chow ring A F n is generated by these divisors over A S1 . ¦ We end by giving a precise description of A pF rnsq when S1 is a complex rational surface. In the second part of this thesis, we focus on finding a characterization of the smooth surfaces S on which a smooth very general curve of genus g embeds as an ample divisor. Our results can be summarized as follows: if the Kodaira dimension of S is κpSq ¡8 and S is not rational, then S is birational to C ¢ P1. If κpSq is 0 or 1, then such an embedding does not exist if the genus of C satisfies g ¥ 22. If κpSq 2 and the irregularity of S satisfies qpSq g, then S is birational to the symmetric square Sym2pCq. We analyze the conditions that need to be satisfied when S is a rational surface. The case in which S is of general type and qpSq 0 remains mainly open; however, we provide a partial answer to our question if S is a complete intersection. Contents

Introduction 1

1 Deﬁnition of the moduli problem 12

2 Construction of the moduli scheme 16

3 Divisors of the moduli scheme 34

4 The Chow ring of the moduli scheme 41

5 The Chow ring of the moduli scheme for rational surfaces 47

6 A Question about Very General Curves 58

7 Curves and Surfaces 61

8 Preliminary results 66

9 Surfaces of Kodaira dimension ¡8 72

10 Surfaces of Kodaira dimension 0 86 10.1 Abelian Surfaces ...... 86 10.2 K3 Surfaces ...... 87

i 10.3 Enriques Surfaces ...... 89 10.4 Bielliptic Surfaces ...... 90

11 Surfaces of Kodaira dimension 1 91

12 Surfaces of Kodaira dimension 2 97 12.1 Surfaces of general type with qpSq g ...... 97 12.2 Surfaces of general type with qpSq 0 ...... 98

Bibliography 103

ii Acknowledgements

I want to start by thanking my advisor Johan de Jong. I look up to Johan as a role model due to his incredible passion for mathematics, his vast knowledge, and his eagerness to share ideas with others. I learned a great amount from him through our many meetings, his research seminars, and the Stacks Project. I thank Johan for his incredible mentorship, through which I found confidence in my mathematical abilities. There were times when I struggled finding my way, and his guidance was key in helping me overcome the obstacles in my mathematical journey. I thank Robert Friedman, Henry Pinkham, Akash Sengupta and Daniel Litt for being part of my thesis committee, and for taking their time to carefully review this work and provide helpful feedback. I especially want to thank Daniel Litt for suggesting to me a project to think about back in 2018. His question and initial ideas were the starting point of this entire body of work. I am deeply grateful for everything I was offered in the Mathematics Department at Columbia, both academically and socially. These past five years have been a grand immersive experience into the world of mathematics research. Among the many avenues to learning, I am thankful for the multitude of seminars offered every semester, which helped me achieve a well-rounded perspective of mathematics. I thank the administrative staff, particularly Nathan Schweer, for all the hard work they put into making sure the graduate students are living their best life in the department. I am thankful for my academic brothers Raymond Cheng, Carl Lian, Dmitrii Pirozhkov, Noah Olander, Shizhang Li, Qixiao Ma, and Remy van Dobben de Bruyn. I learned a lot from our student seminars, our discussions, and from everyone’s different research topics. I particularly want to thank Raymond for reading some of my

iii earlier work and giving me incredibly helpful comments. I want to thank many colleagues in the department who became lifelong friends, including: Clara Dolfen, Renata Picciotto, Raymond Cheng, Elena Giorgi, Laura Hayward, Stanislav Atanasov, Lea Kenigsberg, Sam Mundy, Noah Olander. You made these last ﬁve years some of the best of my life. You proved to me everyday that we can be happy in grad school. Lastly, I want to thank my family for being my support system: my parents, my sister, and my brother-in-law. I thank my partner, Owen, for making my life complete.

iv Introduction

Origin of the thesis

The idea for this thesis started from Daniel Litt. He found Proposition 6.1 in Harris’s and Mumford’s paper “On the Kodaira Dimension of the Moduli Space of Curves” (see [20]), which states the following: if a surface S contains a very general curve of

genus g ¥ 22 that moves in a non-trivial linear system, then S is birational to C ¢P1. Litt thought to analyze a similar question in a different setting: can we characterize the smooth surfaces on which a smooth very general curve C embeds as an ample divisor such that dim |C| 0? As a starting point, de Jong and Litt gave me the idea that Pic0pCq should be a simple abelian variety. To attack this question, we divided the problem into multiple cases, one for each minimal model of the surface S. This is the content of Chapters 6-12. Our result can be summarized as follows: if κpSq ¡8 and S is not rational, then S is birational to C ¢ P1. If the Kodaira dimension of S is 0 or 1, then such an embedding does not exists if the genus of C satisfies g ¥ 21. If κpSq 2 and qpSq g, then S is birational to the symmetric square Sym2pCq. There are a few cases left open. If S is a rational surface, we analyze the conditions that need to be satisfied in this situation (see Prop. 9.5 and 9.10). For the case in which S is of general type and qpSq 0, we prove the following partial result: if

1 S ãÑ Pr is a complete intersection and the composed morphism C ãÑ S ãÑ Pr satisﬁes the Maximal Rank Conjecture, then C is not ample on S if its genus is higher than 16. While working on the proofs above, we encountered the following scenario: say we have a series of morphisms Sn 1 Ñ ¤ ¤ ¤ Ñ S1, where S1 is a smooth minimal surface

and each map Si 1 Ñ Si is the blowup of some point pi P Si. Say, for example, S1 is

a K3 surface, so it has 19 moduli (see Thm. 10.2). Then Sn 1 has 19 2n moduli, since we add 2 moduli each time we blow up a new point. This line of thought made us consider the following scenario: say we ﬁx a smooth surface S1 over Spec R.

Consider all the ordered sequences of morphisms Sn 1 Ñ ¤ ¤ ¤ Ñ S1, where Si 1 Ñ Si is the blowup of Si at a point pi. Can we thoroughly construct a moduli space that parametrizes these objects? The answer is yes. In Chapters 1- 5, we construct a moduli scheme that parametrizes these sequences of blowups Sn 1 Ñ ¤ ¤ ¤ Ñ S1. We prove that the moduli functor is represented by a smooth projective scheme of dimension 2n over Spec R, as expected. We ﬁnd smooth

pnq p qn ÞÑ divisors Di,j that correspond to tuples Si, pi i1 where pj pi under the projection map Sj Ñ Si. When R k is an algebraically closed ﬁeld, we prove that these ¦p r sq ¦p nq divisors generate the Chow ring A F n over A S1 . We end by giving a precise description of this Chow ring in the case where S1 is a complex rational surface. One of the resons why I found this problem interesting is that the moduli scheme has a very beautiful and natural construction, it behaves “as expected”. All the intu- itive guesses we had about these moduli spaces and their properties while developing our theory turned out to be correct. Second of all, this construction is a new step forward in the study of moduli of surfaces. At the moment, one can ﬁnd in the literature various constructions of moduli spaces parametrizing certain minimal surfaces. This moduli scheme stands

2 out because it parametrizes non-minimal surfaces. Lastly, looking at our results regarding the study of very general curves on smooth surfaces, here is what we can say based the open cases: if a curve C of high genus g is embedded on a smooth surface S as an ample divisor, then S is either a surface of general type of irregularity 0 (this is a vast class of surfaces), or it has a very speciﬁc blowup of P2. It is interesting to note that S might be a very general class of surfaces, or a class given by very strict conditions. When it comes to future work, we see some clear ways to expand on the thesis below. First, one can extend the moduli construction over a non-aﬃne base, i.e. replace Spec R by an arbitrary scheme. Another step is to work on the open cases. Ideally, with more work one could conclude that the surface S cannot be rational. One can also work on proving S cannot be a complete intersection (removing the requirement involving the Maximal Rank Conjecture).

Main statements

Let S1 be a ﬁxed smooth projective surface over a commutative ring R. Our goal is to study and parametrize surfaces obtained through a series of n ordered blowups of the base surface S1. More speciﬁcally, we focus our attention on tuples:

pS1,S2,...,Sn 1, p1, p2, . . . , pnq,

where pi P Si and Si 1 is the blowup of Si at pi, for all 1 ¤ i ¤ n.

To parametrize these tuples, we deﬁne the functor Frns FrS1, ns, whose objects are as follows: for any R-scheme B, an element in FrnspBq is a tower of morphisms:

3 πB,n 1 πn πn¡1 π2 ΣB,n 1 ΣB,n ΣB,n¡1 ... ΣB,2 ΣB,1 B ¢ S1

p1 π1

p2 B,

pn¡1 pn

such that the following conditions are satisﬁed:

(1) π1 pr1 is the projection onto the ﬁrst factor;

(2) for each 1 ¤ i ¤ n, the morphism pi : B Ñ ΣB,i is a section of the composed

map ΣB,i Ñ B;

(3) for each 1 ¤ i ¤ n, the morphism πi 1 :ΣB,i 1 Ñ ΣB,i is the blowup of ΣB,i

along the locus pipBq.

Theorem. The functor Frns has a ﬁne moduli scheme F rns. The space F rns, together with all the schemes in its universal family, are smooth and projective over Spec R. Moreover, the top scheme in the universal family of F rns is the moduli space r s p r s ¢ r sq of the functor F n 1 , and it can be identiﬁed with BlM F n F rn¡1s F n .

We think about this moduli scheme F rns inductively. First, it is intuitively easy r s r s r s p ¢ q to see that F 0 Spec R, F 1 S1, and F 2 BlM S1 S1 . Moreover, the

universal family over F r0s is given by the structure morphism S1 Ñ Spec R, and the

r s p ¢ q Ñ ¢ ÝÝÑpr1 universal family over F 1 is BlM S1 S1 S1 S1 S1. Using these base cases as guidance, we form the following hypothesis, which proves to be correct: the top r s p r s ¢ r sq scheme in the universal family over F n is BlM F n F rn¡1s F n , and it is actually the moduli scheme corresponding to the ”next” functor Frn 1s. We prove this claim

4 by constructing a direct 1-to-1 correspondence for every R-scheme B:

p p r s ¢ r sqq r sp q Mor B, BlM F n F rn¡1s F n F n 1 B .

Additionally, since we build these schemes inductively, it is easy to prove at the end of the construction that they are indeed smooth and projective. r s pnq @ ¤ ¤ The moduli scheme F n comes equipped with divisors Di,j , 1 i j n, which can be described as follows: given an R-scheme B, a B-point of such a divisor

pnq p qn Di,j corresponds to a sequence of blow-ups ΣB,i, pi i1 for which pj maps to pi under the projection map ΣB,j Ñ ΣB,i. These divisors arise naturally from the r s r s Ñ n construction of the moduli space F n : the projection map F n S1 decomposes as a sequence of blowups, and these divisors inside F rns are exactly the inverse images of the exceptional divisors corresponding to the blowups. In Proposition 3.5, we show

pnq that the divisors Di,j are smooth. Now assume R k is an algebraically closed ﬁeld. Using a theorem of Keel (see Thm. 4.1), we prove that the Chow ring of the moduli space F rns is generated by

t pnqu ¦p nq the classes of the divisors Di,j 1¤i j¤n over the Chow ring A S1 . Additionally, we prove two key relations among these classes in the Chow ring A¦pF rnsq: let d be any ¦ P ¦p r sq divisor class of S1 and di A F n be the image of d under composed morphism r s Ñ n ÝÑpri @ ¤ ¤ F n S1 S1, 1 i n. Then:

pnq ¦ pnq ¦ @ ¤ ¤ (i) Di,j di Di,j dj , 1 i j n;

pnq pnq pnq pnq @ ¤ ¤ (ii) Di,j Dj,k Di,k Dj,k , 1 i j k n.

r s Ñ n Intuitively, relation (i) above holds because the projection map F n S1 maps pnq n Di,j onto the diagonal Mi,j S1 . Relation (ii) can be explained as follows: for any p qn i j k, the left hand side parametrizes sequences of blowups ΣB,i, pi i1 where pk

5 maps to pj and pj maps to pi. The right hand side parametrizes sequences of blowups p qn ΣB,i, pi i1 where pk maps to pj and pk maps to pi. Hence the two sides agree.

In the case where the base surface S1 is rational and the base field is the complex numbers C, another lemma of Keel (see Prop. 5.2) helps us conclude that the canonical map A¦pF rnsq ÝÑcl H2¦pF rnsq is an isomorphism. With this in mind, we set out to compute the Betti numbers of F rns. Using the spreading out method, we define the r s b surface S1 over a finite field Fq and construct the moduli space F n Fq over this p q new surface. We compute the polynomial Rn q that gives the number of Fq-rational r sb points of F n Fq. Together with the Grothendieck-Lefschetz Trace formula and the identities between the Betti numbers in étaleand singular cohomology, we conclude

that Rnpqq coincides with the Poincare polynomial of F rns, so we immediately recover the Betti numbers we wanted (see Lemma 5.11). Lastly, we use these Betti numbers to conclude that the relations (i) and (ii) above are suﬃcient to give a full description of A¦pF rnsq. For example, in Corollary 9.11 we present the Chow ring of the moduli space

¦ A pF rP2, nsq as follows:

¦ pnq rH ,D s ¤ ¤ ¤ ¤ ¦p r 2 sq Z i j,k 1 i n,1 j k n A F P , n p q p q p q p q p q , x n p n ¡ n q n p ¦ ¡ ¦q ¦3 p¡ n qy Dj,k Di,j Di,k ,Dj,k Hj Hk ,Hi ,Pi,j Di,j

@ ¤ ¤ ¦ P ¦p 2q where, 1 i n, Hi is the image of the hyperplane class H A P under the r 2 s Ñ p 2qn ÝÑpri 2 p q composition F P , n P P , and Pi,j t are certain quadratic polynomials (see Thm. 4.1).

r 2 s Let bn,j be the j-th Betti number of the moduli space F P , n . From the discussion above, we get the following recursive relation:

bn 1,j bn,j pn 1qbn,j¡1 bn,j¡2.

6 Here is how we can interpret the relation above: we have the forgetful map r s Ñ r s ¦ ¦p r sq Ñ Πn 1 : F n 1 F n which induces a map on the Chow rings Πn 1 : A F n A¦pF rn 1sq. Given this map, we can think of the identity above as follows: compared to the moduli space F rns, the space F rn 1s has n 1 extra divisors:

¦ pn 1q pn 1q jp r sq Hn 1,D1,n 1,...Dn,n 1. A generator in A F n 1 is either a class inherited from jp r sq ¦ A F n under the map Πn 1 (this accounts for bn,j generators), or it is a product between a generator class coming from Aj¡1pF rnsq and one of the n 1 new divisor

p q p ¦ q2 classes (this accounts for n 1 bn,j¡1 generators), or it is a product between Hn 1

j¡2 and a generator class coming from A pF rnsq (this accounts for bn,j¡2 generators).

Now we switch gears entirely and go back to the original problem suggested by Daniel Litt. Recall that we want to characterize smooth surfaces S on which a smooth very general curve C of genus g embeds as an ample divisor such that dim |C| 0. Our general strategy will be to break down the problem into multiple cases, one for each minimal model of S. We start this second half of the thesis by recalling some basic facts about curves and surfaces. We recall the Enriques-Kodaira classiﬁcation of complex surfaces, which we will use extensively. Moreover, we recall the basic results of Brill-Noether theory and the Maximal Rank Conjecture. We continue with some preliminary results. Since C is a very general curve, Pic0pCq is a simple abelian variety (see Thm. 8.1). Additionally, since C is ample on S, the pullback map ι¦ : Pic0pSq ãÑ Pic0pCq is injective. Putting these two together, we conclude that either Pic0pSq 0 or the map ι¦ is an isomorphism. Now, we can assume that every (-1)-curve E on S satisﬁes C ¤ E ¥ 2 (if C ¤ E 1,

we can contract E and preserve both the smoothness and ampleness of C). Let S0 be

a minimal model of S and S Sn Ñ ¤ ¤ ¤ Ñ S0 the corresponding sequence of blow-

7 downs. We prove the following inequality, where KS,KS0 are the canonical divisors

of S and S0, respectively:

¤ ¥ ¤ C KS0 C0 KS0 2n.

In addition, if Pic0pSq 0, then the number of moduli of S is at least 3g ¡ 3 (see Lemma 8.7). We analyze ﬁrst the scenario in which a smooth very general curve C of genus g is embedded as an ample divisor on a smooth surface of Kodaira dimension ¡8. If

S is not rational, then it is birational to C ¢ P1 (see Prop. 11.3). The case in which S is rational remains mainly open. However, we analyze the conditions that need to be satisfied in this situation (see Prop. 9.5 and 9.10). While these conditions are not sufficient to conclude such an embedding C ãÑ S exists, we show that there exist cases where this conditions could potentially be satisfied. We make this precise in Chapter 9. Next, we analyze the case in which a smooth very general curve C of genus g is embedded as an ample divisor on a smooth surface of Kodaira dimension 0. We know that the minimal model of S, denoted by S0, is either an abelian surface, a K3 surface, an Enriques surface, or a bielliptic surface. We show that S0 is an abelian surface if and only if the genus of C is g 2 and S is the Jacobian of C. If S0 is bielliptic, such an embedding is not possible. If S0 is a K3 or Enriques surface, then g ¤ 18. To conclude the latter, we combine the following facts: S0 has a fixed (bounded) number of moduli, C ¤ KS is bounded below by 2n, and S has at least 3g ¡ 3 moduli.

When κpSq 1, we show that S is an elliptic ﬁbration over P1 (see Lemma 11.3). We use the moduli spaces constructed by Miranda and Seiler which parametrize these elliptic ﬁbrations (see Thm. 11.7 and 11.9), and proceed with the same analysis as

8 above to conclude that such an embedding does not exist if g ¥ 21. Lastly, we analyze the case in which S is of general type. If qpSq g, then S is birational to the symmetric product Sym2pCq. This result is due to Mendes Lopes and Pardini (see [28]). The case where qpSq 0 remains mainly open. However, we prove the following partial result: if S ãÑ Pr is a complete intersection such that the composed map C ãÑ S ãÑ Pr satisﬁes the Maximal Rank Conjecture, then g ¤ 15. For this part, we mainly use the results of Brill-Noether theory (see Thm. 7.11).

Structure of the thesis

We provide here an outline of each chapter. In Chapter 1, we deﬁne thoroughly the moduli functor whose objects are the

tuples pS1,...,Sn 1, p1, . . . , pnq, where Si 1 is the blowup of Si at pi, @1 ¤ i ¤ n, and

S1 is a fixed smooth projective surface over Spec R. We check that our definition has all the required properties of a contravariant functor. We end with a few remarks on the natural maps that this functor possesses. In Chapter 2, we prove that the aforementioned functor has a fine moduli scheme, which we denote by F rns. We demonstrate that F rns and all the schemes in its universal family are smooth and projective over Spec R. pnq r s ¤ In Chapter 3, we define the divisors Di,j of the moduli scheme F n , where 1 ¤ pnq p q i j n. We prove that each Di,j parametrizes tuples S1,...,Sn 1, p1, . . . , pn for

which pj ÞÑ pi under the projection map Sj Ñ Si, then show they are smooth. In Chapter 4, we let R k be an algebraically closed ﬁeld. We prove that the Chow ring of the moduli space F rns is generated by the classes of the divisors

t pnqu ¦p nq Di,j 1¤i j¤n over the Chow ring A S1 . We end the chapter by proving two key relationships between these generators.

9 In Chapter 5, we give a precise description of the Chow ring A¦pF rnsq in the case where the base surface S1 is rational over Spec C. In Chapter 6, we shift our focus to the study of smooth very general smooth curves on smooth surfaces. We explain our question and its origins, then state the main result. In Chapter 7, we state all the basic facts about curves and surfaces which we need later on. Among these facts is a quick introduction to Brill-Noether theory and the statement of the Maximal Rank Conjecture. In Chapter 8, we start building towards a proof of our main result by presenting some preliminary results. If C is a very general curve that embeds as an ample divisor on a smooth surface S, then Pic0pCq is a simple abelian variety and Pic0pSq is either 0

0 or isomorphic to Pic pCq. We give a lower bound for the intersection number C ¤ KS, where KS is the canonical divisor of S, and ﬁnish by showing that S must have at least 3g ¡ 3 moduli. In Chapter 9, we analyze the case where the surface S has Kodaira dimension

¡8. If S is not rational, then it is birational to C ¢ P1. The case in which S is rational remains mostly open. However, we analyze the conditions that need to be satisﬁed in this situation. In Chapter 10, we analyze the case in which the surface S has Kodaira dimension 0. We show that C cannot be embedded on S as an ample divisor if its genus is higher than 18. In Chapter 11, we analyze the case in which the surface S has Kodaira dimension 1. We show that C cannot be embedded on S as an ample divisor if its genus is higher than 20. In Chapter 12, we analyze the case where the surface S has Kodaira dimension 2. If qpSq g, then S has to be birational to the symmetric product Sym2pCq. The

10 case qpSq 0 remains mostly open. However, we show that if S ãÑ Pr is a complete intersection and the composed morphism C ãÑ S ãÑ Pr satisﬁes the Maximal Rank Conjecture, then C cannot be embedded on S as an ample divisor if its genus is higher than 15.

11 Chapter 1

Deﬁnition of the moduli problem

Let S1 be a ﬁxed smooth projective surface over a ring R. Our goal is to study and parametrize surfaces obtained through a series of n ordered blowups of the base surface S1. More speciﬁcally, we focus our attention on tuples:

pS1,S2,...,Sn 1, p1, p2, . . . , pnq,

where pi P Si and Si 1 is the blowup of Si at pi, for all 1 ¤ i ¤ n.

We deﬁne formally the functor FrS1, ns as follows:

Deﬁnition 1.1. Let S1 be a smooth projective surface over a ring R and n ¥ 0 an integer. Consider the contravariant functor:

Frns FrS1, ns : SchpRq Ñ Sets deﬁned as follows:

For any R-scheme B, an object in FrS1, nspBq is a tower of morphisms:

12 πB,n 1 πn πn¡1 π2 ΣB,n 1 ΣB,n ΣB,n¡1 ... ΣB,2 ΣB,1 B ¢ S1

p1 π1

p2 B,

pn¡1 pn

such that the following conditions are satisﬁed:

(1) π1 pr1 is the projection onto the ﬁrst factor;

(2) for each 1 ¤ i ¤ n, the morphism pi : B Ñ ΣB,i is a section of the composed

map ΣB,i Ñ B;

(3) for each 1 ¤ i ¤ n, the morphism πi 1 :ΣB,i 1 Ñ ΣB,i is the blowup of ΣB,i

along the locus pipBq.

p qn A shortened notation for such a family is ΣB,i, πi, pi i1. Notice that Σn 1 is not included in this notation; however, this scheme is uniquely deﬁned by the data given, so the notation is consistent. For brevity throughout the paper, we will also use the following notation for the same family: ΣB,¤n 1 Ñ B. For every R-scheme B, FrnspBq is the set of all families over B, up to isomorphism. An isomorphism between two families is deﬁned as:

p qn p qn ÝÑp 1 1 1 qn Θ θi i1 : ΣB,i, πi, pi i1 ΣB,i, πi, pi i1,

@ ¤ ¤ Ñ 1 where, 1 i n, θi :Σi Σi are isomorphisms that commute with the maps of the two families.

Let B1,B2 be R-schemes, and let f : B1 Ñ B2 be an R-morphism between the two schemes. There exists a natural contravariant map Fpfq : FrnspB2q Ñ FrnspB1q

13 given as follows: for every family over B2, we obtain a family over B1 by pulling back

the schemes ΣB2,i and the sections pi along f, as in the ﬁgure below. To conclude

that the tower of morphisms over B1 is indeed a valid object in FrnspB1q, we need ¦p q ¤ ¤ to know that ΣB1,i 1 is the blowup of ΣB1,i along the locus pi B1 , for all 1 i n. This is an immediate application of Lemma 2.1 from Chapter 2.

ΣB1,n 1 ΣB2,n 1 x

ΣB ,n ΣB ,n 1 x 2

......

¦ pn pn

ΣB1,1 ΣB2,1 x ¦ p p1 1 f B1 B2.

The identity map on an R-scheme id : B Ñ B corresponds to the identity map on sets Fpidq id : FrnspBq Ñ FrnspBq.

Let B1,B2,B3 be R-schemes. Let f : B1 Ñ B2 and g : B2 Ñ B3 be R-morphisms. Then Fpg ¥ fq Fpfq ¥ Fpgq. This follows from the uniqueness of pullbacks (up to isomorphism): if the middle family is the pullback of the family on the right along the map g, and the family on the left is the pullback of the middle family along the map f, then the family on the left is the pullback of family on the right along the map g ¥ f:

ΣB1,¤n 1 ΣB2,¤n 1 ΣB3,¤n 1 x x

f g B1 B2 B3

Remark 1.2. Let B be a R-scheme, and let ΣB,¤n 1 Ñ B be a family in FrnspBq.

For any point x P B, the ﬁber over x in this family is a sequence Sn 1 Ñ ¤ ¤ ¤ Ñ S1,

14 where Si 1 is the blowup of Si at some point pi, @1 ¤ i ¤ n. Therefore, this ﬁber corresponds to a tuple pS1,...,Sn 1, p1, . . . , pnq as the ones introduced in the beginning of the chapter.

Remark 1.3. For every integer n ¥ 0, there exists a natural forgetful map:

Frn 1s Ñ Frns

which sends families ΣB,¤n 2 Ñ B in Frn 1spBq to families ΣB,¤n 1 Ñ B in FrnspBq, for all R-schemes B.

Remark 1.4. For every integer n ¡ 0, there exists a natural transformation of functors: r s Ñ n F n S1

p qn p q which, for any R-scheme B, maps a family ΣB,i, πi, pi i1 over B to p1, p2,..., pn ,

pi pr2 where pi P S1pBq is the composition map B ÝÑ ΣB,i Ñ ¤ ¤ ¤ Ñ ΣB,1 B ¢ S1 ÝÝÑ S1.

15 Chapter 2

Construction of the moduli scheme

In this section we prove that the functor Frns has a ﬁne moduli scheme, which we denote by F rns. We show that F rns and all the schemes in its universal family are smooth and projective over Spec R. Before we start working towards our main result, note that we reserve the following

notation for the universal family over F rS1, ns:

πn 1,n 1 πn 1,n πn 1,2 Σn 1,n 1 Σn 1,n ... Σn 1,2 Σn 1,1 F rns ¢ S1

σn 1,1 πn 1,1

σn 1,2 F rns,

σn 1,n

Example. We start by constructing F r0s, F r1s (which represent the functors Fr0s and Fr1s, respectively), and their universal families. It is easy to see that F r0s Spec R, since every scheme B P SchpRq comes equipped with a structure morphism to Spec R. Moreover, we notice that the universal family over F r0s is, by deﬁnition,

Σ1,1 Spec R ¢ S1 S1:

16 ΣB,1 Σ1,1 B ¢ S1 S1 x x pr1 pr1 pr1 pr1 B F r0s B Spec R

Next, we want to show that F r1s S1. Intuitively, every object in this moduli space p q is a triple S1,S2, p1 such that S2 Blp1 S1, so this triple is uniquely identiﬁed by the point p1 P S1. More concretely, we need to show that Fr1spBq MorpB,S1q, i.e. that each family over B corresponds uniquely to a morphism B Ñ S1. The equivalence goes as follows: say we start with a family over B, like in the ﬁgure below. The Ñ ¥ corresponding morphism B S1 is f pr2 p1. Conversely, say we start with a morphism f : B Ñ S1. This map gives a section p1 id ¢ f : B Ñ ΣB,1 S1 ¢ B, and ΣB,2 is the blowup of ΣB,1 along this section.

ΣB,2

π2

pr2 S1 ΣB,1 B ¢ S1

π1 p id¢f ¥ 1 f pr2 p1 B

r s p ¢ q The top scheme in the universal family over F 1 is Σ2,2 BlM S1 S1 . This follows immediately from the ﬁgure above, considering the special case where B

F r1s S1 and f id : S1 Ñ S1:

p ¢ q Σ2,2 BlM S1 S1

π2,2 blM

Σ2,1 S1 ¢ S1

p2,1 π2,1 M pr1

F r1s S1

17 Before we give the general construction of the moduli scheme corresponding to the functor Frns, we state and prove the following lemmas, which will be the key ingredients in the construction of the moduli scheme.

Lemma 2.1. Let A, B, ΣB,1 be schemes over R. Let π :ΣB,1 Ñ B be a smooth

morphism, let σ : B Ñ ΣB,1 be a section of π, and let ΣB,2 be the blowup of ΣB,1

¦ along the locus σpBq. Given f : A Ñ B an R-morphism, let ΣA,1 and σ be the

pullbacks along the map f of ΣB,1 and σ, respectively. Then the following statements hold:

(i) The composed morphism ΣB,2 Ñ B is smooth.

¦ (ii) The blowup of ΣA,1 along the locus σ pAq, denoted by ΣA,2, is the pullback of

ΣB,2 along the map ΣA,1 Ñ ΣB,1.

ΣA,2 ΣB,2 x

ΣA,1 ΣB,1 x σ¦ π¦ π σ f A B

Proof. (i) We show that the morphism ΣB,2 Ñ B is smooth by proving that it is flat, locally of finite presentation, and has smooth fibers (see [37], Tag 02K5). Affine locally, on the level of rings, we are given a smooth morphism π : T Ñ T 1, and

1 1 1 σ : T T a section of π. Let I kerpσq T . The blowup of Spec T along I is deﬁned to be:

1 1 2 BlI pSpec T q ProjpT ` I ` I ` ... q.

The map σ is a section of the smooth morphism π, which means that I is a

1 regular ideal. Since I is regular, then BlI pSpec T q is of ﬁnite presentation over Spec T (see [37], Tag 0BIQ).

18 1 2 To show that ΣB,2 Ñ B is flat, it is enough to prove that pT ` I ` I ` ... q is flat over T . We know that T Ñ T 1 is smooth, hence T 1 is flat over T . We show inductively that In is flat over T . By the following short exact sequence, it is enough to prove that T 1{In is T -flat:

0 Ñ In Ñ T 1 Ñ T 1{In Ñ 0.

When n 1, T 1{I T , so the claim is true. For the inductive step, consider another short exact sequence:

0 Ñ In¡1{In Ñ T 1{In Ñ T 1{In¡1 Ñ 0.

Since I is a regular ideal, then In¡1{In is a locally free finite T -module, hence it is flat. By the induction hypothesis, T 1{In¡1 is flat over T . Putting these two facts together, we conclude that T 1{In is flat over T , completing the induction step.

To finish the proof of statement (i), we are left to show that the morphism ΣB,2 Ñ B has smooth fibers. By part (ii) below, the fiber over every point x P B is as follows, where kpxq is the residue field of x:

1 V BlxV

V x

Spec kpxq.

1 The scheme V is smooth over kpxq, hence V BlxV is also smooth over kpxq.

With this, we conclude that the morphism ΣB,2 Ñ B is smooth.

19 (ii) This claim is true because blowups commute with base change. To make this precise, we work aﬃne locally, on the level of rings, where we have the following ﬁgure: 1 1 1 S S bT T T σ σS πS π S T

1 Let πS : S Ñ S be the pullback of π and σS be the pullback of σ. Let IS

1 1 kerpσSq S . The blowup of Spec S along IS is deﬁned to be:

p 1q p 1 ` ` 2 ` q BlIS Spec S Proj S IS IS ... .

On the level of rings, our claim boils down to showing that:

1 2 2 1 p ` ` ` q p ` ` ` q b 1 S IS IS ... S I I ... T S ,

which translates to showing that:

n n 1 n b 1 b IS I T S I T S.

The statement above follows from the following ﬁgure. Note that T 1{In is ﬂat over T , so tensoring the top short exact sequence with S preserves the exactness of the resulting sequence:

n 1 1 n 0 I bT S T bT S T {I bT S 0

n 1 1{ n 0 IS S S IS 0

20 Lemma 2.2. Let A, B, C be R-schemes. Let A ÝÑh C be an R-morphism that factors

f g vC vB as A ÝÑ B ÝÑ C. Given any R-morphism VC ÝÑ C, let VB ÝÑ B be its pullback along

vA 1 g, and VA ÝÑ A be its pullback along h. There exists a unique map f : VA Ñ VB which makes the top triangle commutative and the left square cartesian.

h1 VA VC D!f 1 g1

vA VB vC

vB A h C f g B

1 h f¥vA Proof. First, we are given the maps VA ÝÑ VC and VA ÝÝÝÑ B that form a com-

vC g mutative square with the maps VC ÝÑ C and B ÝÑ C. Since the right square is

1 cartesian, there exists a unique map f : VA Ñ VB making all squares and all triangles commutative. We are left to show that the maps f 1 makes the left square cartesian. Ñ Ñ Consider a scheme M together with morphisms mA : M A, mVB : M VB

f vB that make a commutative diagram with A ÝÑ B, VB ÝÑ B. We need to show there Ñ ¥ 1 ¥ exists a unique map mVA : M VA such that mA vA mVA and mVC f mVA .

mVC M mVB m VA h1 VA VC mA f 1 g1 vA VB vC

vB A h C f g B

m 1¥ ÝÝÝÑVC ÝÑvC ÝÝÑmA ÝÑh Let mVC g mVB . We claim that the maps M VC C and M A C

21 coincide. This is true because:

¥ ¥ 1 ¥ vC mVC vC g mVB ¥ ¥ g vB mVB

g ¥ f ¥ mA

h ¥ mA.

Ñ Now, given that the big square is cartesian, there exists a unique map mVA : M 1 ¥ ¥ VA such that mVC h mVA and mA vA mVA . However, we need to show more 1 ¥ that this, speciﬁcally that mVB f mVA . Assume by contradiction this is not the case; then we have two diﬀerent maps M Ñ VB that commute with the cartesian Ñ square on the right, which is impossible. Hence, we found a map mVA : M VA ¥ 1 ¥ such that mB vA mVA and mVB f mVA . Assume by contradiction this map is not unique; then we would have two maps that commuted with the big cartesian square, which is again impossible. Hence the conclusion holds and the left square is cartesian.

Theorem 2.3. The functor Frns has a ﬁne moduli scheme F rns. Moreover, the following is true:

(a) For every n ¥ 0, the moduli space F rns, together with all the schemes Σn 1,1,

... , Σn 1,n 1 in its universal family, are smooth and projective over Spec R;

(b) For every n ¥ 1, the top scheme Σn 1,n 1 in the universal family of F rns can be identiﬁed as: p r s ¢ r sq Σn 1,n 1 BlM F n F rn¡1s F n ,

where the cartesian product is induced by the forgetful map F rns Ñ F rn ¡ 1s;

22 (c) For every n ¥ 0, the top scheme Σn 1,n 1 in the universal family of F rns is the moduli scheme representing the functor Frn 1s. Under this identiﬁcation, the map

Πn 1 πn 1,1 ¥ ¤ ¤ ¤ ¥ πn 1,n 1 :Σn 1,n 1 F rn 1s Ñ F rns

corresponds to the forgetful functor Frn 1s Ñ Frns.

Proof. We will prove these statements inductively over n. In Example 2 we constructed the moduli schemes F r0s Spec R with its universal family Σ1,1 S1, and

F r1s S1. The map Π1 :Σ1,1 Ñ F r0s is the structure morphism Π1 : S1 Ñ Spec R, so it trivially corresponds to the forgetful functor Fr1s Ñ Fr0s. Hence the statements (a) and (c) are true for n 0. For the inductive step, assume the moduli space F rks exists, for all k n. Assume

F rks, Σk 1,1,..., Σk 1,k 1 are all smooth projective schemes. We prove the following:

(i) Let W Σn,n be the top scheme in the universal family over F rn ¡ 1s. We start by constructing a family over W in FrnspW q. We then show that W is the ﬁne moduli scheme corresponding to the functor Frns, and that the family we deﬁned over W is the universal family;

(ii) We show that F rns, Σn 1,1,..., Σn 1,n 1 are smooth and projective, and that p r s ¢ r sq we can identify Σn 1,n 1 with BlM F n F rn¡1s F n ;

(iii) We prove that the map Πn :Σn,n F rns Ñ F rn ¡ 1s corresponds to the forgetful functor Frns Ñ Frn ¡ 1s.

The ﬁrst step is to construct the family over W in FrnspW q. This family is obtained as follows: let Πn : W Σn,n Ñ F rn ¡ 1s be the composed morphism. Let

ΣW,¤n Ñ W be the pullback of the family Σn,¤n Ñ F rn ¡ 1s along the map Πn (see

23 ﬁgure below). In particular, notice that ΣW,n W ¢F rn¡1s W . To construct the top Ñ ¢ scheme ΣW,n 1, let pW,n : M: W ΣW,n W F rn¡1s W be the diagonal embedding p ¢ q and ΣW,n 1 BlM W F rn¡1s W :

p ¢ q ΣW,n 1 BlM W F rn¡1s W

ΣW,n W ¢F rn¡1s W Σn,n W x

pW,n M ΣW,¤n¡1 Σn,¤n¡1 Πn x

W Πn F rn ¡ 1s.

First, notice that the tower of morphisms ΣW,¤n 1 Ñ W is indeed a family in FrnspW q, as a result of Lemma 2.1. We claim that W is the ﬁne moduli space for Frns, and that the family constructed above is the universal family over F rns. We prove our statement by using this family over W to build the correspondence FrnspBq MorpB,W q, for any R-scheme B. We start by constructing a functor map:

C1 : Frns Ñ Morp¡,W q.

Let B be an R-scheme and ΣB,¤n 1 Ñ B a family in FrnspBq. We need to associate a morphism B Ñ W to this family. The truncated family ΣB,¤n Ñ B is an element of Frn¡1spBq, so it corresponds uniquely to a morphism fn¡1 : B Ñ F rn¡1s that gives the ﬁgure below. Now, recall that the family ΣB,¤n 1 Ñ B comes equipped with a section pn : B Ñ Σn, so we obtain the desired map fW : B Ñ W by composing

pn pr2 B ÝÑ ΣB,n ÝÝÑ W :

24 pr Σ 2 Σ W B,n x n,n

pn ΣB,¤n¡1 Σn,¤n¡1 x

f ¡ B n 1 F rn ¡ 1s.

Second, we construct a functor map:

C2 : Morp¡,W q Ñ Frns.

Let B be an R-scheme and f : B Ñ W an R-morphism. We want to obtain a corresponding family in FrnspBq. To do so, we pull back the family ΣW,¤n 1 Ñ W along f. The resulting tower of morphisms is indeed a family in FrnspBq, as a result of Lemma 2.1: ΣB,¤n 1 ΣW,¤n 1

¦ x p pW, W, B W.

Now we want to show that the maps C1 and C2 are inverses of each other. Say we start with a morphism f : B Ñ W and we construct a family over B by pulling back ΣW,¤n 1 Ñ W along f. Then we obtain the following ﬁgure:

ΣB,n 1 ΣW,n 1 x Σ Σ Σ W B,n x W,n x n,n

pn ΣB,¤n¡1 ΣW,¤n¡1 Σn,¤n¡1 x x f B W F rn ¡ 1s.

Notice the following maps are equivalent, which shows that C1 ¥ C2 id:

25 fW pn rB ÝÝÑ W s rB ÝÑ ΣB,n Ñ ΣW,n Ñ Σn,ns

f pW,n rB ÝÑ W ÝÝÝÑ ΣW,n Ñ Σn,ns

f M pr2 rB ÝÑ W ÝÑ W ¢F rn¡1s W ÝÝÑ W s

rB ÝÑf W s.

Lastly, say we start with a family over B in FrnspBq, denoted by ΣB,¤n 1 Ñ

B. As before, we obtain the corresponding morphism fW : B Ñ W , which is the

pn pr2 composed morphism B ÝÑ ΣB,n ÝÝÑ Σn,n W . We want to show that if we pull back ΣW,¤n 1 Ñ W along fW , we recover the family we started with:

ΣB,¤n 1 ΣW,¤n 1 x p¤n pW,¤n f B W W.

Since the truncated family ΣB,¤n Ñ B is an object in Frn ¡ 1spBq, it corresponds uniquely to a morphism f : B Ñ F rn ¡ 1s which gives this ﬁgure:

ΣB,n 1

pr2 ΣB,n Σn,n x

pn ΣB,¤n¡1 Σn,¤n¡1 x

f B F rn ¡ 1s.

We ﬁrst claim that the map B ÝÑf F rn¡1s factors as B ÝÝÑfW W ÝÝÑΠn F rn¡1s. This is true because of the following equivalence of maps:

26 rB ÝÑf F rn ¡ 1ss rB ÝÑid B ÝÑf F rn ¡ 1ss

pn f rB ÝÑ ΣB,n Ñ B ÝÑ F rn ¡ 1ss

pn pr2 Πn rB ÝÑ ΣB,n ÝÝÑ Σn,n ÝÝÑ F rn ¡ 1ss

rB ÝÝÑfW W ÝÝÑΠn F rn ¡ 1ss.

Since the map B ÝÑf F rn ¡ 1s factors as B ÝÝÑfW W ÝÝÑΠn F rn ¡ 1s, we can use

Lemma 2.2 repeatedly, from the bottom up, to recover the maps Σi Ñ ΣW,i, for all i ¤ n, which make every rectangle in the ﬁgure below cartesian, and every triangle commutative: ΣB,n 1

ΣW,n 1

ΣB,¤n Σn,¤n

ΣW,¤n

f B F rn ¡ 1s fW Πn W

To ﬁnish proving that C2¥C1 id, we need to show that the section pn : B Ñ ΣB,n Ñ ¢ is the pullback along fW of the diagonal embedding pW,n M: W ΣW,n W F rn¡1s

W . This follows from the ﬁgure below: by the deﬁnition of fW , it is the composition

pn B ÝÑ ΣB,n Ñ ΣW,n W ¢F rn¡1s W Ñ W , so all maps in the ﬁgure below commute as expected, and pn is indeed the pullback of pW,n M. By Lemma 2.1, we obtain that ΣB,n 1 is the pullback of ΣW,n 1, and the proof of (i) is complete.

27 fW

fW ¢fW B pn

¢ r ¡ s id ΣB,n W F n 1 W W

M f B W W F rn ¡ 1s

To ﬁnish the proof of part (ii), we show inductively that F rns and all the schemes in its universal family are smooth projective schemes over Spec R. For the base case,

recall that F r0s Spec R and Σ1,1 S1 have this property. Inductively, assume that F rn ¡ 1s and all the schemes in its universal family are smooth and projective

over Spec R. From the arguments above, the top scheme Σn,n over F rn ¡ 1s can be identiﬁed with F rns, so we know that F rns is also a smooth and projective. The ﬁrst

scheme in the universal family over F rns is, by deﬁnition, Σn 1,1 F rns ¢ S1, which

also satisﬁes these properties. Now, by construction, all the maps Σn 1,i Ñ F rns are smooth (this is an application of Lemma 2.1), which means the corresponding sections σn 1,i : F rns Ñ Σn 1,i are regular embeddings. Going up in the tower of morphisms, we can conclude step by step that Σn 1,2,..., Σn 1,n 1 are smooth and projective, since each of them is obtained by blowing up a smooth projective scheme along a smooth projective subscheme. This concludes claim (ii).

Lastly, we need to show that Πn :Σn,n F rns Ñ F rn ¡ 1s corresponds to the forgetful functor Frns Ñ Frn ¡ 1s. For any R-scheme B and any morphism f : B Ñ F rns, let ΣB,¤n 1 Ñ B be its corresponding family in FrnspBq. We need to show that Πn ¥ f : B Ñ F rn ¡ 1s corresponds to the truncated family ΣB,¤n Ñ B in Frn ¡ 1spBq. This follows immediately from the ﬁgure below, and the proof of (iii) is complete:

28 Σn 1 Σn 1,n 1 x

Σ¤n Σn 1,¤n Σn,¤n x x

f B F rns Πn F rn ¡ 1s.

Remark 2.4. As a corollary of the construction outlined in the proof of Theorem 2.3, we obtain the following ascending ladder. Note that each square is cartesian, by construction:

... Σ5,5

π5,5

... Σ5,4 Σ4,4 x π5,4 π4,4

... Σ5,3 Σ4,3 Σ3,3 x x π5,3 π4,3 π3,3

... Σ5,2 Σ4,2 Σ3,2 Σ2,2 x x x π5,2 π4,2 π3,2 π2,2

... Σ5,1 Σ4,1 Σ3,1 Σ2,1 Σ1,1 x x x x π5,1 π4,1 π3,1 π2,1 π1,1 ... F r4s F r3s F r2s F r1s F r0s. Π4 Π3 Π2 Π1

In particular, since Σn,n F rns, for all n ¥ 1, we obtain the following identiﬁca- tion, @1 ¤ i ¤ n 1:

Σn 1,i F rns ¢F ri¡1s F ris. (2.1)

In light of Equation 2.1, we look back to the universal family over F rns and give

29 another description of the projection maps πn 1,¦ and the sections σn 1,¦. Before we do so, we need to establish some notation:

Notation 2.5. Let B be an R-scheme. A point in F rnspBq corresponding to a family p qn p q ΣB,i, πi, pi i1 will simply be denoted as p1, . . . , pn . Similarly, a B-point of Σn 1,i r s ¢ r s p 1 q F n F ri¡1s F i will simply be denoted by p1, . . . , pn; pi , with the understanding p q r sp q p 1 q that p1, . . . , pn is the corresponding point in F n B and p1, . . . , pi¡1, pi is the corresponding point in F ris.

p qn Proposition 2.6. Let B be a R-scheme and Σn 1,i, πn 1,i, σn 1,i i1 be the universal family over the moduli scheme F rns. Using Notation 2.5 above, the morphisms πn 1,¦ and σn 1,¦ map B-points as follows:

(a) σn 1,i : F rnspBq Ñ Σn 1,ipBq

pp1, . . . , pnq ÞÑ pp1, . . . , pn; piq;

(b) πn 1,i :Σn 1,ipBq Ñ Σn 1,i¡1pBq p 1 q ÞÑ p 1 q p1, . . . , pn; pi p1, . . . , pn; pi , 1 Ñ 1 Ñ where pi : B ΣB,i¡1 is the image of pi under the projection map ΣB,i

ΣB,i¡1.

Proof. (a) Let f : F rns Ñ F ris be the natural projection map, which maps B-points

pp1, . . . , pnq ÞÑ pp1, . . . , piq. We obtain the following diagram:

M¥f

F rns σn 1,i

f¢id Σn 1,i F rns ¢F ri¡1s F ris Σi 1,i F ris ¢F ri¡1s F ris id pr1 pr1 f M F rns F ris,

30 where the section σn 1,i : F rns Ñ Σn 1,i is deﬁned to be the unique morphism making the two triangles of the diagram commute. It is clear from the ﬁgure that σn 1,i maps a B-point of F rns, denoted by pp1, . . . , pnq, to pp1, . . . , pn; piq P Σn 1,ipBq.

(b) We start by showing that, for all n, the morphism

πn 1,n 1 :Σn 1,n 1 Ñ Σn 1,n maps B-points as follows:

pp1, . . . , pn, pn 1q ÞÑ pp1, . . . , pn; pn 1q,

where pn 1 is the image of pn 1 under the projection map ΣB,n 1 Ñ ΣB,n.

Since Σn 1,n F rns ¢F rn¡1s F rns, it is enough to show that:

¥ p q Ñ p q pr1 πn 1,n 1 : p1, . . . , pn 1 p1, . . . , pn¡1, pn , ¥ p q Ñ p q pr2 πn 1,n 1 : p1, . . . , pn 1 p1, . . . , pn¡1, pn 1 .

¥ Ñ r s We already know that pr1 πn 1,n 1 Πn 1 :Σn 1,n 1 F n corresponds to the forgetful functor Frn 1s Ñ Frns, as a result of Theorem 2.3.

To show the second identity, let fn 1 : B Ñ F rn 1s be a B-point of F rn 1s,

fn 1 Πn 1 and let fn : B Ñ F rns be the ‘truncated‘ point B ÝÝÝÑ F rn 1s ÝÝÝÑ F rns. The maps fn 1 and fn produce the following diagram, where pn 1 is the pullback of fn 1, and p1, . . . , pn are the pullbacks of σn 1,1, . . . , σn 1,n, respectively:

31 fn 1 B pn 1

r s pn ΣB,n 1 Σn 1,n 1 F n 1 x πn 1,n 1 pr p1 2 Σ Σ Σ F rns B,n x n 1,n n,n

id ......

σn 1,n

Σ Σ Σ B,1 x n 1,1 n,1 σn 1,1 f B n F rns F rn ¡ 1s.

Notice the following maps are equivalent:

fn 1 πn 1,n 1 pr2 rB ÝÝÝÑ Σn 1,n 1 ÝÝÝÝÝÑ Σn 1,n ÝÝÑ Σn,ns

pn 1 rB ÝÝÝÑ ΣB,n 1 Ñ Σn 1,n 1 Ñ Σn 1,n Ñ Σn,ns

pn 1 rB ÝÝÝÑ ΣB,n 1 Ñ Σn Ñ Σn 1,n Ñ Σn,ns

pn 1 rB ÝÝÝÑ ΣB,n Ñ Σn 1,n Ñ Σn,ns.

¥ p q By the equivalence above, it follows that pr2 πn 1,n 1 maps a B-point p1, . . . , pn 1 in Σn 1,n 1 to pp1,..., pn 1q in Σn,n, and this concludes the proof of our initial statement.

The behavior of the more general map πn 1,i becomes clear from the following cartesian square:

Σn 1,i F rns ¢F ri¡1s F ris Σi,i F ris

πn 1,i πi,i

Σn,i¡1 F rns ¢F ri¡2s F ri ¡ 1s Σi,i¡1 F ri ¡ 1s ¢F ri¡2s F ri ¡ 1s

32 We know that the top horizontal map Σn,i Ñ Σi,i is the projection onto the second factor; the morphism Σn,i¡1 Ñ Σi,i¡1 acts as the forgetful map F rns Ñ F ri ¡ 1s onto the ﬁrst factor, and as the identity on the second factor. We know the behavior

of πi,i from the previous paragraph, hence we conclude that πn 1,i maps B-points p 1 q ÞÑ p q 1 1 p1, . . . , pn; pi p1, . . . , pn; pi , where pi is the projection of pi under the map 1 Ñ ΣB,i ΣB,i¡1.

Notation 2.7. As a result of Proposition 2.6, we change notation and denote the

section σn 1,i as Mi,n 1. We will use this notation throughout the rest of the paper.

33 Chapter 3

Divisors of the moduli scheme

r s pnq @ ¤ Deﬁnition 3.1. The moduli scheme F n comes equipped with divisors Di,j , 1 i j ¤ n, which arise naturally from the construction outlined in the previous section.

As we have seen already, we can construct F rns Σn,n as a series of blowups of the r ¡ s ¢ pnq pnq base variety Σn,1 F n 1 S1. We start by deﬁning the divisors D1,n,...,Dn¡1,n.

To do so, recall that, for all 1 ¤ i ¤ n ¡ 1, Σn,i 1 is obtained by blowing up of the p r ¡ sq pnq previous variety Σn,i along the locus Mi,n F n 1 . We deﬁne Di,n on Σn,i 1 to be the exceptional divisor of this blowup:

pnq r ¡ s Di,n BlMi,n Σn,i Σn,i 1 Σn,i F n 1 .

Mi,n

pnq r s By abuse of notation, the divisor Di,n on F n Σn,n is the strict transform of this exceptional divisor coming from Σn,i 1 in the tower of blowups Σn,n Ñ Σn,n¡1 Ñ ¤ ¤ ¤ Ñ

Σn,i 1. We define the other divisors inductively as follows: assume we have defined pn¡1q r ¡ s ¤ ¤ ¡ pnq pn¡1q ¢ Di,j on F n 1 , where 1 i j n 1. Then Di,j Di,j S1 is a divisor r ¡ s ¢ pnq r s on Σn,1 F n 1 S1. By abuse of notation, Di,j F n Σn,n is defined as the strict transform of this divisor in the tower of blowups Σn,n Ñ Σn,n¡1 Ñ ¤ ¤ ¤ Ñ Σn,1.

34 ¥ r s Ñ n Proposition 3.2. For every n 1, the natural projection map F n S1 is a pnq composition of blowups. Under this map, every divisor Di,j is mapped surjectively onto the diagonal Mi,j.

Proof. In the previous section, we constructed a tower of blowups:

πn,n πn,n¡1 πn,2 F rns Σn,n ÝÝÑ Σn,n¡1 ÝÝÝÝÑ¤ ¤ ¤ Ñ Σn,2 ÝÝÑ Σn,1 F rn ¡ 1s ¢ S1.

r s r s Ñ n Inductively, using that F 0 Spec R, we obtain a morphism F n S1 that

decomposes as a series of blowups. Given the behavior of the maps πn,i, outlined in Proposition 2.6, this map coincides with the natural projection map.

r s Ñ n Next, we show inductively on n that this projection morphism F n S1 maps pnq r s p ¢ q surjectively any divisor Di,j onto the diagonal Mi,j. Recall that F 2 BlM1,2 S1 S1 p2q and D1,2 is the exceptional divisor of this blowup, so indeed the projection morphism r s Ñ ¢ p2q Ñ pnq F 2 S1 S1 maps D1,2 M1,2. In the induction step, consider a divisor Di,j of

the moduli space F rns. If j n, then, by construction, the morphism Σn,n Ñ Σn,1 r ¡ s ¢ pnq ÞÑ pn¡1q ¢ F n 1 S1 maps Di,j Di,j S1, so the conclusion follows. If j n, the pnq r s divisor Di,n on F n Σn,n is the strict transform of the exceptional divisor:

pnq r ¡ s Di,n BlMi,n Σn,i Σn,i 1 Σn,i F n 1 .

Mi,n

Given the behavior of the maps πn,i, outlined in Proposition 2.6, the projection r s Ñ Ñ n pnq morphism F n Σn,n Σn,i S1 maps Di,n Σn,n onto the blowup locus p r ¡ sq n Mi,n F n 1 Σn,i, which is then mapped onto the diagonal Mi,n S1 , and the proof is complete.

¤ ¤ ¤ pnq Proposition 3.3. (a) Let 1 i j n, 1 k n. The divisor Di,j Σn,k is the pn¡1q r ¡ s Ñ r ¡ s inverse image of Di,j F n 1 under the projection map Σn,k F n 1 .

35 ¤ ¤ pnq (b) Let 1 i k n. The divisor Di,n Σn,k is the inverse image of the pnq Ñ exceptional divisor Di,n Σn,i 1 under the projection map Σn,k Σn,i 1.

r ¡ s ¢ pnq Proof. (a) When k 1, Σn,1 F n 1 S1. Recall that the divisor Di,j Σn,1 is pn¡1q ¢ pn¡1q deﬁned to be Di,j S1, therefore it is indeed the inverse image of Di,j under the map Ñ r ¡ s πn,1 pr1 :Σn,1 F n 1 .

When k ¡ 1, recall we have the following ﬁgure:

πn,k 1 r ¡ s Σn,k 1 BlMk,n Σn,k Σn,k F n 1 .

Mk,n

pnq The divisor Di,j Σn,k 1 was deﬁned to be the strict transform of the divisor with the same name coming from Σn,k. We claim that the strict transform coincides with the total transform. By construction, the moduli space F rn¡1s is irreducible and the

pnq map Mk,n is a regular embedding, hence the exceptional divisor Dk,n of the blowup

Σn,k 1 Ñ Σn,k is also irreducible. In other words, the only way statement (a) could fail pnq is if the exceptional divisor Dk,n Σn,k 1 is completely contained in the inverse image ¡1 p pnqq p r ¡ sq πn,k 1 Di,j , which could happen only if the blowup locus Mk,n F n 1 Σn,k pnq Ñ r ¡ s was completely contained in Di,j . However, the projection Σn,k F n 1 maps pnq pn¡1q p r ¡ sq r ¡ s Di,j Di,j and Mk,n F n 1 F n 1 , so the induction step is complete and claim (a) is correct. (b) When k ¡ i 1, we have a similar picture as before:

πn,k 1 r ¡ s Σn,k 1 BlMk,n Σn,k Σn,k F n 1 .

Mk,n

36 pnq The divisor Di,n Σn,k 1 is deﬁned to be the strict transform of the divisor with

the same name coming from Σn,k. Using the same argument as in (a), it is enough to p r ¡ sq pnq prove that the blowup locus Mk,n F n 1 is not fully contained in Di,n . We prove this inductively for k i 1, . . . , n. p r ¡ sq Consider the base case k i 1. A B-point of Mi 1,n F n 1 Σn,i 1 p q pnq can be summarized as p1, . . . , pn¡1; pi 1 , and a B-point of Di,n Σn,i 1 can be p 1 q 1 P summarized as p1, . . . , pn¡1; pi 1 , where pi 1 Σi 1 maps to pi under the projection Ñ p r ¡ sq pnq map ΣB,i 1 ΣB,i. Given these descriptions, clearly Mi 1,n F n 1 Di,n . For p r ¡ sq the induction step, recall that a B-point of Mk,n F n 1 can be summarized as p q pnq p1, . . . , pn¡1; pk . By induction, we know that Di,n Σn,k is the inverse image of the Ñ pnq exceptional divisor of the blowup Σn,i 1 Σn,i, hence a B-point of Di,n Σn,k can p 1 q 1 P be summarized as p1, . . . , pn¡1; pk , where pk ΣB,k maps to pi under the projection Ñ p r ¡ sq map ΣB,k ΣB,i. Thus, any point in the blowup locus Mk,n F n 1 of the form pp1, . . . , pn¡1; pkq where pk does not map to pi under the projection map ΣB,k Ñ ΣB,i

pnq is not a point of Di,n , so the induction step is ﬁnished and the proof is complete.

¤ ¤ pnq r s Proposition 3.4. Let 1 i j n and Di,j be a divisor of F n as deﬁned above.

A B-point of F rns, denoted by a tuple pp1, p2, . . . , pnq consistent with Notation 2.5, pnq Ñ is a point of Di,j if and only if the projection ΣB,j ΣB,i maps the point pj to the

point pi.

pnq r s Proof. Let Di,j be a divisor of F n , as constructed above. If j n, then by Proposi- pnq pn¡1q r s Ñ r ¡ s tion 3.3, Di,j is the inverse image of Di,j under the forgetful map F n F n 1 . pn¡1q r ¡ s pnq r s If the claim holds true for Di,j F n 1 , then it also holds true for Di,j F n . In other words, we can assume without loss of generality that j n. pnq r s Let j n; by Proposition 3.3, the divisor Di,n F n Σn,n is the inverse image p r ¡ sq Ñ of the blowup locus Mi,n F n 1 Σn,i under the projection map Σn,n Σn,i:

37 pnq ¡1p p r ¡ sq pr r ¡ s Di,n pr Mi,n F n 1 Σn,n Σn,i F n 1 .

Mi,n

Recall that, by Proposition 2.6, the projection morphism Σn,n Ñ Σn,i maps points as follows:

pp1, . . . , pn¡1, pnq ÞÑ pp1, . . . , pn¡1; pnq,

where pn P ΣB,i is the projection of pn under the map ΣB,n Ñ ΣB,i. On the other r ¡ s Ñ hand, the section Mi,n: F n 1 Σn,i maps points as follows:

pp1, . . . , pn¡1q ÞÑ pp1, . . . , pn¡1; piq.

r s pnq Ñ In conclusion, a B-point F n is a point of Di,n if and only if pn pi under the

map ΣB,n Ñ ΣB,i.

¤ ¤ pnq r s Proposition 3.5. Let 1 i j n. The divisor Di,j F n is smooth over Spec R.

Proof. We will prove this statement inductively over n. When n 2, we have only

p2q one divisor of this form, namely D1,2, which is the exceptional divisor of the blowup

of S1 ¢ S1 along the diagonal, so it is clearly smooth. Inductively, we need to analyze two cases: either j n or j n. pnq Assume j n. We will show inductively over k that Di,j Σn,k is smooth. For pnq r ¡ s¢ the base case k 1, recall that the divisor Di,j Σn,1 F n 1 S1 is deﬁned to be pn¡1q¢ pn¡1q r ¡ s Di,j S1. By the induction hypothesis, Di,j F n 1 is smooth, so the base case pnq is true. Inductively, assume Di,j Σn,k is smooth. Recall that Σn,k 1 BlMk,n Σn,k,

38 pnq and Di,j Σn,k 1 is deﬁned to be strict transform of the divisor with the same name

coming from Σn,k. To show smoothness is preserved, it is enough to prove that the pnqX p r ¡ sq intersection of the divisor with the blowup locus Di,j Mk,n F n 1 inside Σn,k is itself smooth. We prove the following, which, by the induction hypothesis, concludes this step of the proof:

pnqX p r ¡ sq pn¡1q Di,j Mk,n F n 1 Di,j .

pnq As a result of Proposition 2.6 and Proposition 3.3, a B-point of Di,j Σn,k p 1 q Ñ can be summarized as p1, . . . , pn¡1; pk , where pj pi under the projection map Ñ p r ¡ sq ΣB,j ΣB,i. By Proposition 2.6, a B-point of Mk,n F n 1 Σn,k can be p q pnqX p r ¡ sq summarized as p1, . . . , pn¡1; pk . In conclusion, a B-point of Di,j Mk,n F n 1 can be summarized as pp1, . . . , pn¡1; pkq, where pj Ñ pi under the projection map Ñ pnqX ΣB,j ΣB,i. We see a clear correspondence between the B-points of Di,j Mk,n p r ¡ sq pn¡1q P p q F n 1 and those of Di,j , for any B Sch R . This means that the functors of points of the two schemes are isomorphic. By the Yoneda Lemma, the two schemes in question are isomorphic and the induction step is complete. pnq Now assume that j n. We will show inductively that Di,n Σn,k is smooth, for ¤ ¤ pnq all i 1 k n. For the base case k i 1, recall that the divisor Di,n Σn,i 1 p r ¡ sq is the exceptional divisor of the blowup of Σn,i along the locus Mi,n F n 1 . p r ¡ sq By Theorem 2.3, Σn,i is smooth and the blowup locus Mi,n F n 1 is regularly pnq embedded, hence the exceptional divisor Di,n Σn,i 1 is smooth. To prove the pnqX p r ¡ induction step is enough to show, as before, that the intersection Di,j Mk,n F n

1sq Σn,k with the blowup locus is itself smooth. We prove the following, which, by the induction hypothesis, concludes this step of the proof:

39 pnqX p r ¡ sq pn¡1q Di,n Mk,n F n 1 Di,k .

pnq As a result of Proposition 2.6 and Proposition 3.3, a B-point of Di,n Σn,k can p 1 q 1 Ñ Ñ be summarized as p1, . . . , pn¡1; pk , where pk pi under the projection map ΣB,k p r ¡ sq ΣB,i. By Proposition 2.6, a B-point of Mk,n F n 1 Σn,k can be summarized as p q pnqX p r ¡ sq p1, . . . , pn¡1; pk . In conclusion, a B-point of Di,n Mk,n F n 1 can be identiﬁed as pp1, . . . , pn¡1; pkq, where pk Ñ pi under the projection map ΣB,k Ñ ΣB,i. There pnqX p r ¡ sq is a clear correspondence between the B-points of Di,n Mk,n F n 1 and those pn¡1q P p q of Di,k , for any B Sch R , which means that the functors of points of the two schemes are isomorphic. By the Yoneda Lemma, the two schemes in question are isomorphic and the induction step is complete.

40 Chapter 4

The Chow ring of the moduli scheme

In this chapter we assume that the underlying ring R is an algebraically closed field k. In this setup, the moduli space F rns and all the schemes in its universal family are smooth projective varieties. We prove in the main result of this chapter that the Chow ring of the moduli r s t pnqu space F n is generated by the classes of the divisors Di,j 1¤i j¤n over the Chow ¦p nq ring A S1 . We conclude this chapter by proving certain key relations among these classes in the Chow ring A¦pF rnsq. In the next chapter, we prove these relations are sufficient to give a precise description of the Chow ring A¦pF rnsq in the special case where S1 is a rational surface and the base field is the complex numbers C. We start with a theorem by Keel, which is the key in our proof:

Theorem 4.1. Let Y be a variety and let i : X ãÑ Y be a regularly embedded subvariety. Let Y˜ be the blowup of Y along X. Suppose the map of bivariate rings i¦ : A¦pY q Ñ A¦pXq is surjective. Then:

41 A¦pY qrT s A¦pY˜ q , pP pT q,T ¤ kerpi¦qq

where P pT q P A¦pY qrT s is any polynomial whose constant term is rXs and whose

¦ restriction to A pXq is the Chern polynomial of the normal bundle N NX Y , i.e.:

¦ d d¡1 i P pT q T T c1pNq ¤ ¤ ¤ cd¡1pNqT cdpNq,

where d codimpX,Y q. This isomorphism is induced by

π¦ : A¦pY qrT s Ñ A¦pY˜ q

and by sending ¡T to the class of the exceptional divisor.

Proof. See [23], Appendix, Theorem 1.

Remark 4.2. The moduli space F rns is a smooth projective variety, for any n ¥ 1, hence its bivariate ring A¦pF rnsq is isomorphic to its Chow ring CH¦pF rnsq (see [16], Ch. 17).

Corollary 4.3. Let 1 ¤ i n. Let Σn,i 1 and Σn,i be two of the varieties in the

universal family over the moduli space F rn ¡ 1s. The Chow ring of Σn,i 1 has the following description:

¦p qr pnqs ¦ A Σn,i Di,n A pΣ q , n,i 1 x p¡ pnqq pnq ¤ p ¦ qy Pi,n Di,n ,Di,n ker Mi,n

¦ where Pi,n is a quadratic polynomial with coeﬃcients in A pΣn,iq.

Proof. In the universal family over the moduli space F rn ¡ 1s,Σn,i 1 is the blowup p r ¡ sq of Σn,i along the locus Mi,n F n 1 , and the corresponding exceptional divisor is

42 pnq Di,n : pnq r ¡ s Di,n BlMi,n Σn,k Σn,i 1 Σn,i F n 1 .

Mi,n r ¡ s ãÑ By Theorem 2.3, Σn,i is a variety and Mi,n: F n 1 Σn,i is a regularly em- Ñ r ¡ s bedded subvariety. Moreover, since Mi,n is a section of the map Σn,i F n 1 , the ¦ ¦p q Ñ ¦p r ¡ sq corresponding map on Chow rings Mi,n: A Σn,i A F n 1 is surjective. The conclusion follows immediately as an application of Theorem 4.1,

¦ ¦p q Ñ Remark 4.4. By Proposition 3.3, the induced map on Chow rings: πn,i 1 : A Σn,i ¦p q pnq A Σn,i 1 sends the class of any divisor Dj,k to the class of the divisor with the same ¦ name inside A pΣn,iq. This means that the notation remains consistent when we state the next results.

To generalize Corollary 4.5, we ﬁrst need to give an alternative way of deﬁning the

pnq r s r s Ñ n divisors Di,j F n . Recall that the natural projection map F n S1 decomposes as a series of blowups; more speciﬁcally, we encounter the following situation:

r s ¢ n¡j ¢ n¡j r s ¢ n¡j Ñ ¤ ¤ ¤ Ñ n F n Σn,n ... Σj,i 1 S1 Σj,i S1 ... F j S1 S1 .

¢ Mi,j id

¢p qn¡j ¢p qn¡j ¢ The scheme Σj,i 1 S1 is the blowup of Σj,i S1 along the locus Mi,j id. pnq ¢ p qn¡j We can deﬁne Di,j Σj,i 1 S1 to be the exceptional divisor of this blowup. pnq r s By abuse of notation, we deﬁne Di,j F n Σn,n to be the strict transform of this exceptional divisor in the tower of blowups. Following an identical argument as the pnq r s one in Proposition 3.3, more is true: Di,j F n is actually the full inverse image of this divisor in the tower of blowups. Therefore, by Theorem 4.1, we conclude that:

¦p ¢ n¡jqr pnqs ¦ n¡j A Σj,i S1 Di,j A pΣ ¢ S q , j,i 1 1 x p¡ pnqq pnq ¤ p ¦ qy Pi,j Di,j ,Di,j ker Mi,j

43 ¦ n¡j where Pi,j is a quadratic polynomial with coeﬃcients in A pΣj,i ¢ pS1q q. n r s Applying this procedure step by step from S1 all the way up to Σn,n F n , we immediately obtain the Chow ring of the moduli space F rns for any n ¥ 1:

Theorem 4.5. With notation as in Corollary 4.3, the Chow ring of the moduli space

F rS1, ns is as follows:

¦ n pnq A pS qrD s1¤i j¤n A¦pF rnsq 1 i,j . x pnq ¤ p ¦ q p¡ pnqq y Di,j ker Mi,j ,Pi,j Di,j 1¤i j¤n

¦ P ¦p r sq Proposition 4.6. Let d be any divisor class of S1 and di A F n the image of r s Ñ n ÝÑpri d under the composed morphism F n S1 S1. The following relations hold in the Chow ring A¦pF rnsq:

pnq ¦ pnq ¦ @ ¤ ¤ (i) Di,j di Di,j dj , 1 i j n;

pnq pnq pnq pnq @ ¤ ¤ (ii) Di,j Dj,k Di,k Dj,k , 1 i j k n.

Proof. (i) The ﬁrst identity follows immediately from the fact that the natural pro-

r s Ñ n pnq ¤ ¤ jection F n S1 maps Di,j surjectively onto the diagonal Mij, for all 1 i j n. (ii) The second identity requires some work. First, we show that it suﬃces to prove this relation in the case where k n. If i j k n, we can assume inductively that a similar relation holds in the Chow ring of F rS1, n ¡ 1s:

pn¡1q pn¡1q pn¡1q pn¡1q Di,j Dj,k Di,k Dj,k .

r ¡ s¢ pnq pnq pnq Now, recall that inside the variety Σn,1 F n 1 S1, the divisors Di,j ,Di,k ,Dj,k pn¡1q ¢ pn¡1q ¢ pn¡1q ¢ are deﬁned to be Di,j S1,Di,k S1,Dj,k S1, respectively. Hence, it is clear ¦ that relation (ii) holds in the Chow ring A pΣn,1q. Furthermore, by Remark 4.4, the

¦ ¦ projection morphism Σn,n Ñ Σn,1 induces a map on Chow rings A pΣn,1q Ñ A pΣn,nq

44 pnq pnq pnq that sends the classes of the divisors Di,j ,Di,k ,Dj,k to the classes of the divisors with the same name inside Σn,1, which means relation (ii) also holds in the Chow ring

¦ A pF rS1, nsq. pnq We are left to show that the relation (ii) holds true when k n. Recall that Dj,n p r ¡ sq is obtained as the exceptional divisor of Σn,j blown up along Mj,n F n 1 :

pnq r ¡ s Dj,n BlMj,n Σn,j Σn,j 1 Σn,j F n 1 .

Mj,n

By Remark 4.4, it suﬃces to show that relation (ii) holds in the Chow ring of

Σn,j 1. As a result of Theorem 4.1, the Chow ring of Σn,j 1 is as follows:

¦p qr pnqs ¦ A Σn,j Dj,n A pΣ q , n,j 1 x p¡ pnqq pnq ¤ p ¦ qy Pj,n Dj,n ,Dj,n ker Mj,n so we need to show that:

pnq ¡ pnq P p ¦ ¦p q Ñ ¦p r ¡ sqq Di,j Di,n ker Mj,n: A Σn,j A F n 1 .

To prove the relation above, recall that Σn,j F rn ¡ 1s ¢F rj¡1s F rjs. By Propo- pnq pn¡1q ¢ r s sition 3.3, under this isomorphism, Di,j Di,j F rj¡1s F j . Next, we claim that pnq r ¡ s ¢ pjq Di,n F n 1 F rj¡1s Di,j . To see that this isomorphism holds inside Σn,j, we ﬁrst need to check it on Σn,i 1. For this, consider the following diagram:

pnq pjq Di,n Σn,i 1 Σj,i 1 Di,j x

Σ Σ n,i x j,i Mi,n Mi,j F rn ¡ 1s F rj ¡ 1s.

45 The bottom square is cartesian and the section Mi,n is the pullback of Mi,j along r ¡ s Ñ r ¡ s pnq the map F n 1 F j 1 , therefore the exceptional divisor Di,n Σn,i 1 is the pjq pnq r ¡ s ¢ pjq pullback of the exceptional divisor Di,j Σj,i 1, so Di,n F n 1 F rj¡1s Di,j inside Σn,i 1. Clearly, this relation lifts to Σn,j. r ¡ s ¢ r s pnq pn¡1q ¢ In conclusion, inside Σn,j F n 1 F rj¡1s F j , we have Di,j Di,j F rj¡1s r s pnq r ¡ s ¢ pjq r ¡ s Ñ F j , Di,n F n 1 F rj¡1s Di,j . Additionally, we also know that Mj,n: F n 1

Σn,j F rn ¡ 1s ¢F rj¡1s F rjs acts like a ‘truncated‘ diagonal embedding. Putting all this information together, it becomes clear that:

pnq ¡ pnq P p ¦ ¦p q Ñ ¦p r ¡ sqq Di,j Di,n ker Mj,n: A Σn,j A F n 1 , and the proof is complete.

46 Chapter 5

The Chow ring of the moduli scheme for rational surfaces

As a special case of the theory developed above, we give a precise description of the

¦ Chow ring A pF rnsq when the base surface S1 is a smooth projective rational surface over the complex numbers (Spec R Spec C). The result relies on a few key ideas. First, the canonical map A¦pF rnsq ÝÑcl H2¦pF rnsq is an isomorphism. Second, for

l " r s b some prime p and any q p , where l 0, we can define the moduli space F n Fq r sb over the finite field Fq. The number of Fq-points on F n Fq is given by a polynomial

Rnpqq that coincides with the Poincare polynomial of F rns. We use this fact to derive precise formulas for the Betti numbers of F rns; using these formulas, we show that the relations in Proposition 4.6 are enough to give a complete description of A¦pF rnsq.

Deﬁnition 5.1. A scheme X of characteristic zero is called an HI (for Homology Isomorphism) scheme if the canonical map from the Chow groups of X to the homology groups:

cl A¦pXq ÝÑ H2¦pXq

47 is an isomorphism.

Proposition 5.2. Let Y be a variety and i : X ãÑ Y be a regularly embedded subvariety. Let Y˜ be the blowup of Y along X. If X and Y are HI schemes, then so is Y˜ .

Proof. See [23], Appendix, Theorem 2.

Proposition 5.3. Let S1 be a smooth projective variety over an algebraically closed n ¥ r s ﬁeld k of characteristic zero. If S1 is an HI scheme, for all n 0, then so is F n .

Proof. We prove inductively over n something stronger: @n, j ¥ 0, the variety F rns¢

j S1 is an HI scheme. r s r s ¢ j j The base cases are easy. If n 0, then F 0 Spec k, so F 0 S1 S1, which r s r s ¢ j j 1 is an HI scheme by hypothesis. If n 1, then F 1 S1, so F 1 S1 S1 , which is again an HI scheme by hypothesis. r s ¢ j ¤ For the induction step, assume F m S1 is an HI scheme, for all m n and ¥ r s ¢ j ¥ j 0. We want to show that F n 1 S1 is an HI scheme, for all j 0. Recall that we can obtain F rn 1s from F rns ¢ S1 as a series of blowups:

F rn 1s Σn 1,n 1 Σn 1,n ... Σn 1,2 Σn 1,1 F rns ¢ S1

M1,n 1

M2,n 1 F rns.

Mn,n 1

By the induction hypothesis, we know that F rns and F rns ¢ S1 are HI varieties. r s ãÑ By Theorem 2.3, the blowup locus Mi,n 1: F n Σn 1,i is a regular embedding, for all 1 ¤ i ¤ n. Applying Proposition 5.2 repeatedly, we conclude step by step that

Σn 1,2, Σn 1,3,..., Σn 1,n 1 F rn 1s are all HI varieties. More generally, the fact r s ¢ j ¥ that F n 1 S1 is an HI variety, for any j 0, follows immediately if we look instead at the following tower of morphisms:

48 ¢ j ¢ j ¢ j ¢ j r s ¢ ¢ j Σn 1,n 1 S1 Σn 1,n S1 ... Σn 1,2 S1 Σn 1,1 S1 F n S1 S1

r s ¢ j F n S1.

Proposition 5.4. Let k ¥ 1 and S1,...,Sk be complex smooth projective rational ± k surfaces. Then i1 Si is an HI variety.

Proof. We prove this statement inductively over k. As the base case, we show that a complex smooth projective rational surface S is an HI variety. By the Enriques- Kodaira classiﬁcation of complex surfaces (see Thm. 7.9), there exist smooth projective surfaces Sn¡1,...,S1,S0, and morphisms:

S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ S0,

such that each Si i Ñ Si is the contractions of a (¡1)-curve and S0 is a minimal 2 ¥ rational surface (either P or the Hirzebruch surface Fa, for a 0 or a 2).

2 Both P and Fa have algebraic cell decompositions, which means they are HI varieties. Knowing that S0 is an HI surface, we can apply Proposition 5.2 repeatedly, obtaining step by step that S1,S2,...,Sn S are HI varieties, since each of them is obtained by blowing up a smooth HI surface at a smooth point.

Inductively, assume the statement is true for all i k. Let S1,...,Sk be complex ± k smooth projective rational surfaces. We need to show that the product i1 Si is also an HI variety. As before, for each surface Si we have a sequence of morphisms:

49 Ñ Ñ ¤ ¤ ¤ Ñ Ñ Si Si,ni Si,ni¡1 Si,1 Si,0,

such that each Si,j i Ñ Si,j is the contractions of a (¡1)-curve, and Si,0 is a minimal

2 rational surface (either P or the Hirzebruch surface Fa). Putting these morphism together, we obtain the following sequence:

¹k k¹¡1 k¹¡2 ¹k Ñ ¤ ¤ ¤ Ñ ¢ Ñ ¤ ¤ ¤ Ñ ¢ ¢ Ñ ¤ ¤ ¤ Ñ Si,ni Si,ni Sk,0 Si,ni Sk¡1,0 Sk,0 Si,0. i1 i1 i1 i1 ± k Since each surface Si,0 admits a cell decomposition, so does i1 Si,0, which means that this product is an HI variety. The last morphism in the sequence:

¹k ¹k S1,1 ¢ Si,0 Ñ S1,0 ¢ Si,0 i2 i2 ± ± k t u ¢ k P is the blowup of i1 Si,0 along the locus pt i2 Si,0, where pt Si,0 is some smooth point. The base scheme is an HI variety, as noted earlier. By the induction hypothesis, the blowup locus is an HI variety. Thus, as a consequence of Proposi-

tion 5.2, the blown up variety S1,1 ¢ ¤ ¤ ¤ ¢ Sk,0 is also an HI scheme. It is easy to see that we can apply the same proposition in a similar fashion multiple times (from right to left) to conclude that every variety in the main sequence above is HI. The induction step is complete, and the claim is true.

Remark 5.5. In the proof above, we used that the Hirzebruch surface Fa has an

algebraic cell decomposition. One way to see this is to note that Fa is a toric variety, and all toric varieties admit a cell decomposition.

Corollary 5.6. Let S1 be a complex smooth projective rational surface and F rS1, ns

50 F rns its associated moduli variety. There exists a canonical isomorphism:

A¦pF rnsq ÝÑ H2¦pF rnsq.

Proof. This is an immediate result of Proposition 5.3 and Proposition 5.4.

Setup 5.7. Let S be a complex surface. There exists R C a ﬁnitely generated

Z-algebra such that S is deﬁned over R, i.e. there exists a surface SR over Spec R such that the following square is cartesian:

S SR x

Spec C Spec R.

Now, for any m R maximal ideal, the ﬁeld κpmq R{m is ﬁnite, so there exists

l " p q some prime p and q p , where l 0, such that κ m Fq. We obtain the following ﬁgure:

S SR Sκpmq SFq SFq x y y y

p q Spec C Spec R Spec κ m Spec Fq Spec Fq.

For the rest of this section, when we say a complex surface S can be defined as a surface SFq over a finite field Fq, it means we do a procedure as above.

Proposition 5.8. Let S be a complex smooth projective rational surface. There exists a prime integer p and q pl, for some l " 0, such that S can be deﬁned as a smooth

projective rational surface SFq over Spec Fq. For this choice of p and q, there exists a quadratic polynomial rptq with the property that, for any a ¥ 1 and q1 qa, the

1 1 p q number of Fq -points on SFq1 equals r q .

51 Proof. Let S be a complex smooth projective rational surface. By the Enriques- Kodaira classiﬁcation of surfaces, there exist complex smooth projective surfaces

Sn¡1,...,S1,S0, and a sequence of morphisms:

Ñ Ñ ¤ ¤ ¤ Ñ Ñ Ñ S Sn Sn¡1 S1 S0 Spec C,

such that each Si i Ñ Si is the contractions of a (¡1)-curve and S0 is a minimal 2 ¥ rational surface (either P or the Hirzebruch surface Fa, for a 0 or a 2). As in the Setup 5.7 above, we can find R C a finitely generated Z-algebra such that all the surfaces Si are defined over Spec R. Moreover, we can pick R in such a

2 2 way that, if S0 is P or Fa, then the surface S0,R is either PR or Fa,R, respectively:

p q Sn Sn,R Sn,κ m Sn,Fq Sn,Fq1 x y y y ......

S1 S1,R S1,κpmq S1, S1, 1 x y y Fq y Fq

S0 S0,R S0,κpmq S0, S0, 1 x y y Fq y Fq

p q 1 Spec C Spec R Spec κ m Spec Fq Spec Fq .

1 1 2 1 p q2 If S0,R is PR, then the number of Fq -points on S0,Fq1 is q q 1. If S0,R is Fa,R, 1 2 1 1 p q Ñ then the number of Fq -points on S0,Fq1 is q 2q 1. In every blowup Si 1 Si, 1 we replace one smooth point of Si with a copy of P , so the number of Fq1 -points on p 1q SFq1 is given by a polynomial r q that satisﬁes: $ ' &p 1q2 p q 1 2 q n 1 q 1, if S0 P , rpq1q ' %p 1q2 p q 1 q n 2 q 1, if S0 Fa.

52 Proposition 5.9. Let S1 be a complex smooth projective rational surface. Let p be a prime integer and q pl, as in Proposition 5.8. Let rptq be the quadratic polynomial

1 a corresponding to S1 from Proposition 5.8. For any a ¥ 1 and q q , the number of

1 r s r s p q Fq -points on the moduli space F n Fq1 F S1,Fq1 , n is given by a polynomial Rn q which has the following formula:

n¹¡1 1 1 1 Rnpq q prpq q iq q. i0

Proof. We prove the statement inductively, using the fact that the number of Fq1 - p 1q 1 points of S1,Fq1 blown up at n points equals r q nq . 1 r s p q 1 When n 1, F 1 Fq1 S1,Fq1 has exactly r q points over Fq . Assume the statement of the theorem is true for all k ¤ n. We want to show that

1 1 1 1 Rn 1pq q Rnpq qprpq q nq q.

r s Ñ r s Recall that we have a forgetful map F n 1 Fq1 F n Fq1 . Using Notation 2.5, r s p q every point in F n Fq1 is of the form x p1, . . . , pn , and its ﬁber under the forgetful

map is isomorphic to Sn 1,Fq1 . The surface Sn 1,Fq1 is obtained by blowing up S1,Fq1 at n consecutive points, hence it has rpq1q nq1 rational points. Since every ﬁber of

1 1 1 p q the forgetful map has the same number of Fq -points, namely r q nq , the equation above is true and the inductive step is complete.

Deﬁnition 5.10. Let X be a smooth, irreducible complex algebraic variety. The Poincar´epolynomial of X is:

2 dim¸X i PX pqq biq , i1

53 th ip q where bi is the rank of the i singular homology group H X, Z .

Lemma 5.11. Let S1 be a complex smooth projective rational surface. Let p be a prime integer and q pl, as in Proposition 5.8. The Poincar´epolynomial of

F rns F rS1, ns, denoted by Pnpqq, coincides with the polynomial Rnpqq which gives r s the number of Fq-points on the moduli space F n Fq .

r s Proof. Let S1 be a smooth projective rational surface over Spec C. Let X F n r s F S1, n be the moduli space corresponding to S1 over Spec C. We can regard S1 as

a rational surface S1,Fq over Fq, as in Proposition 5.8 above. Let XFq be the moduli

space associated to S1,Fq . Since X is smooth and projective, the Betti numbers corresponding to the l-adic cohomology (where l 0 mod p) are independent of l, and they coincide with the Betti numbers corresponding to the ordinary (integral) cohomology of the topological space X (see [30]):

b rk HipX, q rk HipX, q rk Hi pX , q. i Z Q et´ Fp Ql

One the other hand, we have the Grothendieck-Lefschetz Trace Formula (see [30], Thm. 13.4, p. 292), which states the following:

¸2n #Xp q p¡1qitrpFrob |HipX , qq. Fq q c Fq Ql i0

Since X is proper, HipX , q HipX , q. As a consequence of the Weil c Fq Ql Fq Ql conjectures (see [8]), we have:

p | ip b qq ¤ ¤ ¤ tr Frobq H X Fq, Ql zi,1 zi,bi ,

where zi,1, . . . , zi,bi are the eigenvalues of the Frobenius map. These eigenvalues satisfy

54 { 1 | | i 2 1 a zi,j q , for all j. Now, if we replace the ﬁeld Fq by Fq , where q q , then:

i a a trpFrob a |H pX , qq z ¤ ¤ ¤ z . q Fqa Ql i,1 i,bi

° ° ¥ p aq 2n p¡ qi bj a | | Thus, we have that for all a 1, Rn q i1 1 j0 zi,j, where zi,j qi{2, @i, j. It follows immediately that H2i 1pX , q 0, for all i and l 0 mod p, Fqa Ql i and z2i,j q , for all i, j. With this, we conclude our statement:

¸n i Rnpqq Pnpqq b2iq . i1

p q 2 ¦p q Let r q q kq 1 be the Poincar´epolynomial of S1. This means that A S1, Z

is generated in degree 1 by k classes d1, . . . , dk, and by one class in degree 2.

Theorem 5.12. Let S1 be a complex smooth projective rational surface. Let Π: r s Ñ n n Ñ F n S1 be the natural projection map and pri : S1 S1 the projection onto the @ ¤ ¤ ¦ ¥ ¦ ¦p q Ñ p r sq i-th copy, 1 i n. Let pri Π : A S1 A F n be the induced map on Chow

rings and di,1, . . . , di,k be the images of the classes d1, . . . , dk, respectively. The Chow ring of the moduli space A¦pF rnsq is:

¦ bn pnq pA pS1qq rD s1¤i j¤n A¦pF rnsq i,j . x pnqp pnq ¡ pnqq pnqp ¡ q p¡ pnqqy Dj,k Di,j Di,k ,Dj,k di,j di,k ,Pi,j Di,j

p ¦p qqbnr pnqs A S1 Di,j 1¤i j¤n Proof. Let R p q p q p q p q p q . We claim the following compo- x n p n ¡ n q n p ¡ q p¡ n qy Dj,k Di,j Di,k ,Dj,k di,j di,k ,Pi,j Di,j sition of morphisms is an isomorphism, after tensoring by Q:

¦ ¦ ¦ R A pF rns, Zq ÝÑ H pF rns, Zq H pF rns, Qq.

55 ¦p r sq ¦p nq By Theorem 4.5, we know that A F n is generated over A S1 by the classes of t pnqu the divisors Di,j 1¤i j¤n. Moreover, by Proposition 4.6, we know that the following relations hold in the Chow ring A¦pF rnsq:

pnqp ¡ q Dj,k di,j di,k 0 pnqp pnq ¡ pnqq Dj,k Di,j Di,k 0 (5.1) p¡ pnqq Pi,j Di,j 0.

We show the relations above are suﬃcient by looking at the Betti numbers of the moduli space F rns. By deﬁnition, the j-th Betti number of F rns gives us the

¦ number of codimension j linearly independent generators of A pF rnsq as a Z-module. Recall the formula from Proposition 5.9 describing the relation between the Poincar´e polynomial of F rn 1s and that of F rns:

2 Pn 1pqq pq pn kqq 1qPnpqq.

Let bn,j be the j-th Betti number of the moduli space F rns. The relation above translates to an equality between the Betti numbers as follows:

bn 1,j bn,j pn kqbn,j¡1 bn,j¡2.

We give the following interpretation to the relation above: recall we have the forgetful map Πn 1 : F rn 1s Ñ F rns, which induces a map on the Chow rings ¦ ¦p r sq Ñ ¦p r sq Πn 1 : A F n A F n 1 . Given this map, we can think of the identity above as follows: compared to the moduli space F rns, the space F rn 1s has n k

pn 1q pn 1q jp r sq extra divisors: dn 1,1, . . . , dn 1,k,D1,n 1,...Dn,n 1. A generator in A F n 1 is

56 jp r sq ¦ either a class inherited from A F n under the map Πn 1 (this accounts for bn,j generators), or it is a product between a generator class coming from Aj¡1pF rnsq and

one of the n k new divisor classes (this accounts for pn kqbn,j¡1 generators), or it is a product between a generator class coming from Aj¡2pF rnsq and the one generator

2p q n 1 Ñ class coming from A S1 under the projection map prn 1 : S1 S1 (this accounts for bn,j¡2 generators). It is easy to see that these are the only generators, since the

pnq divisors Di,j satisfy the identities in 5.1.

Corollary 5.13. When S 2 , the Chow ring of the moduli space A¦pF r 2, nsq is: 1 PC P

¦ pnq rH ,D s ¤ ¤ ¤ ¤ ¦p r 2 sq Z i j,k 1 i n,1 j k n A F P , n p q p q p q p q p q , x n p n ¡ n q n p ¦ ¡ ¦q ¦3 p¡ n qy Dj,k Di,j Di,k ,Dj,k Hj Hk ,Hi ,Pi,j Di,j

@ ¤ ¤ ¦ P ¦p 2q where, 1 i n, Hi is the image of the hyperplane class H A P under the

pri composition F rP2, ns Ñ pP2qn ÝÑ P2.

57 Chapter 6

A Question about Very General Curves

We now switch gears entirely and go back to the original problem suggested by Daniel Litt. We refer to the Introduction for more information. Recall that we want to characterize smooth surfaces S on which a very general curve C of genus g embeds as an ample divisor.

Notation. Throughout the rest of the thesis, a curve (resp. surface) is a complex projective variety of dimension 1 (resp. 2), reduced and irreducible.

An early result related to this question is the following Proposition from the paper “On the Kodaira Dimension of the Moduli Space of Curves” by Harris and Mumford (see [20]):

Proposition 6.1. Assume for some g that the Kodaira dimension of Mg is at least 0.

Then if C is a very general curve of genus g (i.e. the corresponding point rCs P Mg

lies in no subvariety deﬁned over Q), and S is an algebraic surface containing C on which C moves in a non-trivial linear system, then S is birational to C ¢ P1.

58 Proof. See [20].

The study of the Kodaira dimension of the moduli space of genus g curves Mg has a long history. We recall the main results here, noting that not much is known when the genus satisﬁes 17 ¤ g ¤ 21:

for g ¤ 16, Mg is uniruled, therefore κpMgq ¡8 (see [4], [5], [6], [15], [33], [36], [35], [38]);

for g ¥ 22, Mg is of general type, so the Kodaira dimension satisﬁes κpMgq 3g ¡ 3 (see [11], [19], [20]).

Corollary. Let S be a surface. Assume S contains a very general curve C of genus g ¥ 22 such that dim |C| ¥ 1. Then the surface S is birational to C ¢ P1.

We thought to analyze a similar question in a diﬀerent setting:

Question. Let C be a very general smooth curve of genus g which embeds on a smooth surface S as an ample divisor such that dim |C| 0. What can we say about the surface S? Is S always birational to C ¢ P1?

On this question, we will show the following:

Theorem 6.2. Let C be a very general smooth curve of genus g which embeds on a smooth surface S as an ample divisor such that dim |C| 0. The following statements hold:

(i) If the Kodaira dimension of S satisﬁes κpSq ¡8 and S is not rational, then

S is birational to C ¢ P1;

(ii) If the Kodaira dimension of S is 0 or 1, then such an embedding does not exist if the genus of C is g ¥ 21;

59 (iii) If S is of general type and its regularity is qpSq g, then S is birational to the symmetric square Sym2pCq.

There are a few cases left open. If S is a rational surface, we analyze of the conditions that need to be satisﬁed (see Prop. 9.5 and 9.10). For the case in which

S is of general type and qpSq 0, we prove the following partial result: if S ãÑ Pr is a complete intersection and the composed morphism C ãÑ S ãÑ Pr satisﬁes the Maximal Rank Conjecture, then C is not ample on S if its genus is higher than 15.

60 Chapter 7

Curves and Surfaces

In this chapter we recall some well-known statements related to the study of smooth curves and surfaces. This section has three parts. First, we recall some basic results about curves on surfaces. Second, we outline the Enriques-Kodaira classiﬁcation of surfaces. Lastly, we sketch the main results of Brill-Noether theory, ﬁnishing with the statement of the Maximal Rank Conjecture. Here are a few basic facts about curves and surfaces:

Proposition 7.1 (Projection formula). Let f : pX, OX q Ñ pY, OY q be a morphism

of ringed spaces. Let F be an OX -module and E be a locally free OY -module of ﬁnite p b ¦ q b rank. Then there exists a natural isomorphism f¦ F OX f E f¦F OY E.

Proof. See [21], Exercise II.5.1(d).

Theorem 7.2 (Torelli). Let C,C1 be complete smooth curves over an algebraically

1 closed ﬁeld k. Assume there exists an isomorphism pJpCq, λC q pJpC q, λC1 q between the canonically polarized Jacobian varieties of C and C1. Then C and C1 are isomorphic.

Proof. See, for example, [29].

61 Proposition 7.3 (Genus formula). Let S be a smooth surface and KS its canonical

divisor. Let C be a smooth genus g curve on S. Then CpKS Cq 2g ¡ 2.

Proof. See [21], Proposition V.1.5.

Proposition 7.4 (Nakai-Moishezon). Let S be a smooth surface and D a divisor on S. Then D is ample if and only if D2 ¡ 0 and D ¤ B ¡ 0, for all irreducible curves B on S.

Proof. See [21], Proposition V.1.10.

We now turn our focus to the Enriques-Kodaira classiﬁcation of surfaces:

Deﬁnition 7.5. Let S be a surface. A curve E on S is called a (-1)-curve if it is smooth, rational, and E2 ¡1. The surface S is called minimal if it doesn’t contain any (-1)-curves.

Theorem 7.6 (Castelnuovo Contractibility Criterion). Let S be a smooth surface. Let E be a (-1)-curve on S. There exists a smooth surface S1 and a morphism π : S Ñ S1 such that π contracts E to a point p and pS1, πq is isomorphic to the blowup of S at p.

Proof. See [2], Theorem II.17.

Proposition 7.7. Let S be a smooth surface. Let E be a (-1)-curve on S and π : Ñ ¦ S S the contraction of E. Then KS π KS E, where KS and KS are the canonical divisors of S and S, respectively, and π¦ : PicpSq Ñ PicpSq is the pullback map on the Picard groups.

Proof. See [21], Proposition V.3.3.

62 Every time we contract a (-1)-curve E on the surface S, the rank of the Neron- Severi group NSpSq decreases by 1. This means that we can obtain a minimal surface starting from S in a ﬁnite number of steps. Moreover, the following stronger results holds:

Theorem 7.8. Let S be a surface. Then S is birational to a minimal surface. More- over, if S is non-ruled, this minimal surface is unique.

Proof. See [2], Theorem V.19.

Before we invoke the well-known Enriques-Kodaira classiﬁcation of smooth minimal surfaces, we recall the deﬁnition of the Kodaira dimension of a smooth surface S: ¡ ¡ 0 © © h pS, nKSq κpSq min c P N| is bounded from above . nc n¥1

0 By convention, when the plurigenera h pS, nKSq vanish for all n " 0, one sets κpSq ¡8. We are now ready to state the Enriques-Kodaira classiﬁcation of complex smooth minimal surfaces, focusing on speciﬁc properties of these surfaces which we will use later. For more details, see [1]:

Theorem 7.9 (Enriques-Kodaira Classiﬁcation). Let S be a smooth minimal surface. The following holds:

(a) if κpSq ¡8, then S is one of the following:

(i) S P2;

(ii) S is a ruled surface onto a smooth curve B, such that all ﬁbers are iso-

1 morphic to P . If the genus of B is 0, then S is a Hirzebruch surface Fa, with a 0 or a ¥ 2;

63 (b) if κpSq 0, then S is one of the following:

(i) Abelian surface: these satisfy qpSq 2, pgpSq 1, KS 0;

(ii) K3 surface: these satisfy qpSq 0, pgpSq 1, KS 0;

(iii) Enriques surface: these satisfy qpSq pgpSq 0, KS 0, 2KS 0;

(iv) Bielliptic suface: these satisfy qpSq 1, pgpSq 1, nKS 0 for n P t2, 3, 4, 6u;

(d) if κpSq 2, then S is called a surface of general type.

In this last part of this chapter, we state the main statements of Brill-Noether theory. For more informations, see [18]:

Deﬁnition 7.10. Let C be a smooth curve. A map f : C Ñ Pr is called nondegenerate if the image of C does not lie in a hyperplane.

Theorem 7.11. Let C be a general smooth genus g curve. There exist a nondegenerate map f : C Ñ Pr of degree d if and only if the Brill-Noether number ρpg, d, rq is nonnegative: ρpg, d, rq pr 1qd ¡ rg ¡ rpr 1q ¥ 0.

Proof. See [18].

2p q Theorem 7.12. Let C be a general smooth genus g curve. Let Gd C be the (smooth) projective variety parametrizing all linear systems of degree d and dimension 2 on C.

2p q Ñ 2 A general point of Gd C corresponds to a morphism f : C P which maps C birationally onto a plane curve with only nodal singularities.

Proof. See [18].

64 As seen in [26], for ﬁxed d, g, r satisfying ρpd, g, rq ¥ 0, there exists a unique

p r q component of the Kontsevich space of stable maps M g P , d that dominates the

moduli space of curves M g, whose general member is non-degenerate and whose p q p rq relative dimension over M g is ρ d, g, r dim Aut P . A stable map corresponding to points in this component is called a Brill-Noether curve.

Theorem 7.13 (Maximal Rank Conjecture). If r ¥ 3 and C Pr is a general Brill- Noether curve of degree d, then the dimension of the space of polynomials of degree k which vanish on C is given by: $ ' ¨ ¨ &' r k ¡ p ¡ q ¡ ¤ r k ¥ k kd 1 g , if kd 1 g k and k 2; ' %0 , otherwise.

Proof. See [26].

65 Chapter 8

Preliminary results

In this chapter we prove some preliminary facts that will be very useful in the proof of Theorem 6.2. If C is a very general curve of genus g, then Pic0pCq is a simple abelian variety. Additionally, if C embeds on a smooth surface S as an ample divisor, then either Pic0pSq 0 or Pic0pSq Pic0pCq. We ﬁnish the chapter by

giving a lower bound on the intersection number C ¤ KS, where KS is the canonical divisor of S. Lastly, we show that when Pic0pSq 0 and dim |C| 0, S must have at least 3g ¡ 3 moduli. To start, we focus our attention on the Picard group of the curve C and of the surface S, more speciﬁcally on the component of the identity of the Picard group Pic0p¡q. An abelian variety X is called simple if 0 and X are the only abelian subvarieties of X. We use the following theorems of Koizumi and Mumford to derive a very useful corollary:

Theorem 8.1. Let C be a very general smooth curve. Then Pic0pCq is a simple abelian variety.

Proof. The proof of this statement can be found in [25].

66 Theorem 8.2. Let S be a smooth surface. Let ι : C ãÑ S be an ample divisor on S. Then the pullback morphism ι¦ : Pic0pSq ãÑ Pic0pCq is injective.

Proof. See [32, p. 99].

Proposition 8.3. Let S be a smooth surface. Let C ãÑ S be a very general smooth curve that embeds as an ample divisor on S. Then one of the following conditions must hold:

(i) Pic0pSq Pic0pCq;

(ii) Pic0pSq 0.

Proof. By Theorem 8.2, the inclusion ι : C ãÑ S induces an injective map ι¦ : Pic0pSq ãÑ Pic0pCq, so Impι¦q is an abelian subvariety of Pic0pCq. By Theorem 8.1, Pic0pCq is simple, so Impι¦q is either Pic0pCq or 0. This means that ι¦ is either an isomorphism, which corresponds to case (i), or Pic0pSq 0, which corresponds to case (ii).

Out strategy in later chapters will be to prove Theorem 6.2 in several steps, analyzing every possible minimal model of S. Before we get to that part, we ﬁrst prove some lemmas that will be very helpful for some of the cases:

Lemma 8.4. Let S be a smooth surface, C be a smooth ample curve on S, and E be a (-1)-curve on S. Let π : S Ñ S be the contraction of E and C ImpCq. Then C is an ample divisor. Moreover, if C ¤ E 1, then C is also smooth.

Proof. We start by showing that C is ample using the Nakai-Moishezon Criterion for Ampleness. Let π¦ : PicpSq Ñ PicpSq be the pullback map on the Picard groups. If C ¤ E r, then π¦C C rE, and:

2 C pπ¦Cq2 pC rEq2 C2 2rC ¤ E r2E2 C2 r2 ¡ 0. (8.1)

67 Now, let D S to be any irreducible curve on S. Say π¦D D nE, for some integer n ¥ 0, where D is the strict transform of D on the surface S. We obtain:

C ¤ E π¦C ¤ π¦D pC rEqpD nEq C ¤ D rn ¡ 0. (8.2)

As a consequence of inequalities 8.1 and 8.2, we conclude that C S is an ample divisor. For the second part, assume C ¤ E 1. Since C and E don’t have any common irreducible components, then

¸ C ¤ E pC.EqP , P PCXE

where we deﬁne the intersection multiplicity pC.EqP of C and E at P to be the length of OP,X {pf, gq, where f, g are local equations for C,E at P . Since C ¤ E 1, there exists a unique point P P C X E, with intersection multiplicity 1. This means that contracting E does not aﬀect the smoothness of C, i.e. C is still smooth.

By Lemma 8.4, we can contract all (-1)-curves E on S that satisfy C ¤ E 1 and still preserve the smoothness of C. Therefore, we can assume without loss of generality that every (-1)-curve E on S satisﬁes C ¤ E ¥ 2. We are now ready to state

and prove the lower bound for the intersection number C ¤ KS. Here is our setup:

Setup 8.5. Let S be a smooth surface and S0 a smooth minimal model of S. There

exists an integer n ¥ 0 and a sequence of morphisms S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ

S0 with the property that each map Si Ñ Si¡1 is the contraction of a (-1)-curve of Si. Let C be a smooth ample divisor on S such that C ¤ E ¥ 2, for all (-1) curves E S.

Let C0 be the image of C on S0.

68 Lemma 8.6. Let S,S0, n, C, C0 be as in the Setup 8.5 above. Let KS,KS0 be the canonical divisors of S and S0, respectively. The following inequality holds:

¤ ¥ ¤ C KS C0 KS0 2n. (8.3)

Proof. We prove this statement inductively. Let E S be a (-1)-curve on S and π : S Ñ S its contraction. Let C be the image of C under π. By Lemma 8.4, we know that C is an ample divisor. Using Proposition 7.7, we obtain:

¤ ¤p ¦ q ¤ ¦ ¤ ¤ ¤ ¤ ¤ ¥ ¤ C KS C π KS E C π KS C E π¦C KS C E C KS C E C KS 2.

To conclude the statement, we need one more fact. Let E1 be another (-1)-curve on S, and let E1 be its image in S. We claim that if C ¤ E1 ¥ 2, then C ¤ E1 ¥ 2. Assume π¦E1 E1 tE, for some t ¥ 0. We obtain the following inequality, which completes the proof:

C ¤ E1 π¦C ¤ π¦E1 pC rEqpE1 tEq

C ¤ E1 rE ¤ E1 tC ¤ E rtE2

C ¤ E1 2rt ¡ rt ¥ C ¤ E1 ¥ 2.

To ﬁnish the chapter, we give a lower bound for the number of moduli of S when Pic0pSq 0:

Lemma 8.7. Let S be a smooth surface with irregularity qpSq 0, and C a very general curve of genus g on S satisfying dim |C| 0. Then the number of moduli of S is at least 3g ¡ 3.

69 Proof. We can find a smooth projective family C S over a base scheme B of finite type over Q such that C S is a C-valued point of B. Moreover, for every t P B, the 0 fiber Ct St satisfies Pic pStq dim |Ct| 0. This family gives the following figure:

rCSs Spec C B MS,

rCs Spec C Mg

where Mg is the coarse moduli space of genus g curves and MS is a “moduli space of surfaces”. We will make this more precise in the remark below.

Now, since C S is a C-valued point of B and C is a very general curve, then Ñ Ñ the image of the map Spec C Mg is the generic point, which means B Mg is

dominant. Additionally, we observe that the map B Ñ MS must have 0-dimensional ﬁbers. To see this, assume by contradiction that there exists a 1-dimensional ﬁber. Thus there exists a surface S1 that admits a 1-dimensional family of curves on it. Since Pic0pS1q 0, then these curves must move in the same linear system, but that

contradicts dim |Ct| 0, for any Ct in the family, so we are done.

In conclusion, the moduli space MS has dimension at least 3g ¡ 3, as claimed.

Remark 8.8. We want to make the deﬁnition of the “moduli of surfaces” MS above more precise. In our applications below, the surface S is obtained by blowing up a

smooth minimal surface S0 at n consecutive points, and the minimal surface is always

a K3 surface, an Enriques surface, or an elliptic ﬁbration over P1. As we will see in the next chapters, all these minimal surfaces are parametrized by coarse moduli spaces. Our claim is that S admits a moduli space, as well. To see this, recall that in the previous chapters, we constructed the moduli space

F rns parametrizing n-fold blowups of a smooth surface S0 over an aﬃne base Spec R (see Thm. 2.3). We can generalize this construction such that the aﬃne base is

70 replaced by the moduli space of K3 surfaces, or the moduli space of Enriques surfaces, and so on. At least for the purpose of our dimension count, this conclusion follows easily.

71 Chapter 9

Surfaces of Kodaira dimension ¡8

In this chapter we analyze the scenario in which a very general curve C of genus g is embedded as an ample divisor on a smooth surface of Kodaira dimension ¡8 such that dim |C| 0. We show that either S is birational to C ¢P1, or its minimal model 2 is P or the Hirzebruch surface Fa, where a 1. The latter case remains mainly open. However, we analyze the conditions that need to be satisfied in this situation. While these conditions are not sufficient to conclude that such an embedding C ãÑ S exists, we show that there exist cases where this conditions could potentially be satisfied.

To start, let S be a smooth surface satisfying κpSq ¡8. Let S0 be a smooth minimal model of S. By the Enriques-Kodaira classiﬁcation of minimal surfaces, we

2 know that S0 is either P or a ruled surface, i.e. it admits a surjective morphism onto a smooth curve B such that every ﬁber is isomorphic to P1. We show ﬁrst that, if S is not rational, then it is birational to C ¢ P1.

Lemma 9.1. Let π : S0 Ñ B be a ruled minimal surface over a smooth curve B.

¦ 0 0 ¦ Let π : Pic pBq Ñ Pic pS0q be the pullback map on the Picard groups. Then π is injective.

Proof. First, notice that OB π¦OS0 . This is true because every ﬁber is connected,

72 P 0p q ¦ so the result follows by base change. Let L Pic B such that π L OS0 . We apply the projection formula and conclude that π¦ is injective:

¦ L O b L π¦O b L π¦pO b π Lq π¦O O . B OB S0 OB S0 OS0 S0 B

Lemma 9.2. Let S be a smooth surface and C ãÑ S a very general smooth curve embedded on S as an ample divisor. If S has Kodaira dimension κpSq ¡8 and it

is not rational, then it is birational to C ¢ P1.

Proof. From the Enriques-Kodaira classiﬁcation of surfaces (see Thm. 7.9), we know

Ñ 1 that the minimal model of S is a ruled surface S0 B, where B P . As a consequence of Theorem 8.2 and Lemma 9.1, we have the following sequence of mor-

0 0 0 0 phisms: Pic pBq ãÑ Pic pS0q ÝÑ Pic pSq ãÑ Pic pCq. On the other hand, Lemma 8.1 gives us that Pic0pCq is simple, therefore either Pic0pBq 0 or the composed map

0 0 Pic pBq Ñ Pic pCq is an isomorphism. Since B P1, then we are in the second case. By the Torelli theorem (see Thm. 7.2), we conclude that the morphism C Ñ B is an isomorphisms, hence S is birational to C ¢ P1.

We now begin our analysis of the case in which S is a rational surface, i.e. its

2 minimal model S0 is either P or the Hirzebruch surface Fe, where e 1. Here is a partial result to our question, which follows from our research below:

Lemma 9.3. Let S be a smooth surface and C ãÑ S a very general smooth curve embedded on S as an ample divisor. If S is rational and the genus of C satisﬁes g ¥ 22, then the image of C on the smooth minimal model of S is not a nodal curve.

Proof. This is a consequence of Lemmas 9.11 and 9.6 below.

73 2 Assume we are in the case in which S0 P . To start, notice that a very general smooth curve C of genus g can be embedded on a smooth surface whose minimal

model is P2. To see this, recall that any curve can be mapped birationally to P2 such that its image has only nodal singularities (see Thm. 7.12). Once we blow up these nodal singularities, the curve C will be embedded on the resulting blown up surface. The question we try to tackle next is whether C can ever be an ample divisor on S. This question seems diﬃcult to answer in the most general of circumstances. In this paper, we decided to focus on a special case, as follows:

Setup 9.4. Assume that C ãÑ S is a very general curve embedded on S as an ample divisor such that dim |C| 0, where S is obtained by blowing up P2 at n distinct points

p1, . . . , pn. Let E1,...,En be the corresponding exceptional divisors. The Picard group

`n 1 of S is isomorphic to Z , and it is generated by the classes H,E1,...,En, which ° 2 2 ¡ ¤ ¤ ¡ n satisfy H 1, Ei 1, H Ei Ei Ej 0. Let C dH i1 miEi be the divisor class of C.

Proposition 9.5. Let S and C be as in the Setup 9.4 above. Then the following conditions need to be satisﬁed:

(a) d ¡ 0 and mi ¥ 2, for all i 1, . . . , n; ° p ¡ q ¡ n p ¡ q ¡ (b) d d 3 i1 mi mi 1 2g 2; ° 2 2 ¡ n 2 ¡ (c) C d i1 mi 0; ° ¡ n ¡ ¡ (d) 3d i1 mi 2 2g; ° ¡ n ¤ ¡ (e) 3d i1 mi 1 g;

¥ 2g 6 (f) d 3 ;

74 ¡ ¥ g 2 (g) d mi 2 ;

(h) 2n ¥ 3g 5; ° p q2 p ¡ q2 ¡ n p ¡ q2 ¥ ¡ (i) C KS d 3 i1 mi 1 g 2;

t un ¥ g (j) max mi i1 9 .

Proof. (a) Since C is ample, then d ¡ 0. As a result of Lemma 8.4, we can assume

without loss of generality that C ¤ Ei ¥ 2, for all i 1, . . . , n. ° ¡ n (b) This is the genus formula combined with the fact that KS 3H i1 Ei.

2 (d) This is true because C ¤ KS 2g ¡ 2 ¡ C 2g ¡ 2.

(e) Consider the space of homogeneous polynomials of degree d in three variables

which vanish to order at least mi at the point pi, for all i 1, . . . , n. The naive ¨ ¨ ° d 2 ¡ n mi 1 dimension count for this space is 2 i1 2 . The actual dimension is always at least the expected one. Since we are assuming that dim |C| 0, this translates to the following:

¢ ¢ d 2 ¸n m 1 ¡ i ¤ 1. 2 2 i1

Combining the equation above with the genus formula, we obtain the desired inequality.

(f) This is an immediate application of Brill-Noether theory (see Theorem 7.11).

Ñ 2 (g) Fix a point pi among the n singular points of C P . When we project from Ñ 2 Ñ 1 the point pi onto a line, we obtain a composed map C P P of degree

75 d ¡ mi. The inequality now follows immediately from Brill-Noether theory (see Thm. 7.11).

(h) This follows from a moduli count. From Lemma 8.7, we know that S needs to

have at least 3g ¡ 3 moduli. On the other hand, since S is the blowup of P2 at n distinct points, the number of moduli of S is 2n ¡ dimpAutpP2qq 2n ¡ 8, and the conclusion follows.

(i) Consider the following short exact sequence:

0 Ñ OSpKSq Ñ OSpKS Cq Ñ OC pKSq Ñ 0,

which gives the following long exact sequence in cohomology:

0 0 0 0 Ñ H pS, OSpKSqq Ñ H pS, OSpKS Cqq Ñ H pC, OC pKSqq Ñ ...

0 0 0 Since h pS, OSpKSqq 0 and h pC, OC pKSqq g, then h pS, OSpKS Cqq g.

Now, pick two very general global sections in ΓpOC pKSqq that have no common

1 zeros and let σ, σ P ΓpOSpKS Cqq be very general lifts of these sections. Let D V pσq S. The following short exact sequence:

0 Ñ OS Ñ OSpKS Cq Ñ loooooooomoooooooonODpKS C|Dq Ñ 0

gives a long exact sequence on cohomology:

0 0 0 0 Ñ H pS, OSq Ñ H pS, OSpKS Cqq Ñ H pD, Lq Ñ ...

0 0 Since h pS,OSq 1 and h pS, Lq g, we obtain:

76 h0pD, Lq ¥ g ¡ 1. (9.1)

2 We note here that L is a line bundle on D of degree pKS Cq . Now, recall

1 0 2 that we picked another general section σ P H pS, OSpKS Cqq. Let σ be its image in H0pD, Lq and D1 V pσ1q S. By our initial assumptions about σ and σ1, we know that σ2 vanishes at a ﬁnite number of points. Now consider the following short exact sequence:

σ2 0 Ñ OD ÝÑ L Ñ L|DXD1 Ñ 0, which gives the following long exact sequence on cohomology:

0 0 0 1 0 Ñ H pD, ODq Ñ H pD, Lq Ñ H pD X D , L|DXD1 q Ñ ...

1 0p q 0p X | 1 q We claim that H D, OD C. Together with the fact that H D D , L DXD

2 is a vector space of dimension pKS Cq , we conclude our statement:

2 0 1 pKS Cq ¥ H pD, Lq ¥ g ¡ 1.

0p q Hence, we are left to show that H D, OD C. By the following short exact

1 sequence, it suﬃces to show that h pS, OSp¡KS ¡ Cqq 0:

0 Ñ OSp¡KS ¡ Cq Ñ OS Ñ OD Ñ 0.

If we show that KS C is big and nef on the surface S, then the condition above follows from a strong version of Kodaira Vanishing Theorem (see [27], Theorem

77 4.3.1). The divisor KS C is nef because the base locus of its linear system

2 has dimension 0. Given that it is nef, it suﬃces to show that pKS Cq ¡ 0 to conclude that it is also big (see [27], Remark 4.3.2).

2 1 1 Now, we know that pKS Cq D ¤ D ¥ 0, since D and D don’t have any

2 components in common. Thus, we need to rule out the case when pKS Cq

1 1 D ¤ D 0, i.e. D and D do not meet. If this is the case, then KS C is

0 base point free and h pD, ODq ¥ g ¡ 1. Since D is a reduced CM 1-dimensional scheme over the complex numbers, this means that D has g ¡ 1 ¡ 1 connected components. Thus, we get a base point free linear system whose general member is disconnected, which implies that the image of our linear system is a curve (see [22], Theorem 7.1): | | Ñ g¡1 KS C : S P .

On the other hand, this curve has to be the canonical image of C because

pKS Cq|C KC . Hence, we get morphisms C Ñ S Ñ C whose composition is the identity. Denote S Ñ C1 Ñ C the Stein factorization of the second morphism. Then C1 is a curve, but that means that C C1, i.e. the ﬁbers S Ñ C are connected, which gives us the contradiction we wanted. ° ° 1 n 2 1 n 2 (j) Let mi n i1 mi and mi n i1 mi . The genus formula gives us the following:

¸n ¸n ¡ 2 ¡ ¡ 2 3 ¡ ¡ 2 ñ 2g 2 d 3d mi mi d 3d nmi nmi i1 i1 1 ñ n pd2 ¡ 3d ¡ 2g 2q. 2 ¡ mi mi

On the other hand, C is ample on S, which means C2 ¡ 0:

78 m2 d2 ¡ nm2 d2 ¡ i pd2 ¡ 3d ¡ 2g 2q ¡ 0 ñ i 2 ¡ mi mi ñ ¡ 2 2 2p ¡ q ¡ mid 3mi d mi 2g 2 0.

m2 Let α i . By Brill-Noether theory (see Thm. 7.11), we know that d ¥ 2g 6 , mi 3 so the conclusion follows:

2g 6 2g 6 d2 ¡ 3αd ¡ αp2g ¡ 2q 0 ñ p qp ¡ 3αq ¤ dpd ¡ 3αq αp2g ¡ 2q 3 3 g pg 3q2 ñ α ¤ maxtm un . 9 9pg 1q i i1

Now that we have established some preliminary conditions that need to be satis- ﬁed, we start by ruling out one of the easiest cases, where all the multiplicities are

the same: m1 ¤ ¤ ¤ mn m. We show that C ãÑ S cannot be ample:

Lemma 9.6. Let S be a rational surface obtained by blowing up P2 at n distinct

points p1, . . . , pn, as in the Setup 9.4 above. Assume C ãÑ S is a very general curve ° ¥ ¡ n of genus g 22 such that C dH i1 mEi is its divisor class on S. Then C is not an ample divisor of S.

Proof. Assume by contradiction that C is ample. From the proof of Proposition 9.5 above, we know that d2 m ¡ . 3d 2g ¡ 2

On the other hand, from Proposition 9.5(f) we know that 3d ¥ 2g 6. Combining

79 these two facts, we obtain:

d2 d2 d m ¡ ¡ ¡ . 3d 2g ¡ 2 6d ¡ 6 6

Lastly, since C is assumed to be ample, it should satisfy C2 d2 ¡ nm2 ¡ 0, but that produces a contradiction:

nd2 3g 5 d2 ¡ nm2 ¡ ñ n 36 ñ g 22. 36 2

Now, Brill-Noether theory gives us a bit more (see Theorem 7.12). A general point

2p q Ñ 2 of the variety Gd C corresponds to a map C P that maps C birationally onto a plane curve with only nodes as singularities. Thus, we quickly deduce the following corollary:

Corollary 9.7. Let C be a very general curve of genus g ¥ 22. Assume C is embedded on a rational surface S as in Setup 9.4 above. Let d be the degree of the composed

Ñ Ñ 2 2p q map C S P . Then this map corresponds to a non-general point of Gd C .

Remark 9.8. Using a computer program, we were able to find numbers n, d, m1, . . . , mn for every g " 0 such that all the conditions in Proposition 9.5 are satisfied. From our simulations, it seems like we can always find solutions that have the following format: 8 points have the same (high) multiplicity m, and n2 n ¡ 8 nodal points, i.e. of multiplicity 2. Here are a few examples:

g 50 d 241 n2 70 m 85

g 50 d 285 n2 70 m 102

80 g 100 d 651 n2 145 m 230

g 100 d 733 n2 158 m 259

g 150 d 948 n2 221 m 335 (9.2)

g 150 d 1030 n2 228 m 364

g 150 d 1112 n2 231 m 393

g 200 d 1245 n2 306 m 440

g 200 d 1327 n2 307 m 469.

Further discussion. Ideally, we want to ﬁnd a way to use that C is ample to its

full potential. We already know that the points p1, . . . , pn are not in general position. However, let’s assume every subset of 9 points are in general position. Then there exist inﬁnitely many (-1)-curves on S. More speciﬁcally, for every set of indices

ti1, . . . , i9u t1, . . . , nu, there are inﬁnitely many (-1)-curves E whose divisor class is ° ¡ 9 of the form eH r1 nir Eir . Since every such curve has genus 0 and self-intersection -1, then the following system of equations has an inﬁnite number of solutions: $ ' ° & 2 ¡ 9 2 ¡ e r1 nir 1 ' ° % 2 ¡ ¡ 9 p ¡ q ¡ e 3e r1 nir nir 1 2

p q Conversely, given a tuple e, ni1 , . . . , ni9 that satisfies the equations above, the ° ¡ 9 § corresponding divisor E eH r1 nir Eir need not be a (-1)-curve (see [10], 1). ¨ ° ¨ 9 0p p qq e 2 ¡ nir 1 However, the “expected dimension” of h S, OS E is 2 r1 2 1, which means that E is effective and hence C ¤ E ¡ 0. Here are a few example ¡ ¡ ¡ ¡ ¤ ¤ ¤ ¡ of effective divisors with these properties: H Ei1 Ei2 , 2H Ei1 Ei5 , ¡ ¡ ¡ ¤ ¤ ¤ ¡ 3H 2Ei1 Ei2 Ei7 . These divisors impose the following (further) restrictions

81 on C:

¡ ¡ ¡ d mi1 mi2 0 ¡ ¡ ¡ ¡ ¡ ¡ 2d mi1 mi2 mi3 mi4 mi5 0 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 3d 2mi1 mi2 mi3 mi4 mi5 mi6 mi7 0.

We note that all the divisors C of the form found in (9.2) satisfy the inequalities above. However, it would be interesting to know if other divisors E like the ones above could impose further restrictions on d, m1, . . . , mn, leading to a contradiction together with the other conditions.

We now switch to do a similar analysis of the case in which S0 is the Hirzebruch ¥ surface Fe, for some e 0 or e 2. As before, we will focus on a special case, as follows:

Setup 9.9. Let C ãÑ S be a very general curve embedded on S as an ample divisor such that dim |C| 0. Assume S is obtained by blowing up the Hirzebruch surface S0 Fe at n distinct points p1, . . . , pn, and E1,...,En are the corresponding exceptional divisors. The Picard group of S is generated by the classes σ, f, E1,...,En, 2 ¡ 2 ¤ 2 ¡ ¤ ¤ ¤ which satisfy σ e, f 0, σ f 1, Ei 1, σ Ei f Ei Ei Ej 0. Let ° ¡ n C aσ bf i1 miEi be the divisor class of C.

Proposition 9.10. Let S and C be as in Setup 9.9 above. Then the following conditions need to be satisﬁed:

(a) mi ¥ 2, for all i 1, . . . , n;

(b) a ¡ 0 and b ¡ ae;

82 ° ¡ 2 ¡ p q ¡ ¡ n p ¡ q ¡ (c) a e 2ab a 3e 2 2b i1 mi mi 1 2g 2; ° 2 ¡ 2 ¡ n 2 ¡ (d) C a e 2ab i1 mi 0;

¥ g 2 (e) a 2 ;

(f) 2n ¥ 3g e 2.

Proof. (a) We explained in Lemma 8.4 why we can assume without loss of generality

that C ¤ Ei ¥ 2, for all i 1, . . . , n.

(b) Let C0 aσ bf be the image of C on S0 Fe. Since C0 is ample on Fe, it must satisfy a ¡ 0 and b ¡ ae (see [21], Corollary V.2.18);

(d) Since C is ample, then C2 ¡ 0.

(e) This is an immediate application of Brill-Noether theory (see Thm. 7.11).

(f) This follows from a moduli count. From Lemma 8.7, we know that S needs to ¡ have at least 3g 3 moduli. On the other hand, since S is the blowup of Fe at n ¡ p p qq ¡ p q distinct points, the number of moduli of S is 2n dim Aut Fe 2n e 5 , and the conclusion follows.

Now that we have established some preliminary conditions that need to be satis- ﬁed, we start by ruling out one of the easiest cases, in which all the multiplicities are the same: m1 ¤ ¤ ¤ mn m. We show that C ãÑ S cannot be ample if g ¥ 22:

Lemma 9.11. Let S be a rational surface obtained by blowing up the Hirzebruch surface S0 Fe at n distinct points p1, . . . , pn, as in the Setup 9.9 above. Assume

83 ° ãÑ ¥ ¡ n C S is a very general curve of genus g 22 such that C aσ bf i1 mEi is its divisor class in S. Then C is not an ample divisor of S.

Proof. Assume by contradiction that C is ample. From Proposition 9.10(c) above, we have: ¡a2e 2ab ¡ ap3e 2q ¡ 2b ¡ nmpm ¡ 1q 2g ¡ 2,

which means that:

1 n p¡a2e 2ab ¡ ap3e 2q ¡ 2b ¡ 2g 2q. mpm ¡ 1q

On the other hand, from Proposition 9.10(d), we have:

C2 ¡a2e 2ab ¡ nm2 ¡ 0 m ñ ¡a2e 2ab ¡ p¡a2e 2ab ¡ ap3e 2q ¡ 2b ¡ 2g 2q ¡ 0 m ¡ 1 ñ2 e ¡ 2ab mp3e 2gq ¡ 0,

which translates to:

2ab ¡ a2e m ¡ . (9.3) 3e 2g

Now, combining Equation 9.3 with C2 ¡ 0, we obtain:

Combining Inequality 9.4 above with the fact that b ¡ ae (see Prop. 9.10(b)), we obtain: nab p3e 2gq2. (9.5)

84 If e ¡ 0, then we combine the results (b), (e), (f) of Proposition 9.10 with Equation 9.5 to get the following inequality, which leads to a contradiction, since g ¥ 22:

3g e 2 g 2 pg 2qe ¤ ¤ p3e 2gq2 ñ 3g3e g2e2 122e2 82g2. 2 2 2

1 ¢ 1 Lastly, if e 0, then S0 P P . In this situation we have two projections onto 1 ¥ g 2 P , which means we have the added condition b 2 , by Brill-Noether theory (see Thm. 7.11). Since g ¥ 22, Proposition 9.10(f) implies that n ¥ 36. Using the same inequalities as before, we get a contradiction, and the proof is complete:

85 Chapter 10

Surfaces of Kodaira dimension 0

In this chapter we analyze the case in which a very general curve C of genus g is embedded as an ample divisor on a smooth surface S of Kodaira dimension 0 such that dim |C| 0. We know that the minimal model of S is either an abelian surface, a K3 surface, an Enriques surface, or a bielliptic surface. We show that if g ¥ 19, such an embedding does not exist.

10.1 Abelian Surfaces

Theorem 10.1. Let C ãÑ S be a very general curve embedded on a smooth surface whose minimal model is an abelian surface. Then C has genus 2 and S is the Jacobian of C.

Proof. Let S be a smooth surface whose minimal model S0 is an abelian surface. Let

S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ S0 be the corresponding sequence of blow-downs. By the Enriques-Kodaira classiﬁcation (see Thm. 7.9), we know that S0 satisﬁes:

p q p q pg S0 1, q S0 2,KS0 0.

86 By Theorem 8.3, we know that qpSq must be either 0 or g. Since the irregularity of a surface is a birational invariant, we must have qpSq qpS0q g 2. Thus, we are in the situation where the genus of the curve C is 2. Now, by the moduli count, we get: ¡ ¤ 2 ¡ ¤ ¥ ¤ 2 2g 2 C KS C C KS C0 KS0 2n 2n, (10.1) which means that n 0, i.e C is embedded on a simple abelian surface. We have a natural candidate for such a surface, which is the Jacobian of C. Given that S is an abelian surface, the inclusion C ãÑ S factors through the Jacobian of C: C ãÑ J ÝÑh S. Since S is simple, the map h must be surjective. Assume by contradiction that h is not an isomorphism, i.e. there exists x P J such that hpxq 0. We know that the canonical map C Ñ J is a closed immersion and the image of C is an ample divisor on J. In particular, this means that the curves x C and C intersect inside J, which means there exist y, z P C such that x y z. We conclude: hpzq hpx yq hpxq hpyq hpyq, but this contradicts the map C ãÑ S being an embedding, so we are done.

10.2 K3 Surfaces

Lemma 10.2. The moduli space of polarized K3 surfaces of a given degree 2d is a quasi-projective variety of dimension 19.

Proof. See [1].

Theorem 10.3. There exists no smooth surface S whose minimal model is a K3

87 surface, that contains a smooth ample very general curve C of genus ¥ 19 satisfying dim |C| 0.

Proof. Assume by contradiction we can ﬁnd such an embedding C ãÑ S, where C is a very general curve of genus g ¥ 19. Let S0 be the minimal K3 surface associated to S and S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ S0 be the corresponding sequence of blow-downs.

By the Enriques-Kodaira classiﬁcation (see Thm. 7.9), we know that S0 satisﬁes:

p q p q pg S0 1, q S0 0,KS0 0.

The curve C is ample on S, hence C2 ¡ 0. Using this fact together with the genus formula and Lemma 8.6, we obtain:

¡ ¤ 2 ¡ ¤ ¥ ¤ 2g 2 C KS C C KS C0 KS0 2n 2n. (10.2)

Now we count moduli: by Lemma 10.2 stated above, the number of moduli of S0 is 19. It follows that the number of moduli of S is 19 2n, since we add 2 moduli for each point we blow up from S0 all the way up to Sn. On the other hand, Lemma 8.7 states that the number of moduli of S needs to be at least 3g ¡ 3. Putting this information and Equation 10.2 together, we obtain a contradiction:

3g ¡ 3 ¤ 19 2n ¤ 19 p2g ¡ 4q 2g 15 ñ g ¤ 18.

88 10.3 Enriques Surfaces

Lemma 10.4. The moduli space of Enriques surfaces is an irreducible smooth Artin stack of dimension 10.

Proof. See [9].

Theorem 10.5. There exists no smooth surface S whose minimal model is an En- riques surface, that contains a smooth ample very general curve C of genus ¥ 10 satisfying dim |C| 0.

Proof. Assume by contradiction we can ﬁnd such an embedding C ãÑ S, where C is a very general curve of genus g ¥ 10. Let S0 be the minimal Enriques surface

associated to S and S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ S0 be the corresponding sequence of blow-downs. By the Enriques-Kodaira classiﬁcation (see Thm. 7.9), we know that

S0 satisﬁes: p q p q pg S0 q S0 0,KS0 0, 2KS0 0.

The curve C is ample on S, hence C2 ¡ 0. Using this fact together with the genus formula and Lemma 8.6, we obtain:

¡ ¤ 2 ¡ ¤ ¥ ¤ 2g 2 C KS C C KS C0 KS0 2n 2n. (10.3)

Now we count moduli: by Theorem 10.2 stated above, the number of moduli of

S0 is 10. It follows that the number of moduli of S is 10 2n, since we add 2 moduli for each point we blow up from S0 all the way up to Sn. On the other hand, Lemma 8.7 states that the number of moduli of S needs to be at least 3g ¡ 3. Putting this information and Equation 10.3 together, we obtain a contradiction:

3g ¡ 3 ¤ 10 2n ¤ 10 p2g ¡ 4q 2g 6 ñ g ¤ 9.

89 10.4 Bielliptic Surfaces

Theorem 10.6. There exists no smooth surface S whose minimal model is a bielliptic surface, that contains a smooth ample very general curve C.

Proof. Assume by contradiction we can ﬁnd such an embedding C ãÑ S. Let S0 be the minimal bielliptic surface associated to S and S Sn Ñ Sn¡1 Ñ ¤ ¤ ¤ Ñ S1 Ñ S0 be the corresponding sequence of blow-downs. By the Enriques-Kodaira classiﬁcation

(see Thm. 7.9), we know that S0 satisﬁes:

p q p q P t u pg S0 0, q S0 1, nKS0 0, n 2, 3, 4, 6 .

By Theorem 8.3, we know that qpSq must be either 0 or g. Since the irregularity of a surface is a birational invariant, we must have that qpSq qpS0q g 1. Thus, we are in the situation where the genus of the curve C is 1. Using this fact together with the genus formula and Lemma 8.6, we obtain the following contradiction:

¡ ¤ 2 ¡ ¤ ¥ ¤ 0 2g 2 C KS C C KS C0 KS0 2n 2n.

90 Chapter 11

Surfaces of Kodaira dimension 1

In this chapter we analyze the scenario in which a very general curve C of genus g is embedded as an ample divisor on a smooth surface S of Kodaira dimension 1 such that dim |C| 0. We show that if g ¥ 21, such an embedding does not exist.

Setup 11.1. Let S be a smooth surface of Kodaira dimension 1 and S0 its unique smooth minimal model. By the Kodaira-Enriques classification of surfaces (see The- orem. 7.9), we know that S0 is an elliptic surface, i.e. there exists a surjective morphism S0 Ñ B onto a smooth curve, such that all but finitely many fibers are smooth irreducible curves of genus 1. As before, let C be our ample very general curve on S satisfying dim |C| 0, and let C0 be the image of C in S0. We will be working with the following morphisms:

ãÑ C S S0 B. (11.1)

¦ 0 Lemma 11.2. Let π : S0 Ñ B be an elliptic surface over B and π : Pic pBq Ñ

0 ¦ Pic pS0q the pullback morphism on the Picard groups. Then π is injective.

Proof. First, notice that OB π¦OS0 . This is true because every ﬁber is connected,

91 P 0p q ¦ so the result follows by base change. Consider L Pic B such that π L OS0 . We apply the projection formula and conclude that π¦ is injective:

¦ L O b L π¦O b L π¦pO b π Lq π¦O O . B OB S0 OB S0 OS0 S0 B

1 Lemma 11.3. Let C,S,S0,B as in the Setup 11.1 above. Then B P .

Proof. As a consequence of Theorem 8.2 and Lemma 11.2, we have the following

0 0 0 0 sequence of morphisms: Pic pBq ãÑ Pic pS0q ÝÑ Pic pSq ãÑ Pic pCq. By Lemma 8.1, since C is a very general curve, we have that Pic0pCq is simple, therefore either Pic0pBq 0 or the composed map Pic0pBq Ñ Pic0pCq is an isomorphism. If the composed map Pic0pBq Ñ Pic0pCq is an isomorphism, then by the Torelli theorem for curves (see Thm. 7.2), the composed map C Ñ B is an isomorphism.

However, if this is the case, we get a minimal elliptic surface S0 Ñ C with a section σ. As a consequence of Theorem 11.5 below, we obtain C2 ¡σ2 ¤ 0, which contradicts

0 the ampleness of C. In conclusion, we must have that Pic pBq 0, i.e. B P1.

1 As a consequence of the Lemma 11.3 above, S0 is an elliptic fibration over P . Now we need to analyze two scenarios: either the fibration admits a section, or it doesn’t. The simpler case is when the fibration has a section, so we will analyze that case first. The second case builds upon the first one, as we will see momentarily.

Ñ 1 Proposition 11.4. Let π : S0 P be a minimal elliptic surface. Then the irregularity of S is zero.

Proof. See [31], Corollary 2.4.

92 Ñ 1 Now, assume S0 P is an elliptic ﬁbration that admits a section σ. In this case, we have some nice properties, summarized by Miranda (see [31]):

Ñ 1 1 Theorem 11.5. Let π : S0 P be a minimal elliptic surface over P with section

1 1 p¡ q ¥ σ. Then R π¦OS0 OP N for some N 0. The surface S0 is a product if and only if N 0. Moreover, if N ¡ 0, then:

(i) σ2 ¡N

p ¡ q (ii) KS0 N 2 F , where F is the class of the ﬁber of π.

Proof. See [31], Corollary 2.4.

Remark 11.6. Observe that, in our setup, we are in the case where N ¡ 0. To see

¢ 1 this, assume by contradiction that N 0 and S0 E P , where E is an elliptic 0 0 curve. Recall we have the following sequence of morphisms: Pic pS0q ÝÑ Pic pSq ãÑ 0p q 0p q 0p ¢ 1q 0p q 0p q Pic C . Since Pic S0 Pic E P Pic E , this contradicts Pic C being a simple abelian surface of dimension g ¥ 2.

In his paper (see [31]), Miranda constructed a coarse moduli space parametrizing

Weierstrass ﬁbrations over P1, which are in a 1-to-1 correspondence with minimal elliptic surfaces over P1 that admit a section σ. We refer the reader to the original paper for more information. Here is the main result we need:

Theorem 11.7. Let N be a positive integer. There exists a coarse moduli space that

parametrizes minimal elliptic surfaces π : S Ñ P1 that admit a section and satisfy ¡ 1 ¡ N deg RπOS. The dimension of this moduli space is 10N 2.

Proof. See [31].

Generalizing Miranda’s work, Seiler constructed the moduli space of polarized

minimal elliptic surfaces (see [34]). To deal with the missing section σ : P1 Ñ S, he

93 worked instead with the associated Jacobian of such a ﬁbration, which is obtained as follows:

Deﬁnition 11.8. Let f : S Ñ B be an elliptic surface over an algebraically closed

field k. Let K kpBq be the function field of B. The general fiber SK of f is

a curve of genus 1 over K, therefore its Jacobian JK is an elliptic curve over K. This Jacobian extends to an elliptic surface j : J Ñ B, which is called the Jacobian ﬁbration associated to f. This ﬁbration comes equipped with a section, namely the closure of the zero divisor on SK .

Here is Seiler’s main result:

Theorem 11.9. Let N be a positive integer. There exists a coarse moduli space that parametrizes minimal elliptic surfaces π : S Ñ P1 that have k multiple ﬁbers and ¡ 1 Ñ 1 satisfy N deg RpOJ , where p : J P is the associated Jacobian ﬁbration. The dimension of this moduli space is 10N k ¡ 2.

Now that we have the moduli count, we are almost ready to prove the main statement. Before we do so, we need two more lemmas from Seiler:

Theorem 11.10. Let f : S Ñ C be an elliptic ﬁbration with multiple ﬁbers miDi,

1 _ where 1 ¤ i ¤ k. Let L pR f¦OSq . Then:

¸ ¦ ωS f pL b ωC q b OSp pmi ¡ 1qDiq. i

Proof. See [34], Theorem 1.1.

Lemma 11.11. Let f : S Ñ C be an elliptic surface and j : J Ñ C be the corre-

1 1 sponding Jacobian ﬁbration. Then R f¦OS R j¦OJ .

Proof. See [34], Lemma 1.3.

94 Now that we stated all the necessary prerequisites, we turn to the main theorem of this section:

Theorem 11.12. There exists no smooth surface S of Kodaira dimension 1 on which one can embed a smooth very general curve C of genus g ¥ 21 as an ample divisor satisfying dim |C| 0.

Proof. Assume by contradiction that such a surface S exists. Let S0 be its unique

smooth minimal model and S Sn Ñ ¤ ¤ ¤ Ñ S0 the corresponding sequence of

1 blow-downs. From Lemma 11.3, we know that S0 is an elliptic surface over P with

(possible) multiple fibers m1D1, . . . , mkDk, where k ¥ 0. Ñ 1 ¡ 1 Let p : J P be the Jacobian fibration associated to S0 and N deg RpOJ . By Remark 11.6, we know that N ¡ 0, so the moduli space parametrizing such surfaces S0 has dimension 10N k¡2. As before, this means that S has 10N k¡2 2n moduli. On the other hand, since C is a very general curve on S and Pic0pSq 0, Lemma 8.7 gives us that S must have at least 3g ¡ 3 moduli, so we conclude our first inequality: 3g ¡ 3 ¤ 10N k ¡ 2 2n. (11.2)

On the other hand, let d be the degree of the composed map C ãÑ S Ñ S0 Ñ

P1. Since C is a very general curve, Brill-Noether theory (see Thm. 7.11) gives the following bound on d:

g 2 ρpg, 1, dq 2d ¡ g ¡ 2 ¥ 0 ðñ d ¥ . (11.3) 2

¤ Now, as a consequence of Theorem 11.5 and Lemma 11.11, we obtain C0 KS0 ° p ¡ q k p ¡ q ¤ d N 2 i1 mi 1 C0 Di. Combining this fact with the genus formula, Lemma 8.6, and Equation 11.3, we conclude:

95 ¡ 2 ¤ ¡ ¤ ¥ ¤ 2g 2 C C KS C KS C0 KS0 2n ¸k pN ¡ 2qd pmi ¡ 1qC0 ¤ Di 2n i1 pN ¡ 2qpg 2q ¥ pN ¡ 2qd k 2n ¥ k 2n, 2 from which we derive that:

pN ¡ 2qpg 2q 2n k 2g ¡ 2 ¡ . (11.4) 2

Finally, we can put equations 11.2 and 11.4 together to obtain:

pN ¡ 2qpg 2q 3g ¡ 3 ¤ 10N ¡ 2 k 2n ¤ 10N ¡ 2 2g ¡ 2 ¡ , (11.5) 2 which is a contradiction when g ¥ 21. In conclusion, there exists no smooth surface S of Kodaira dimension 1 on which one can embed a smooth very general curve C of genus g ¥ 21 as an ample divisor.

96 Chapter 12

Surfaces of Kodaira dimension 2

In this chapter we analyze the scenario in which a very general curve C of genus g is embedded as an ample divisor on a surface of general type. From Proposition 8.3, we know that the irregularity qpSq of the surface must be either 0 or g. If qpSq g, then S is birational to the symmetric product Sym2pCq. The case where qpSq 0 remains mainly open. However, we prove the following statement: if S ãÑ Pr is a complete intersection such that the composed map C ãÑ S ãÑ Pr satisﬁes the Maximal Rank Conjecture, then g ¤ 15.

12.1 Surfaces of general type with qpSq g

In the case where S is of general type and satisﬁes qpSq g, we have a complete characterization due to Mendes Lopes and Pardini (see [28]):

Theorem 12.1. Let C be a very general genus g curve embedded on a smooth surface S of general type with irregularity qpSq g. Then S is birational to Sym2pCq.

Proof. We follow the proof of Theorem 1.1 in [28]. Let S0 be the (unique) smooth minimal model of S. We ﬁrst claim that S0 is not a product of curves. To prove

97 this, assume by contradiction that S0 C1 ¢ C2, where gpC1q gpC2q g and

gpC1q, gpC2q ¥ 2. By Theorem 8.2, we obtain the following sequence of morphisms:

0 0 0 0 0 Pic pC1q ¢ Pic pC2q Pic pS0q ÝÑ Pic pSq ãÑ Pic pCq,

but this contradicts the simpleness of Pic0pCq (see Thereom 8.1). Therefore, we can

assume that S0 is not a product of curves. Let d C2 ¡ 0. There exists a d-dimensional system of curves C on S which are numerically equivalent to C (see [28], Proposition 4.3). Moreover, all smooth elements of C are isomorphic to C. Now, if d ¡ 1, then S is not of general type (see [3], §0). Therefore, C2 1 and the conclusion follows by [3], Proposition 0.18.

Remark 12.2. We remark here that the situation above can happen. Fix x P C and

2 consider the composed morphism fx : C ãÑ C ¢ C Ñ Sym pCq that maps y ÞÑ x y. 2p q 2 Let Cx be the image of C in Sym C . First of all, it is easy to see that Cx 1. 2 Moreover, Cx is ample on Sym pCq because its inverse image in C ¢ C is ample (it is the union of C ¢ txu and txu ¢ C).

12.2 Surfaces of general type with qpSq 0

While this case remains mostly unsolved, we would like to discuss a simple scenario in which qpSq 0, i.e. when S is a complete intersection. Our result is the following:

Theorem 12.3. Let C be a very general smooth curve. Let S be a complete inter-

r p q ¤ ¤ ¤ ¤ ¤ ¤ section in P of multidegree d1, . . . , dr¡2 , where 2 d1 dr¡2. Assume C is embedded in S as an ample divisor such that the composed map C ãÑ Pr satisﬁes the Maximal Rank Conjecture. Then the genus of C satisﬁes g ¤ 15.

98 Proof. Assume by contradiction that we can embed a very general curve C of genus

g ¥ 16 on a complete intersection S, where S has multidegree pd1, . . . , dr¡2q and ¤ ¤ ¤ ¤ ¤ ¤ ¤ Ñ r 2 d1 d2 dr¡2, such that C P is a general degree d Brill-Noether curve. ° p r¡2 ¡ ¡ q Since S is a complete intersection, then ωS OS i1 di r 1 , which means:

r¸¡2 C ¤ KS dp di ¡ r ¡ 1q. (12.1) i1

On the other hand, C Ñ Pr is a non-degenerate map of degree d, so by Brill- Noether theory (see Thm. 7.11), we obtain:

r d ¥ g r. (12.2) r 1

Now, we claim the following inequality holds:

r¸¡2 di ¡ r ¡ 1 ¤ 2. (12.3) i1 ° r¡2 ¡ ¡ ¥ To prove this, assume by contradiction that i1 di r 1 3. This means that

C ¤ KS ¥ 3d. We combine the genus formula, the ampleness of C, and Equations 12.1 and 12.2 to obtain a contradiction:

r 2g ¡ 2 C2 C ¤ K ¡ C ¤ K ¥ 3d ¥ 3p g rq. S S r 1

Since Inequality 12.3 fails for r ¥ 8, we are left to analyze the cases where 3 ¤ r ¤ 7. The composed map C Ñ Pr satisﬁes the Maximal Rank Conjecture (see Thm. 12.4), which means that in order to have a degree k polynomial vanishing on

C ãÑ Pr, the following inequality needs to be satisﬁed:

99 ¢ k r ¡ kd 1 ¡ g. (12.4) k

We combine Inequalities 12.4 and 12.2 with the fact that g ¥ 16 to obtain:

¢ k r r2 17r ¡ k ¡ 15. (12.5) k r 1

p q 3 If r 3, then S is a complete intersection of type d1 on P . Equation 12.5 gives us the following bound:

¢ 3 k ¡ 15k ¡ 15 ñ k ¥ 6, k

which means that the degree d1 of S is at least 6. On the other hand, Inequality 12.3 implies d1 ¤ 6. Hence d1 must be exactly 6. Combining Inequalities 12.4 and 12.2, we obtain: ¢ 9 7 ¥ g 19 ñ g ¤ 18. 6 2

Therefore, we only need to analyze the cases where d1 6 and g 16, 17, 18. Inequality 12.2 gives us the following bounds on d:

g 16 ñ d ¥ 15

g 17 ñ d ¥ 16

g 18 ñ d ¥ 17, but all these bounds contradict the ampleness of C:

100 2 0 C 2g ¡ 2 ¡ C ¤ KS 2g ¡ 2 ¡ 2d ¤ 0.

p q 4 If r 4, then S is a complete intersection of type d1, d2 on P . Equation 12.5 gives us the following bound:

¢ 4 k 84 ¡ k ¡ 15 ñ k ¥ 4, k 5

which means that the degrees d1, d2 of S are at least 4, but that contradicts Inequal- ity 12.3.

p q ãÑ ãÑ 5 If r 5, then S is a complete intersection of type d1, d2, d3 and C S P . Equation 12.5 gives us the following bound:

¢ 5 k 55 ¡ k ¡ 15 ñ k ¥ 3, k 3

which means the degrees d1, d2, d3 of S are at least 3, but that contradicts Inequal- ity 12.3. ° 4 If r 6, then Inequality 12.3 implies that i1 di 10. This means that either

d1 d2 d3 d4 2 or d1 d2 d3 2 and d4 3. If we are in the ﬁrst case, the space of polynomials of degree 2 that vanish on C must be at least 4, which by the Maximal Rank Conjecture (see Thm. 12.4), translates to:

¢ 6 2 ¡ p2d 1 ¡ gq ¥ 4 ñ 23 g ¥ 2d. 2

Combining the above with Equation 12.2, we obtain that g ¤ 15, which is a contradiction. If we are in the second case, the space of polynomials of degree 2

101 that vanish on C must be at least 3, which by the Maximal Rank Conjecture (see Thm. 12.4), translates to:

¢ 6 2 ¡ p2d 1 ¡ gq ¥ 3 ñ 24 g ¥ 2d. 2

Combining the above with equation 12.2, we obtain that g ¤ 16. In the special case when g 16, we apply Inequality 12.2 again and get the following contradiction:

6 30 2g ¡ 2 ¡ K ¤ C 2d ¥ 2p ¤ 16 6q ¥ 39. S 7

° 5 If r 7, then Inequality 12.3 implies that i1 di 11. This means that d1

¤ ¤ ¤ d5 2, so the space of polynomials of degree 2 that vanish on C must be at least 5, which by the Maximal Rank Conjecture (see 12.4), translates to:

¢ 7 2 ¡ p2d 1 ¡ gq ¥ 5 ñ 30 g ¥ 2d. 2

Combining the above with equation 12.2, we obtain that g ¤ 21. On the other hand, we apply Inequality 12.2 again and get the following contradiction, ﬁnishing the proof: 7 2g ¡ 2 ¡ C ¤ K 2d ¥ 2p g 7q ñ g ¡ 64. S 8

Remark 12.4. We note that Verra ([38]) proved the unirationality of the moduli

space of genus g curves Mg for 11 ¤ g ¤ 14 by embedding a general smooth curve C on a complete intersection, hence this situation does happen.

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