Towards a General Theory of Weierstrass Points on Curves

Sami Douba Department of Mathematics & Statistics McGill University, Montr´eal June 2017

A thesis submitted to McGill University in partial fulﬁllment of the requirements of the degree of Masters of Science

c Sami Douba 2017 2 Abrégé Nous présentons plusieurs généralisationsalgébro-géométriquesde la notion classique d’un point de Weierstrass sur une surface de Riemann. Nous montrons qu’une large part de l’intuition acquise par la théorie classique est vraie en caractéristique 0 et en grande caractéristique.Finalement, nous calculons et examinons les points de Weierstrass de la quartique de Klein en caractéristiques2 et 3 pour démontrer que ces points peuvent faire preuve d’un comportement intéressant en petite caractéristique.

Abstract We present various algebro-geometric generalizations of the classical notion of a Weierstrass point on a Riemann surface. We show that much of the intuition gained from the classical theory holds in characteristic 0 and in high characteristic. Finally, we compute and examine the Weierstrass points of the Klein quartic in characteristics 2 and 3 to demonstrate that these points may exhibit interesting behaviour in low characteristic.

3 4 Preface

The deﬁnitions and the proofs in Chapter 1 are modelled very closely after Hershel Farkas and Irwin Kra’s exposition of Weierstrass points in [FK80], but the discussion is slightly more general. The theory in Chapter 2 and Section 3.1 is taken entirely from the papers [Lak81], [Lak84] by Dan Laksov, and [LT94] by Laksov and Anders Thorup. Section 3.2 makes extensive use of Noam Elkies’ paper [Elk99] on the Klein quartic, but the results in that section are, to the author’s knowledge, original.

Acknowledgements I would like to express my immense gratitude to my supervisor, Prof. Eyal Goren, for his support, his guidance, and his patience. I would also like to thank Prof. Jacques Hurtubise for agreeing to examine my thesis. Last but not least, I am indebted to my fellow students, with and from whom I gained much of the background knowledge necessary to complete this thesis.

5 6 Contents

1 A na¨ıve approach to Weierstrass points 13 1.1 The gap theorems ...... 13 1.2 Weierstrass points and wronskians ...... 17 1.3 Weierstrass points on hyperelliptic curves ...... 23 1.4 Higher Weierstrass points ...... 26

2 Laksov’s Weierstrass points on curves 29 2.1 The bundle of principal parts ...... 30 2.2 Weierstrass points ...... 35 2.3 The wronskian ...... 36 2.4 Local computations ...... 40 2.5 The classical situation ...... 43

3 Weierstrass points in families 45 3.1 Weierstrass points on schemes ...... 45 3.2 Reductions of the Klein quartic ...... 50

7 8 Introduction

Since elliptic curves have non-trivial automorphisms, one cannot hope to construct a ﬁne moduli space parametrizing isomorphism classes of these objects. However, an elliptic curve E is endowed with distinguished points — the group E[N] of its N-torsion points. We can use this group to endow an elliptic curve with additional structure, namely, a full level N structure, and then stipulate that isomorphisms between elliptic curves preserve this structure. The resulting moduli problem admits a ﬁne moduli space for large enough N; it is the modular curve Y (N). Now what about curves of genus greater than 1? Is there a “non-abelian” analogue of the set E[N]?

In the early 1860s, Weierstrass proved his so-called Lückensatz, or “gap theorem”, which can be stated as follows: For any point on a compact Riemann surface of genus g, there are exactly g integers m, called “gaps”, such that there is no meromorphic function on that surface having a pole of order m at that point as its only singularity. For all but a finite number of special points, the surface’s so-called Weierstrass points, these gaps are the integers 1 ≤ m ≤ g. Furthermore, it is possible to attach weights to these points so that the sum of their weights is a constant that depends only on g. From the perspective of classifying curves of a fixed genus, this seems promising. Since compact Riemann surfaces can be thought of as algebraic curves over C, it is natural to ask if one can talk about Weierstrass points on more general algebraic curves, and if one can use these points to “rigidify” spaces of such curves and construct moduli spaces, as is done with torsion points on elliptic curves.

In the ﬁrst chapter, we provide a na¨ıve theory of Weierstrass points on smooth projective curves over algebraically closed ﬁelds. This approach is completely analogous to

9 10

the classical theory of Weierstrass points on compact Riemann surfaces. However, this theory breaks down when the characteristic of the ground field is low relative to the genus. Indeed, in low enough characteristic, there are examples of curves C of genus g with the property that each point on C has a non-classical gap sequence, that is, a gap sequence that differs from 1, . . . , g (see Example 2.0.1). In the second chapter, we resolve this issue by introducing Laksov’s theory of Weier- strass points on curves. Instead of defining a Weierstrass point as a point whose gap sequence is non-classical, Laksov associates to each curve a generic gap sequence, and defines a point on that curve to be Weierstrass if the gap sequence at that point differs from the generic gap sequence. Under this approach, a curve has finitely many Weier- strass points regardless of the characteristic of the ground field. However, the weighted number of Weierstrass points may vary between curves of a fixed genus (see Section 3.2). This is already an indication that Weierstrass points may not be satisfactory candidates for higher-genus analogues of torsion points on elliptic curves. In the final chapter, we expose Laksov and Thorup’s notion of Weierstrass points on families of curves over arbitrary base schemes. This generalizes the theory of the previous chapter. We end with a thorough examination of the Klein quartic. We show that none of the points obtained as reductions mod 3 of Weierstrass points of the classical Klein quartic is a Weierstrass point of the reduction in the sense of Laksov. This shows that Laksov’s Weierstrass points have another drawback: their behaviour in algebraic families appears to be subtle.

Literature Review For the following discussion, let g ≥ 2 be an integer, and denote by X a compact Riemann surface of genus g. Furthermore, let Mg be the moduli space

of Riemann surfaces of genus g, and let Mg be the Mumford-Deligne compactiﬁcation of

Mg ([DM69]). The study of Weierstrass points dates back to the nineteenth century, when Weier- strass stated and proved the L¨uckensatz (discussed above). The theorem ﬁrst appeared in the dissertation of Weierstrass’s student Schottky in 1875 (later published as [Sch77]). Through their work on adjoints of plane curves, Brill and Noether implicitly proved in 11

1873 that the set of Weierstrass points on X is nonempty and has size at most g3 − g

([BN74]). In 1882, Noether proved the Lückensatz for any sequence of points P1,P2,... on X, generalizing Weierstrass’s result ([Noe84]; see 1.1.4 for a modern — and more general — version of this theorem). He also proved the finiteness of Aut(X) using Weierstrass points ([Noe82]), but the first proof of this fact is probably due to Klein ([dC08, p. 42]).

In his landmark 1893 paper [Hur92], H¨urwitzintroduced the wronskian and the notion of “weights” to the study of Weierstrass points. He even introduced the concept of “higher” Weierstrass points, although the ﬁrst time Weierstrass points were thus labelled was in a paper by Haure from 1896 ([Hau96]). Segre adopted this label when he discussed “punti di Weierstrass” in his short 1899 publication [Seg99], in which he proved that the maximal weight of a Weierstrass point on a nonhyperelliptic curve of genus g is (g − 1)(g − 2)/2 + 1.

The first half of the twentieth century is considered a period of relative dormancy in the study of Weierstrass points, but it was in this period that a paper of great relevance to this thesis was published. In his 1939 paper [Sch39], Schmidt used the wronskian method to extend the notion of Weierstrass points to curves defined over fields of positive characteristic. The gap sequences and Weierstrass points defined by Laksov in [Lak81], whose modern approach to the topic is the focus of this thesis, are the same as those given by Schmidt.

Rauch ushered in a period of renewed interest in Weierstrass points with the publication of his paper [Rau59] in 1959. Ahlfors had already endowed the set of isomorphism

classes Mg of Riemann surfaces of genus g with the structure of a (3g − 3)-dimensional complex analytic space ([Ahl60]). In his 1959 paper, Rauch proved that the Riemann surfaces of genus g carrying a Weierstrass point with a prescribed ﬁrst non-gap constitute

a complex-analytic subvariety of Mg. Some years later, expanding on the work of Rauch,

Arbarello examined the closure W n,g in Mg of the set of Riemann surfaces of genus g possessing a Weierstrass point x such that h0(C, nx) ≥ 2, where 2 ≤ n ≤ g (so, for

example, W 2,g is the closure of the hyperelliptic locus, and in general W n,g contains the space of Riemann surfaces of genus g possessing a Weierstrass point whose ﬁrst non-gap 12

is n). Among other things, Arbarello proved that W 2,g is an irreducible analytic variety of dimension 2g + n − 3 ([Arb74]). From the 1970s onwards, questions about Weierstrass points have tended to involve the existence of Weierstrass points of a given type, or the relationship between Weierstrass points and automorphisms of Riemann surfaces (see, for example [Lax75], [Far73]). There has been relatively little eﬀort to generalize this theory to accommodate algebraic curves over ﬁelds of arbitrary characteristic. The work of Laksov and Thorup ([Lak81], [LT94], [LT95]) is exceptional in this regard. Chapter 1

A na¨ıve approach to Weierstrass points

The theory presented in this chapter is “na¨ıve” in the sense that it is modelled very closely after the classical theory of Weierstrass points on Riemann surfaces. For the following discussion, by curve we will mean a projective non-singular algebraic curve over some fixed algebraically closed field k. We fix a curve C of genus g. Let

1 k(C) be its function ﬁeld, and let OC and ΩC be its structure sheaf and its canonical sheaf, respectively. Given a divisor D on C, denote by O(D) the invertible OC -module associated to D and by H0(C,D) the space of global sections of O(D). This is a ﬁnite dimensional k-vector space whose dimension we denote by h0(C,D) or `(D). Denote by

0 1 −1 i(D) the dimension of H (C, ΩC ⊗ O(D) ), that is, i(D) = `(K − D), where K is a canonical divisor on D. For the rest of this chapter, we will assume points are closed unless otherwise stated.

1.1 The gap theorems

The discussion in this section is based on that in [FK80, pp. 78-80].

Lemma 1.1.1. Let P a point on C and D a divisor on C. Let D0 = D + [P ]. Then `(D0) − `(D) is either 0 or 1.

13 Chapter 1. A na¨ıve approach to Weierstrass points 14

Proof. Let K be a canonical divisor on C. By Riemann-Roch, we have

`(D0) − i(D0) = deg D0 − g + 1

`(D) − i(D) = deg D − g + 1

Subtracting the second equation from the ﬁrst, we obtain

`(D0) − `(D) = 1 + i(D0) − i(D)

Now, since D0 ≥ D, we have H0(C,D) ⊆ H0(C,D0), so `(D0) − `(D) ≥ 0. Furthermore, since D0 ≥ D, we have i(D0) − i(D) ≤ 0. Thus, from the above equation,

0 ≤ `(D0) − `(D) ≤ 1

Remark 1.1.2. Note that this proof also shows that i(D) − i(D + [P ]) is either 0 or 1.

Definition 1.1.3. Let (P1,P2,P3,...) an infinite sequence of points on C. Define a sequence of divisors Dm on C as follows: set D0 = 0, and for m ≥ 1, define

m X Dm = [Pi] i=1

We say that an integer m is a Noether gap w.r.t. the sequence (Pi) if `(Dm)−`(Dm−1) = 0, 0 0 i.e., if H (C,Dm) = H (C,Dm−1). We say m is a Weierstrass gap at a point P on C if it is a Noether gap w.r.t. to the constant sequence P = P1 = P2 = ... = Pi = .... Note that m is a Weierstrass gap at P if and only if there does not exist a morphism ϕ : C → P1 of degree m with ϕ−1(∞) = {P }.

Theorem 1.1.4. Any sequence of points on C gives rise to precisely g Noether gaps, and they are all less than 2g.

Proof. Let (P1,P2,...) be a sequence of points on C,(D0,D1,D2,...) the associated

sequence of divisors, and let AN be the number of Noether gaps ≤ N. For every m ≥ 1, we know from the above lemma that either `(Dm) − `(Dm−1) = 0, in which case m is a 15 1.1. The gap theorems

Noether gap, or `(Dm) − `(Dm−1) = 1. Thus, for any N ≥ 1, the number of integers ≤ N that are not Noether gaps is

N X (`(Dm) − `(Dm−1)) = `(DN ) − `(D0) m=1

= N + `(K − DN ) − `(K) by Riemann-Roch

= N + `(K − DN ) − g

and so AN = N−(N+`(K−DN )−g) = g−`(K−DN ). Thus, for every N ≥ max(1, 2g−1),

we have AN = g (since deg K = 2g − 2). This shows that there are precisely g Noether gaps and they are all less than 2g.

Remark 1.1.5. One consequence of the above theorem is that any sequence of points on a curve of genus 0 gives rise to no Noether gaps. This is straightforward to verify since every such curve is isomorphic to P1. It is also worth noting that for a curve C of positive genus and any sequence of points on C, we have that 1 is a Noether gap; otherwise, we would obtain an isomorphism from C to P1, which is impossible.

Corollary 1.1.6. Given a point P on C, there are precisely g Weierstrass gaps at P , and they are all less than 2g. Furthermore, if g > 0, then 1 is a Weierstrass gap at P .

From this point on, we will refer to Weierstrass gaps simply as gaps, and to integers that are not gaps as non-gaps. We note that the non-gaps at a point P on C are closed under addition; indeed, i, j are non-gaps if and only if ∃ ϕ, ψ ∈ k(C)× whose divisors have polar parts −i[P ], −i[P ], respectively, in which case ϕψ has polar part −(i + j)[P ], and so i + j is a non-gap. From this, we can easily conclude that if n is a gap, then at least half of the positive integers strictly less than n are also gaps, for if n is a gap and m < n is a non-gap, then n − m must be a gap.

Proposition 1.1.7. Suppose g > 0, and let 1 < α1 < . . . < αg = 2g be the g smallest non-gaps at a point P on C. Deﬁne α0 := 0. Then

(i) αj + αg−j ≥ αg = 2g for 0 ≤ j ≤ g;

(ii) if α1 = 2, then αj = 2j for 0 ≤ j ≤ g; Chapter 1. A na¨ıve approach to Weierstrass points 16

(iii) αj + αg−j = 2g for 0 ≤ j ≤ g if and only if α1 = 2; Pg (iv) j=1 αj ≥ g(g + 1), with equality if and only if α1 = 2.

Proof. Clearly, α0 + αg−0 = 0 + αg = αg. Now suppose αj + αg−j < αg for some j strictly between 0 and g. Then, since non-gaps are closed under addition, α1 +αg−j, α2 +

αg−j, . . . , αj + αg−j are j non-gaps that are strictly between αg−j and αg. But there are only j − 1 non-gaps strictly between αg−j and αg, namely, αg−j+1, . . . , αg−1. This proves (i).

If α1 = 2, then, by closure of non-gaps under addition, α1 < 2α1 < . . . < gα1 are g non-gaps that are ≤ 2g, hence (ii). In this case, for 0 ≤ j ≤ g, αj +αg−j = 2j +2(g −j) = 2g.

Now suppose αj + αg−j = 2g for 0 ≤ j ≤ g . We prove by strong induction that αj is a multiple of α1 for each j, which is only possible if α1 = 2. Clearly, α0 = 0 and α1 are multiples of α1. Now suppose α0, α1, . . . , αj are all multiples of α1 for some j between 1 and g − 1. We have

αg−(j+1) = 2g − αj+1 < (2g − αj+1) + αj = 2g − (αj+1 − αj) < 2g = αg

Since (2g − αj+1) + αj = αg−(j+1) + αj is a non-gap, by the above inequality it must be one of

αg−j = 2g − αj, αg−(j−1) = 2g − αj−1, . . . , αg−1 = 2g − α1

By the induction hypothesis, this implies that αj+1 is a multiple of α1. It remains to prove (iv). We have

g g g X X X 2 αj = (αj + αg−j) ≥ 2g by (i) j=0 j=0 j=0 = 2g(g + 1)

By (iii), the above inequality is an identity if α1 = 2 and a strict inequality otherwise.

Finally, we note that α0 = 0 to obtain statement (iv) on the nose. 17 1.2. Weierstrass points and wronskians

1.2 Weierstrass points and wronskians

We are now in a position to introduce the notion of a Weierstrass point on C. Throughout this section, we assume g > 0. The discussion in this section is modelled after that in [FK80, pp. 81-86].

Deﬁnition 1.2.1. Let 1 = n1 < . . . < ng ≤ 2g − 1 be the g gaps at a point P on C. We deﬁne g X τ(P ) := (nj − j) j=1

(Note that τ(P ) ≥ 0.) We call τ(P ) the weight of the point P . If P has positive weight, we call P a classical Weierstrass point or simply a Weierstrass point.

Remark 1.2.2. Note that a curve of genus 1 has no Weierstrass points.

Proposition 1.2.3. Suppose g ≥ 2. For any point P on C, the following are equivalent: (i) P is a Weierstrass point; (ii) at least one of the integers 2, . . . , g is a non-gap; (iii) `(g[P ]) ≥ 2; (iv) i(g[P ]) > 0.

Proof. First, we recall that 1 is necessarily a gap. The equivalence between (i) and (ii) is clear. Now, (ii) holds if and only if there is a function in k(C) with a pole at P of order ≤ g, that is, if and only if there is a non-constant function f ∈ k(C) s.t. div(f) ≥ −g[P ]. This shows that (ii) and (iii) are equivalent. The equivalence of (iii) and (iv) follows directly from the Riemann-Roch Theorem.

Now ﬁx a point P on C and let t ∈ k(C) be a uniformizer at P . Let Ωk(C)/k be the

module of relative diﬀerentials of k(C) over k and denote by d : k(C) → Ωk(C)/k the associated derivation. Note that d is simply the map on stalks at the generic point of C

1 induced by the canonical map OC → ΩC . Chapter 1. A na¨ıve approach to Weierstrass points 18

Deﬁnition 1.2.4. Given a sequence of functions ϕ1, . . . , ϕn ∈ k(C), we deﬁne

ϕ1 . . . ϕn

dϕ1 dϕn ... dt dt Wt(ϕ1, . . . , ϕn) := . . . (1.1) . .. .

dn−1ϕ dn−1ϕ 1 ... n dtn−1 dtn−1 (For any function f ∈ k(C), we write df/dt to denote the unique function h ∈ k(C) s.t. df = h · dt.) We call wt(ϕ1, . . . , ϕn) the wronskian of ϕ1, . . . , ϕn with respect to t. From now on, when the context is clear, we will denote the “derivative” df/dt of a function f ∈ k(C) with respect to t simply as f 0. Note that, from the properties of the determinant, for any f ∈ k(C) we have

n Wt(fϕ1, . . . , fϕn) = f wt(ϕ1, . . . , ϕn) (1.2)

(See [FK80, pp. 82-83] for details.) Also note that for any aij ∈ k,

Wt(a11ϕ1 + ... + an1ϕn, . . . , a1nϕ1 + ... + annϕn) = det(aij)Wt(ϕ1, . . . , ϕn) (1.3)

In particular, if ϕ1, . . . , ϕn and a11ϕ1+...+an1ϕn, . . . , a1nϕ1+...+annϕn are both linearly independent over k, then the wronskian of the latter is a non-zero constant multiple of the wronskian of the former.

× Lemma 1.2.5. Let ϕ1, . . . , ϕn ∈ k(C) . Denote ordP ϕi by µi and suppose

µ1 < µ2 < . . . < µn

Suppose further that we have char(k) - µj − µ1 for j = 1, . . . n. Then the wronskian

Φ := Wt(ϕ1, . . . , ϕn) is non-zero, and n X ordP Φ = (µj − j + 1) j=1 Proof. We proceed by induction on n. The result is clearly true for n = 1. Now assume it is true for n = k. From the above, we have

k+1 Wt(ϕ1, . . . , ϕk+1) = ϕ1 Wt(1, ϕ2/ϕ1, . . . , ϕk+1/ϕ1)

k+1 0 0 = ϕ1 Wt((ϕ2/ϕ1) ,..., (ϕk+1/ϕ1) ) 19 1.2. Weierstrass points and wronskians

where the second identity is obtained by expanding the determinant Wt(1, ϕ2/ϕ1, . . . , ϕk+1/ϕ1) along the ﬁrst column. Note that ordP (ϕj/ϕ1) = µj − µ1, and so

0 < ordP (ϕ2/ϕ1) < . . . < ordP (ϕk+1/ϕk)

0 In particular, by the condition on char(k), this implies that ordP ((ϕj/ϕ1) ) = µj − µ1 − 1 and

0 0 ordP ((ϕ2/ϕ1) ) < . . . < ordP ((ϕk+1/ϕk) )

Thus, the induction hypothesis gives

k k+1 0 0 X ordP ϕ1 Wt((ϕ2/ϕ1) ,..., (ϕk+1/ϕ1) ) = (k + 1)µ1 + ((µj+1 − µ1 − 1) − j + 1) j=1 k X = µ1 + (µj+1 − (j + 1) + 1) j=1 k+1 X = (µj − j + 1) j=1

Now, following the notation in [FK80], let H(C) ⊂ Ωk(C)/k denote the space of holo- 0 1 1 morphic diﬀerentials on C, that is, the space H (C, ΩC ) of global sections of ΩC . This

is a k-vector space of dimension g. We can always construct a basis ω1, . . . , ωg for H(C) s.t.

ordP ω1 < . . . < ordP ωg

If a basis for H(C) has the above property, we say it is adapted to P . In the following lemma, we show that if we have such a basis ω1, . . . , ωg, then ordP ωj = nj − 1, where

1 = n1 < . . . < ng ≤ 2g − 1 are the g Weierstrass gaps at P .

Lemma 1.2.6. Let 1 = n1 < . . . < ng ≤ 2g − 1 be the g gaps at P . Then

{ordP ω : ω ∈ H(C)\{0}} = {n1 − 1, . . . , ng − 1}

In particular, there is a holomorphic diﬀerential ω on C that does not vanish at P (i.e., ordP ω = 0). Chapter 1. A na¨ıve approach to Weierstrass points 20

Proof. By Lemma 1.1.1, n is a gap at P if and only if i((n − 1)[P ]) − i(n[P ]) = 1, that

is, if and only if there is a diﬀerential ω ∈ H(C)\{0} s.t. ordP ω = n − 1.

Definition 1.2.7. Given a differential ω ∈ Ωk(C)/k, let ω/dt = f where f ∈ k(C) is the unique function satisfying ω = f · dt. For any sequence of differentials ω1, . . . , ωn ∈ ΩC , we define

Wt(ω1, . . . , ωn) := Wt(ω1/dt, . . . , ωn/dt)

We call Wt(ω1, . . . , ωn) the wronskian of ω1, . . . , ωn with respect to t. It is easy to see that the wronskian for diﬀerentials inherits properties 1.2 and 1.3 from the wronskian for functions.

Lemma 1.2.8. Let t, t˜ ∈ k(C) uniformizers at points P , P˜ on C, respectively. For any sequence of diﬀerentials ω1, . . . , ωn ∈ Ωk(C)/k, we have

dt˜n(n+1)/2 W (ω , . . . , ω ) = W (ω , . . . , ω ) t 1 n t˜ 1 n dt

Proof. Let ϕi = dωi/dt andϕ ˜i = dωi/dt˜ for i = 1, . . . , n. By the properties of the determinant, we have

ϕ˜1 ... ϕ˜n dϕ˜ dϕ˜ Pn j 1 n n(n+1)/2 dt˜ j=1 ... dt˜ dt˜ dt˜ Wt(ω1, . . . , ωn) = . . . = Wt˜(ω1, . . . , ωn) dt . .. . dt

dn−1ϕ˜ dn−1ϕ˜ 1 ... n dt˜n−1 dt˜n−1

Deﬁnition 1.2.9. Let C be a curve, and consider the k(C)-vector space

⊗m Ωk(C)/k = Ωk(C)/k ⊗k(C) ... ⊗k(C) Ωk(C)/k | {z } m times

1 ⊗m (Note that this is the stalk of the invertible sheaf (ΩC ) at the generic point of C.) We ⊗m call an element of this space an m-diﬀerential on C. Since Ωk(C)/k is one-dimensional 21 1.2. Weierstrass points and wronskians

over k(C), we have that every m-diﬀerential ζ on C is of the form ω1 ⊗ ... ⊗ ωm for some

ω1, . . . , ωm ∈ Ωk(C)/k. Assuming ζ is non-zero, we may deﬁne

m X ordP ζ := ordP ωi i=1 In fact, for any uniformizer t at P , there is a unique function f ∈ k(C) s.t.

f(dt)m := f dt ⊗ dt ⊗ ... ⊗ dt = ζ | {z } m times and so m X ordP ωi = ordP f i=1 Qm The function f = i=1(dωi/dt).

We associate to ζ a divisor (ζ) on C as follows:

m X X div(ζ) = (ordP ζ)[P ] = div(ωi) P ∈C i=1

Note that deg((ζ)) = m(2g − 2).

Deﬁnition 1.2.10. Let ω1, . . . , ωn ∈ Ωk(C)/k, and set m = n(n + 1)/2. We deﬁne the

wronskian of ω1, . . . , ωn to be the m-diﬀerential

m W (ω1, . . . , ωn) := Wt(ω1, . . . , ωn)(dt)

where t is a uniformizer at any point P on C.

Remark 1.2.11. Note that the wronskian is independent of the choice of point and uniformizer. Indeed if t˜ is a uniformizer at a point P˜ on C, by Lemma 1.2.8, we have

dt˜m W (ω , . . . , ω )(dt)m = W (ω , . . . , ω ) (dt)m = W (ω , . . . , ω )(dt˜)m t 1 n t˜ 1 n dt t˜ 1 n

Lemma 1.2.12. Suppose that char(k) = 0 or char(k) > 2g − 2. Then for any basis

ω1, . . . , ωg of H(C), the wronskian W (ω1, . . . , ωg) is non-zero, and for any point P on C, we have

ordP (W (ω1, . . . , ωg)) = τ(P ) Chapter 1. A na¨ıve approach to Weierstrass points 22

Proof. Let 1 = n1 < . . . < ng ≤ 2g − 1 be the g gaps at P , and let t be any uniformizer

at P . By property (2), we may assume that the basis ω1, . . . , ωg is adapted to P , i.e., that for j = 1, . . . , g, we have

ordP (ωj/dt) = nj − 1

Since nj −n1 = nj −1 ≤ ng −1 ≤ (2g−1)−1 = 2g−2 for j = 1, . . . , g and char(k) > 2g−2, we can apply Lemma 1.2.5 to obtain ordP (W (ω1, . . . , ωg)) = ordP (Wt(ω1, . . . , ωg)) = ordP (Wt(ω1/dt, . . . , ωg/dt)) g X = (ordP (ωj/dt) − j + 1) j=1 g g X X = ((nj − 1) − j + 1) = (nj − j) = τ(P ) j=1 j=1

Proposition 1.2.13. Suppose char(k) = 0 or char(k) > 2g − 2. Then X τ(P ) = (g − 1)g(g + 1) = g3 − g P ∈C In particular, C has at most g3 − g Weierstrass points.

Proof. For each point P on C, let 1 = nP,1 < . . . < nP,g ≤ 2g − 1 be the g gaps at P ,

let ω1, . . . , ωg be any basis for H(C), and let m = g(g + 1)/2. Since W (ω1, . . . , ωg) is an m-diﬀerential, we have

3 deg((W (ω1, . . . , ωg))) = m(2g − 2) = g − g

On the other hand, by Lemma 1.2.12, g X X X X deg((W (ω1, . . . , ωg))) = ordP (W (ω1, . . . , ωg)) = (nP,j − j) = τ(P ) P ∈C P ∈C j=1 P ∈C

Theorem 1.2.14. For any point P on C, g(g − 1) τ(P ) ≤ 2 with equality if and only if the smallest non-gap at P is 2. 23 1.3. Weierstrass points on hyperelliptic curves

Proof. Let 1 < α1, . . . , αg = 2g be the g smallest non-gaps and 1 = n1 < . . . < ng ≤ 2g−1 the g gaps at P . Then we have g g g 2g g ! g X X X X X X τ(P ) = (nj − j) = nj − j = j − αj − j j=1 j=1 j=1 j=1 j=1 j=1 2g g X X = j − αj j=g+1 j=1 g g(3g + 1) X = − α 2 j j=1 g(3g + 1) g(g − 1) ≤ − g(g + 1) = 2 2 The inequality follows from Prop. 1.1.7 (iv), and so does the fact that equality holds if

and only if α1 = 2.

Corollary 1.2.15. If g ≥ 2, and char(k) = 0 or char(k) > 2g − 2, then C has at least 2g + 2 and at most g3 − g Weierstrass points.

1.3 Weierstrass points on hyperelliptic curves

In this section, we discuss the situation when C is hyperelliptic, that is, when we have a P1 morphism ϕ : C → k of degree 2. The discussion here is based on [FK80, pp. 99-102] and [SS15, Prop. 11,12]. Throughout the section, we assume g ≥ 2, and that char(k) = 0 or char(k) > 2g − 2.

Proposition 1.3.1. The following are equivalent: (i) C is hyperelliptic. (ii) C has precisely 2g + 2 Weierstrass points. (iii) At every Weierstrass point on C, the gaps are 1, 3,..., 2g − 1. (iv) At some Weierstrass point on C, the gaps are 1, 3,..., 2g − 1. Furthermore, in this case, the ramiﬁcation points of any double cover C → P1 are precisely the Weierstrass points of C.

Proof. Suppose that C is hyperelliptic, i.e., that there is a map ϕ : C → P1 of degree 2. Since, under the given conditions, char(k) does not divide 2, we can apply Hurwitz’ genus Chapter 1. A na¨ıve approach to Weierstrass points 24

formula (see, for example, [Har77, IV Cor. 2.4]) to obtain that the map ϕ is ramified over precisely 2g + 2 points on C, each with ramification index 2. Take any such point P . Let ψ : P1 → P1 be an automorphism mapping ϕ(P ) to ∞. Thenϕ ˜ = ψ ◦ ϕ is a morphism of degree 2 withϕ ˜−1(∞) = {P }. This shows that the smallest non-gap at any ramification point is 2, and so every ramification point is a Weierstrass point of weight g(g − 1)/2 by Theorem 1.2.14. Thus, if we sum the weights of all 2g + 2 ramification points, we obtain g3 − g. By Prop. 1.2.13, this means that there are no more Weierstrass points, so we have shown that (i) implies (ii). Now suppose C has precisely 2g + 2 Weierstrass points. Then it follows directly from Prop. 1.2.13 and Theorem 1.2.14 that each Weierstrass point P has weight g(g − 1)/2, and so, by Theorem 1.2.14 again, the smallest non-gap at each such P is 2. By Prop. 1.1.7, this is equivalent to saying that the g smallest non-gaps at P are 2, 4,..., 2g, i.e., that the g gaps at P are 1, 3,..., 2g − 1. So we have shown (ii) implies (iii). It is clear that (iii) implies (iv). Now suppose that at some point P on C, the gaps are 1, 3,..., 2g − 1. Then 2 is a non-gap; in particular, there is a morphism ϕ : C → P1 of degree 2. So we have shown (iv) implies (i).

Corollary 1.3.2. Let P be a Weierstrass point on C, and suppose we have a double cover ϕ : C → P1. Let Q, R be the pre-images of a single point in P1 under ϕ (we allow Q = R). Then [Q] + [R] ∼ 2[P ], where ∼ denotes linear equivalence.

Proof. We may assume without loss of generality that Q, R are the pre-images of ∞. By the previous proposition, P is a ramification point of ϕ. Thus, if ϕ(P ) = ∞, then Q = R = P , and we are done. Otherwise, we note that if we define f = ϕ − ϕ(P ), then 2[P ] = [Q] + [R] + (f), again using the fact that P is a ramification point of ϕ.

Lemma 1.3.3. If C is hyperelliptic, then the double cover C → P1 is unique up to automorphisms on P1.

Proof. Let ϕ, ϕ˜ : C → P1 be two double covers. We would like to show thatϕ ˜ = ψ ◦ ϕ, where ψ is an automorphism on P1. Let [P ] + [Q], [P˜] + [Q˜] be the polar parts of (ϕ), (ϕ ˜), respectively. Then, by the preceding corollary, we have [P ]+[Q] ∼ [P˜]+[Q˜]. This implies 25 1.3. Weierstrass points on hyperelliptic curves

that there is some function h ∈ k(C)× such that multiplication by h is a k-vector space isomorphism H0(C, [P ] + [Q]) → H0(C, [P˜] + [Q˜])

By Lemma 1.1.1 (and the fact that 1 is a gap at every point), we have that {1, ϕ} and {1, ϕ˜} are bases for H0(C, [P ] + [Q]),H0(C, [P˜] + [Q˜]), respectively. Thus, we may write

ϕ˜ = aϕh + bh

1 = cϕh + dh

for some a, b, c, d ∈ k s.t. ad 6= bc, and so we have

ϕ˜ aϕh + bh aϕ + b ϕ˜ = = = 1 cϕh + dh cϕ + d at + b Thus,ϕ ˜ = ψ ◦ ϕ, where ψ : P1 → P1 is the automorphism mapping t ∈ P1 to . ct + d Lemma 1.3.4. A non-trivial automorphism of C ﬁxes at most 2g + 2 points.

Proof. Let θ be an automorphism on C that is not the identity. Then there is a point P on C that is not ﬁxed by θ. Let m ∈ {1, 2, . . . , g + 1} be a non-gap at P (there must be a non-gap among these integers by Corollary 1.1.6). This means there is a morphism ϕ : C → P1 with a single pole of order m at P . Now let ψ = ϕ − ϕ ◦ θ. Every ﬁxed point of θ is a zero of ψ. The poles of ψ are P and θ−1(P ), and each pole is of order m. Thus, ψ has at most 2m ≤ 2(g + 1) = 2g + 2 zeroes.

Remark 1.3.5. Note that Lemma 1.3.4 holds regardless of the characteristic of k.

Proposition 1.3.6. Let Aut(C) be the group of automorphisms on C, and Σ the permutation group on the set of Weierstrass points of C. The group homomorphism

λ : Aut(C) → Σ

induced by the action of Aut(C) on the Weierstrass points of C is injective if and only if C is not hyperelliptic. If C is hyperelliptic, ker(λ) = {Id, ι}, where ι is the hyperelliptic involution on C. Chapter 1. A na¨ıve approach to Weierstrass points 26

Proof. By Prop. 1.3.1, if C is not hyperelliptic, then it has more than 2g + 2 Weierstrass points, and so no non-identity automorphism on C fixes all of them by the preceding lemma. Thus, in this case, ker(λ) = {Id}. Now suppose C is hyperelliptic. Take a double cover ϕ : C → P1. Since the Weierstrass points of C are the ramification points of ϕ, the hyperelliptic involution fixes them. Now suppose we have an automorphism θ on C that fixes all 2g + 2 Weierstrass points of C. By Lemma 1.3.3, ϕ = ψ ◦ ϕ ◦ θ, where ψ is an automorphism on P1. But, by the given, ψ fixes 2g + 2 ≥ 6 points of P1, and thus it must be the identity. Hence, ϕ = ϕ ◦ θ, and so θ must be the identity or the hyperelliptic involution.

1.4 Higher Weierstrass points

In this section, we brieﬂy discuss higher Weierstrass points. The discussion here is based on [FK80, pp. 84-85]. For the rest of this section, we ﬁx a positive integer q and a canonical divisor K on C, and we suppose g > 0.

Denote by iq(D) the dimension of

0 1 ⊗q −1 ∼ n ⊗q o H (C, (ΩC ) ⊗ O(D) ) = ξ ∈ Ωk(C)/k : div(ξ) ≥ D

Note that

iq(D) = `(qK − D)

Denote by Hq(C) the space of holomorphic q-diﬀerentials on C,

q 0 1 ⊗q H (C) = H (C, (ΩC ) )

q and let r = dimk H (C).

Deﬁnition 1.4.1. Fix a point P on C. A positive integer n is called a q-gap at P if ` n[P ] − (q − 1)K − ` (n − 1)[P ] − (q − 1)K = 0 (1.4)

Remark 1.4.2. Note that, by Lemma 1.1.1, the left-hand side of 1.4 is either 0 or 1. If it is 1, we say n is a non-q-gap. Furthermore, by the Riemann-Roch Theorem, we have ` n[P ] − (q − 1)K − ` (n − 1)[P ] − (q − 1)K = 1 + iq(n[P ]) − iq((n − 1)[P ]) 27 1.4. Higher Weierstrass points

so n is a q-gap if and only if iq((n − 1)[P ]) − iq(n[P ]) = 1, i.e., if and only if there is a

holomorphic q-diﬀerential ζ with ordP ζ = n − 1. A 1-gap at P is, of course, simply a (Weierstrass) gap at P .

Proposition 1.4.3. There are precisely r q-gaps at P . Furthermore, 1 is a q-gap and all q-gaps are less than 2(q(g − 1) + 1).

Proof. By Lemma 1.2.6, there is a holomorphic diﬀerential ω ∈ H(C) s.t. ordP ω = 0. ⊗q Now let ζ = ω . ζ is a holomorphic q-diﬀerential with ordP ζ = 0. By the above remarks, this means that 0 + 1 = 1 is a q-gap at P .

For N ≥ 1, let AN be the number of q-gaps ≤ N. The number of non-q-gaps ≤ N is

N X ` n[P ] − (q − 1)K − ` (n − 1)[P ] − (q − 1)K = N + iq(N[P ]) − iq(0) n=1

= N + iq(N[P ]) − r

and so

AN = N − (N + iq(N[P ]) − r) = r − iq(N[P ]) = r − `(qK − N[P ])

Now we note that for N ≥ q(2g − 2) + 1, we have deg(qK − N[P ]) < 0, and so for such N we have `(qK − N[P ]) = 0. This shows that there are precisely r q-gaps and they are all less than q(2g − 2) + 2 = 2(q(g − 1) + 1).

Deﬁnition 1.4.4. Let 1 = n1 < . . . < nr < 2(q(g − 1) + 1) be the r q-gaps at P . We deﬁne r X τq(P ) := (nj − j) j=1

(Note that τq(P ) ≥ 0.) We call τq(P ) the q-weight of the point P . If P has positive q-weight, we call it a q-Weierstrass point.

⊗q Definition 1.4.5. Let t be a uniformizer at P , and ζ1, . . . , ζr ∈ Ωk(C)/k a sequence of q-differentials on C. For i = 1, . . . , r , let ϕi ∈ k(C) be the unique function satisfying q ζi = ϕi(dt) . We define

Wt(ζ1, . . . , ζr) := Wt(ϕ1, . . . , ϕr) Chapter 1. A na¨ıve approach to Weierstrass points 28

We call Wt(ζ1, . . . , ζr) the wronskian of ζ1, . . . , ζn with respect to t. The wronskian for q-diﬀerentials inherits properties 1.2 and 1.3 from the wronskian for functions, and so we have the following result.

Proposition 1.4.6. Let each of t, t˜ ∈ k(C) be a uniformizer at some point on C. For ⊗q any sequence ζ1, . . . , ζr ∈ Ωk(C)/k of q-diﬀerentials on C, we have

dt˜r(2q−1+r)/2 W (ζ , . . . , ζ ) = W (ζ , . . . , ζ ) t 1 r t˜ 1 r dt

Definition 1.4.7. Let m = r(r + 2q − 1)/2. We define the wronskian W (ζ1, . . . , ζr) of a ⊗q m sequence ζ1, . . . , ζr ∈ Ωk(C)/k of q-differentials to be the m-differential Wt(ζ1, . . . , ζr)(dt) , where t is a uniformizer at some point on C. This definition is independent of the choice of point and uniformizer, since, by the above proposition, we have

dt˜m W (ζ , . . . , ζ )(dt)m = W (ζ , . . . , ζ ) (dt)m = W (ζ , . . . , ζ )(dt˜)m t 1 r t˜ 1 r dt t˜ 1 r for any other uniformizer t˜.

The proof of the following proposition is identical to that of Prop. 1.2.13, so we omit it.

Proposition 1.4.8. Suppose that char(k) = 0 or char(k) > 2q(g − 1). Then for any

q basis ζ1, . . . , ζr of H (C), the Wronskian W (ζ1, . . . , ζr) is non-zero, and

ordP (W (ζ1, . . . , ζr)) = τq(P ) for any point P on C.

Corollary 1.4.9. Let C, g, d, k be as in the above proposition. Then

X τq(P ) = (g − 1)r(r + 2q − 1) P ∈C

In particular, C has at most (g − 1)r(r + 2q − 1) q-Weierstrass points, and the curves that do not have q-Weierstrass points are the curves of genus 1. Chapter 2

Laksov’s Weierstrass points on curves

While the definition of a Weierstrass point that we provided in the previous chapter is perhaps the most natural one from a classical perspective, it presents some problems in low characteristic. Pathologies may occur if the base field of a curve C is of characteristic smaller than 2g − 2, where g is the genus of C. It is possible for such a curve to have infinitely many Weierstrass points; indeed, in the following example, we provide a curve with the property that each of its points is a Weierstrass point in the na¨ıve sense.

Example 2.0.1. A point P on a smooth plane curve C is called a flex if the tangent line to C at P intersects C at P with multiplicity at least 3. In the case that C is a quartic curve, the flexes of C are precisely its Weierstrass points. Indeed, a smooth plane quartic C is a curve of genus 3, so by Prop. 1.2.3, we have that P is a Weierstrass point if and only if there exists a canonical divisor K on C such that K ≥ 3[P ]. As C is canonically embedded, this is equivalent to saying that P is a flex, since the effective P P2 canonical divisors on C are the divisors of the form P iP (L, C) where L is a line in and iP (L, C) is the intersection multiplicity of L with C at P . Now consider the smooth plane quartic C, known as the Klein quartic, given by

X3Y + Y 3Z + Z3X = 0

29 Chapter 2. Laksov’s Weierstrass points on curves 30

over a field k of characteristic 3. This curve has the remarkable property that each of its points is a flex. We verify this. By symmetry, it suffices to show that every point on the

3 3 affine curve C1 given by x y + y + x = 0 is an inflection point in characteristic 3. We note that this curve intersects the line x = 0 solely at the origin with multiplicity 3. Now ˜ let P = (α, β) be a point on C1 with α 6= 0, and let C1 be the affine curve given by

f(x, y) := ((y + x) + α)3 −α−3(y − x) + β + −α−3(y − x) + β3 + ((y + x) + α) = 0

This curve is the image of C1 under a linear change of coordinates. The tangent line to ˜ C1 at the origin is x = 0, and the intersection multiplicity at P of C1 with the tangent ˜ line to C1 at P is equal to the intersection multiplicity at the origin of C1 with the line x = 0. In characteristic 3, we have

f(x, y) = y3 −α−3y + β − α−9 + xg(x, y)

for some g ∈ k[x, y], and so this multiplicity is at least 3.

As the above example illustrates, we will need a different definition of a Weierstrass point if we would like finiteness to be preserved in low characteristic. In this chapter, we expose Laksov’s definition of a Weierstrass point. Under his definition, a curve C has finitely many Weierstrass points regardless of the characteristic of the base field k (and their weighted number is a constant that depends only on the genus of C). Furthermore, Laksov associates Weierstrass points to any effective divisor D on C of positive degree; these are precisely the na¨ıve Weierstrass points defined in the previous chapter when char(k) > 2g − 2 and D is a canonical divisor.

2.1 The bundle of principal parts

Let C be a non-singular curve of genus g over an algebraically closed ﬁeld k and D an eﬀective divisor on C of positive degree d and projective dimension r, so that h0(C,D) = r + 1. Note that if D is a canonical divisor, then g ≥ 2 by these assumptions. All points in this chapter are assumed to be closed, unless otherwise stated. 31 2.1. The bundle of principal parts

If F is an OC -module and x is a point of C, we denote the ﬁber of F at x by F(x).

This is the k-vector space Fx ⊗OC,x k, where Fx and OC,x are the stalks of F and OC at x.

If V is a k-vector space, denote by VC the OC -module V ⊗k OC . This is the pullback of V (thought of as an OSpec(k)-module) via the morphism C → Spec(k).

Denote by I the sheaf of ideals defining the diagonal in C ×k C. For m ≥ −1, denote th by ∆m the m infinitesimal neighbourhood of the diagonal; this is the subscheme of m+1 0 C ×k C defined by the sheaf of ideals I . We set I = OC×C .

Denote by p and q the projections of C ×k C onto the ﬁrst and second coordinates, respectively. The short exact sequence

m+1 0 → I → OC×C → O∆m → 0

∗ of OC×C -modules remains exact after tensoring over OC×C by q O(D). After applying

p∗ we obtain a long exact sequence

m+1 ∗ ∗ ∗ 0 → p∗(I ⊗ q O(D)) → p∗q O(D) → p∗(O∆m ⊗ q O(D))

1 m+1 ∗ 1 ∗ → R p∗(I ⊗ q O(D)) → R p∗q O(D) → 0 (2.1)

Here we have zero to the right because the restriction of p to the closed subscheme ∆m

of C ×k C is aﬃne (topologically, it is a homeomorphism).

∗ th Deﬁnition 2.1.1. We call the OC -module p∗(O∆m ⊗ q O(D)) the sheaf of m order principal parts of D and denote it by Pm(D).

For each m ≥ 0, we have an exact sequence

m m+1 0 → I /I → O∆m → O∆m−1 → 0 (2.2)

∗ Applying p∗(· ⊗ q O(D)) to the above exact sequence, we obtain the exact sequence

1 ⊗m m m−1 0 → ΩC ⊗ O(D) → P (D) → P (D) → 0 (2.3) Chapter 2. Laksov’s Weierstrass points on curves 32

of OC -modules. Indeed, we have

m m+1 ∗ ∼ ∗ m m+1 ∗ p∗ I /I ⊗ q O(D) = ∆ I /I ⊗ q O(D) (2.4)

∼= ∆∗ Im/Im+1 ⊗ ∆∗q∗O(D)

∼= ∆∗ Im/Im+1 ⊗ (q∆)∗O(D)

∼ 1 ⊗m = ΩC ⊗ O(D)

We justify the existence of isomorphism (2.4). We can cover C with open aﬃne subsets

m m+1 ∗ Ui = SpecAi, where Ai is a k-algebra, such that the restriction of I /I ⊗ q O(D) to

Ui ×k Ui is the sheaf associated to some Ai ⊗k Ai-module Mi annihilated by the kernel Ii of

the multiplication map Ai ⊗k Ai → Ai. Now (2.4) follows from the fact that under these

conditions, as shown in the following lemma, we have compatible Ai-module isomorphisms ∼ Mi ⊗Bi Ai −→ Mi.

Lemma 2.1.2. Let A be a k-algebra, B = A ⊗k A, and I the kernel of the multiplication

map B → A. Let M be a B-module with the property that IM = 0. Then M and M ⊗B A are naturally isomorphic as A-modules, where M is viewed as an A-module via the map A → B, a 7→ a ⊗ 1.

Proof. We have natural B-module isomorphisms

∼ ∼ ∼ M ⊗B A −→ M ⊗B (B/I) −→ M/IM −→ M

where A is viewed as a B-module via the multiplication map B → A. These maps are all A-linear, hence the statement in the lemma.

We now return to sequence (2.3). The exactness of (2.2) is preserved after applying

∗ ∗ p∗(· ⊗ q O(D)) since q O(D) is a locally free sheaf and the sheaves in (2.2) are supported on the diagonal. By induction and the fact that P0(D) = O(D), we see that Pm(D) is a

locally free OC -module of rank m + 1.

i ∗ i By ﬂat base change ([Har77, III Prop. 9.3]), we have R p∗q O(D) = H (C,D)C . m 0 m m m Let v : H (C,D)C → P (D) be the map deﬁned by (2.1), and let B (D) and V (D) m be the image and cokernel of v . Since it is a subsheaf of the locally free OC -module Pm(D), we have that Bm(D) is also locally free (see, for example, [lP97, Lemma 5.2.1]). 33 2.1. The bundle of principal parts

Since the maps vm are compatible with the maps Pm(D) → Pm−1(D), we obtain a commutative diagram with exact rows

0 −−−→ Bm(D) −−−→ Pm(D) −−−→ Vm(D) −−−→ 0       y y y (2.5) 0 −−−→ Bm−1(D) −−−→ Pm−1(D) −−−→ Vm−1(D) −−−→ 0

where the vertical maps are surjective. Moreover, we obtain from (2.1) a commutative diagram of OC -modules

m 1 m+1 ∗ 1 0 −−−→ V (D) −−−→ R p∗(I ⊗ q O(D)) −−−→ H (C,D)C −−−→ 0     y y (2.6) m−1 1 m ∗ 1 0 −−−→ V (D) −−−→ R p∗(I ⊗ q O(D)) −−−→ H (C,D)C −−−→ 0

Since C is a curve, we have an isomorphism

1 m+1 ∗ ∼ 1 m+1 ∗ R p∗(I ⊗ q O(D))(x) −→ H (C × C)x, (I ⊗ q O(D))x

of k-vector spaces for each point x of X, where (C ×C)x = C is the ﬁber of q above x and m+1 ∗ m+1 ∗ (I ⊗q O(D))x is the pullback of I ⊗q O(D) to that ﬁber ([Har77, III 11.2,12.11]). Furthermore, we have

m+1 ∗ ∼ (I ⊗ q O(D))x = O(D − (m + 1)x)

(see [LT94, p. 413]) and so we obtain from diagram (2.6) a commutative diagram of k-vector spaces

0 −−−→ Vm(D)(x) −−−→ H1(C,D − (m + 1)x) −−−→ H1(C,D) −−−→ 0     y y (2.7) 0 −−−→ Vm−1(D)(x) −−−→ H1(C,D − mx) −−−→ H1(C,D) −−−→ 0

1 Here the horizontal sequences are exact because H (C,D)C is free and so

OC,x 0 Tor1 H (C,D)C,x, k = 0 Chapter 2. Laksov’s Weierstrass points on curves 34

Proposition 2.1.3 ([Lak81, Prop. 1]). There are integers 0 = b0 < b1 < . . . < br ≤ d s.t.  m + 1 bm ≤ j < bm+1, 0 ≤ m < r rk Bj(D) =

r + 1 j ≥ br

Proof. It follows directly from the Riemann-Roch Theorem that, for m ≥ d,

h1(C,D − (m + 1)x) = g − d + m

1 m+1 ∗ In particular, the function x 7→ dimk R p∗(I ⊗ q O(D))(x) is constant, and so (by 1 m+1 ∗ [Har77, III 12.8, 12.9]) the sheaf R p∗(I ⊗ q O(D)) is locally free on X. Thus, when m ≥ d, the sheaf Vm(D) is locally free of rank g − d + m − h1(C,D) = m − r, and so

rk Bm(D) = (m + 1) − (m − r) = r + 1

for such m. Now the ring of global sections of B0(D) is the full ring of global sections of the line bundle P0(D) = O(D), and this is not trivial since D is eﬀective. Thus, B0(D) is of rank 1, and so for any m the rank of Bm(D) lies between 1 and r + 1. From diagram (2.5) we see that rk Bm+1(D) − rk Bm(D) ≤ 1 since rk Pm+1(D) − rk Pm(D) = 1. Thus, in

0 1 d the chain B (D) ← B (D) ← ... B (D) there are exactly r jumps b1, . . . , br in the ranks, each jump increasing the rank by 1.

Deﬁnition 2.1.4. The sequence b0 < b1 < . . . < br of integers in Corollary 2.1.3 is called the generic gap sequence of D. For each m = 0, 1, . . . , r, we denote Vbm (D) by Wm(D) and call it the mth Weierstrass module of D.

Remark 2.1.5. Let D0 be a divisor on C linearly equivalent to D. Then we have a canonical isomorphism O(D) ∼= O(D0) from which we obtain isomorphisms Bm(D) ∼= Bm(D0) for each m. Thus, the generic gap sequence of a divisor depends only on the linear equivalence class of that divisor. 35 2.2. Weierstrass points

2.2 Weierstrass points

By inspecting diagram (2.7), we deduce the following proposition.

Proposition 2.2.1 ([Lak81, Prop. 2]). Fix an integer m ≥ 0 and a closed point x of C. TFAE: (i) the canonical surjection H1(C,D − (m + 1)x) → H1(C,D − mx) is a surjection; (ii) the surjection Vm(D)(x) → Vm−1(D)(x) of diagram (2.7) is an isomorphism; (iii) the kernel of the map Pm(D)(x) → Pm−1(D)(x) is contained in the image of vm(x), i.e., the image of Bm(D)(x) in Pm(D)(x).

Deﬁnition 2.2.2. An integer m + 1 ≥ 1 satisfying the three equivalent conditions of Prop. 2.2.1 is called a gap of D at x.

Remark 2.2.3. We note that, from the above deﬁnition, an integer is a gap of D at x if and only if that integer is a gap of D0 at x for every D0 linearly equivalent to D.

Remark 2.2.4. As previously noted, when m ≥ d, we have h1(C,D−(m+1)x) = g−d+m for any point x by the Riemann-Roch theorem, and so if m + 1 is a gap at x then 1 ≤ m + 1 ≤ d + 1. Since we know that h1(C,D − (m + 1)x) − h1(C,D − mx) ≤ 1, the number of gaps at x is equal to

d + 1 − (h1(C,D − (d + 1)x) − h1(C,D)) = d + 1 − (g − (g − d + r)) = r + 1

We denote them by

1 ≤ g1(x) < g2(x) < . . . < gr+1(x) ≤ d + 1

Deﬁnition 2.2.5. If gi+1(x) = bi(x) + 1 for i = 0, . . . , r, we call x a D-ordinary point. A point that is not D-ordinary is called a D-Weierstrass point. If the divisor D is not speciﬁed, it is assumed that D is a canonical divisor.

Remark 2.2.6. We note that, by Remarks 2.1.5 and 2.2.3, the Weierstrass points associated to a divisor depend only on the linear equivalence class of that divisor.

Proposition 2.2.7 ([Lak81, Prop. 3]). The following assertions hold: Chapter 2. Laksov’s Weierstrass points on curves 36

(i) gi+1(x) ≥ bi + 1 for i = 0, . . . , r and for all points x of C; m (ii) a point x of C is D-Weierstrass if and only if dim W (D)(x) > bm − m for some m = 0, . . . , r.

Proof. (i) Fix a point x of C and let 0 ≤ i ≤ r. By the deﬁnition of b0, . . . , br, we have

rk Bbi−1(D) = i. Since tensoring by k is right exact, by passing to ﬁbers from (2.5) we get

bi−1 bi−1 bi−1 dim V (D)(x) ≥ rk P (D)(x) − rk B (D)(x) = ((bi − 1) + 1) − i = bi − i

Now let j be the unique integer satisfying gj(x) ≤ bi < gj+1(x). By the deﬁnition of a gap, we have

bi−1 dim V (D)(x) = ((bi − 1) + 1) − j = bi − j

It follows that i ≥ j, and so gi+1(x) ≥ gj+1(x) > bi. (ii) By the deﬁnition of a gap, for any point x of C and m = 0, . . . , r, we have

m dim W (D)(x) = bm + 1 − j, where j is the unique integer satisfying gj(x) ≤

bm + 1 < gj+1(x).

m Now suppose that for some m = 0, . . . , r, we have dim W (D)(x) > bm − m. Then

m ≥ j, and so gm+1(x) ≥ gj+1(x) > bm + 1, and so x is a Weierstrass point.

Conversely, if x is a Weierstrass point then for some m = 0, . . . , r we have

gm+1(x) > bm + 1 ≥ gj(x)

This implies that m + 1 > j and so

m dim W (D)(x) = bm + 1 − j > bm − m

2.3 The wronskian

In this section, we introduce a map of invertible modules on C whose zeroes are precisely the Weierstrass points of C. 37 2.3. The wronskian

Lemma 2.3.1. Let A be a commutative ring, and

0 → L → M → N → 0

a short exact sequence of A-modules, where L, N are free of ranks rL, rN , respectively. Then there is a canonical isomorphism

rL ! rN ! rL+rN ^ ^ ∼ ^ L ⊗ N −→ M

Proof. Deﬁne a map as follows

(l1 ∧ ... ∧ lrL ) ⊗ (n1 ∧ ... ∧ nrN ) 7→ l1 ∧ ... ∧ lrL ∧ n˜1 ∧ ... ∧ n˜rN

wheren ˜i is any lift of ni. We show that this is map is well-deﬁned. If the lj do not form

a basis of L, then the element on the left gets mapped to 0 regardless of the choice ofn ˜i.

If the lj do form a basis of L, then eachn ˜i diﬀers from any other lift of ni by a linear

combination of the lj, and so the element on the right is again independent of the choice

ofn ˜i.

Now to verify that this map is an isomorphism, note that if l1, . . . , lrL is a basis for

L and n1, . . . , nrN is a basis for N, then l1, . . . , lrL , n˜1,..., n˜rN is a basis for M, so that

VrL+rN l1 ∧ ... ∧ lrL ∧ n˜1 ∧ ... ∧ n˜rN is a basis for M.

Corollary 2.3.2. Let X be a ringed space, and

0 → F → G → H → 0

a short exact sequence of OX -modules, where F, H are free of ranks rF , rH, respectively. Then there is an isomorphism

rF ! rH ! rF +rH ^ ^ ∼ ^ F ⊗ H −→ G of line bundles on X.

Theorem 2.3.3 ([Lak84, Cor. 8]). There is a canonical map

r+1 ^ 0 1 ⊗(b0+b1+...+br) ⊗(r+1) w : H (C,D)C → ΩC ⊗ O(D) such that w(x) = 0 if and only if x is a D-Weierstrass point. Chapter 2. Laksov’s Weierstrass points on curves 38

Proof. Deﬁne b−1 = −1. From sequence (2.3) and diagram (2.5), we obtain for each 0 ≤ m ≤ r a commutative diagram

1 ⊗bm bm bm−1 0 −−−→ (ΩC ) ⊗ O(D) −−−→ P (D) −−−→ P (D) −−−→ 0 x x   (2.8)  

bm bm−1 0 −−−→ Km −−−→ B (D) −−−→ B (D) −−−→ 0

bm bm−1 with exact rows, where Km is the kernel of the map B (D) → B (D). Note that

bm bm−1 rk B (D) = m + 1 and rk B (D) = m, and so Km is a line bundle. Furthermore, the maps in the sequence

Bbm−1(D) → Bbm−2(D) → ... → Bbm−1 (D) are surjections of locally free sheaves of equal rank, and so they are all isomorphisms.

By the commutativity of the right square in diagram (2.8), the map Pbm (D) →

bm−1 1 ⊗bm P (D) is trivial on Km, from which we get an inclusion Km → (ΩC ) ⊗ O(D). We thus obtain the following commutative diagram

1 ⊗bm bm bm−1 0 −−−→ (ΩC ) ⊗ O(D) −−−→ P (D) −−−→ P (D) −−−→ 0 x x x    (2.9)   

bm bm−1 0 −−−→ Km −−−→ B (D) −−−→ B (D) −−−→ 0

Vm bm−1 The bottom row of (2.9) provides us with an isomorphism B (D) ⊗ Km →

Vm+1 bm B (D). Composing the inverse of this isomorphism with the inclusion Km →

1 ⊗bm (ΩC ) ⊗ O(D) gives us a map

m+1 m ! ^ bm ^ bm−1 1 ⊗bm vm : B (D) → B (D) ⊗ ΩC ⊗ O(D)

The induced map vm(x) on the ﬁbers is the zero map if and only if the same is true

1 ⊗bm 1 ⊗bm of the map Km(x) → (ΩC ) ⊗ O(D) (x). Since the map Km → (ΩC ) ⊗ O(D) is 1 ⊗bm an inclusion of line bundles, the map Km(x) → (ΩC ) ⊗ O(D) (x) is trivial for only

ﬁnitely many x, and so we have in particular that vm is non-zero. 39 2.3. The wronskian

From (2.9), we get a commutative diagram 1 ⊗bm bm bm−1 0 −−−→ (ΩC ) ⊗ O(D) (x) −−−→ P (D)(x) −−−→ P (D)(x) −−−→ 0 x x x      

bm bm−1 0 −−−→ Km(x) −−−→ B (D)(x) −−−→ B (D)(x) −−−→ 0 (2.10)

where the rows are exact since Pbm−1(D), Bbm−1 (D) are locally free. We observe from (2.10) that if the vertical map to the right is injective, then the vertical map in the

1 ⊗bm middle is injective if and only if the map Km(x) → (ΩC ) ⊗ O(D) (x), and hence

vm(x), is trivial. To say that the right vertical map is injective is equivalent to saying

bm−1 bm−1 dim P (D)(x)/im B (D)(x) = bm − m

which is guaranteed if

m−1 bm−1 bm−1 dim W (x) = dim P (D)(x)/im B (D)(x) = bm−1 − (m − 1)

via the commutativity of the diagram

Pbm−1(D)(x) Pbm−1 (D)(x)

Bbm−1 (D)(x) To say that the middle vertical map in (2.10) is injective is equivalent to saying

m bm bm dim W (x) = dim P (D)(x)/im B (D)(x) = bm − m

m−1 In summary, we have that if dim W (x) = bm−1 − (m − 1), then

m vm(x) = 0 ⇔ dim W (x) > bm − m

Now for m = 0, 1, . . . , r, deﬁne maps

m+1 ! m ! Pr Pr ^ bm 1 i=m+1 bi r−m ^ bm−1 1 i=m bi r+1−m um : B (D) ⊗ ΩC ⊗O(D) → B (D) ⊗ ΩC ⊗O(D) by Pr 1 i=m+1 bi r−m um = vm ⊗ identity map on ΩC ⊗ O(D) Chapter 2. Laksov’s Weierstrass points on curves 40

Vr+1 0 and let w be the map u0u1 . . . ur pre-composed with the isomorphism H (C,D)C →

Vr+1 br B (D). Since the vm’s are non-zero, so are the um’s, and hence so is w (indeed, the

um’s are injective since they are non-zero maps between line bundles). Now if x is not a Weierstrass point, then by Prop. 2.2.7(ii), we have dim Wm(x) =

bm − m for m = 0, 1, . . . , r. Thus, by the above, v1(x), . . . , vr(x) are non-zero, and hence so is w(x).

m Conversely, if x is a Weierstrass point, then dim W (x) > bm − m for some m =

0, 1, . . . , r, and so, by the above, vm(x) = 0, which implies w(x) = 0.

Deﬁnition 2.3.4. The canonical section of the line bundle

r+1 !−1 ^ 0 1 ⊗(b0+...+br) ⊗(r+1) H (C,D)C ⊗ ΩC ⊗ O(D) given by the map w described in Theorem 2.3.3 is called the wronskian of the divisor D. We will abuse notation and denote this section by w as well. We denote the order to which w vanishes at a point x by d(x) and call it the weight of the point x.

Theorem 2.3.5 ([Lak81, Thm. 6]). A curve C has ﬁnitely many D-Weierstrass points and the sum of their weights is

r X (2g − 2) bi + (r + 1)d i=0

2.4 Local computations

m 0 m In this section, we express the maps v : H (C,D)C → P (D) in local coordinates. Fix

a point x of C. Let R = OC,x be the local ring of C at x, let t ∈ R be a uniformizer at

x, and let I be the kernel of the multiplication map R ⊗k R → R. We have

m ∼ m+1 P (D)x = (R ⊗k R)/I

m+1 as R-modules, where we view (R ⊗k R)/I as an R-module via the projection map p,

i.e., via the map R → R ⊗k R, f 7→ f ⊗ 1. For each m ≥ 0, by the exact sequence

m m+1 m+1 m 0 → I /I → (R ⊗k R)/I → (R ⊗k R)/I → 0 41 2.4. Local computations

and the fact that Im/Im+1 is a free R-module with basis (1 ⊗ t − t ⊗ 1)m, we have that

m+1 m (R ⊗k R)/I is a free R-module with basis 1 ⊗ 1, (1 ⊗ t − t ⊗ 1),..., (1 ⊗ t − t ⊗ 1) . i m+1 i We will denote (1 ⊗ t − t ⊗ 1) ∈ (R ⊗k R)/I by (dt) . m m+1 Denote by dR : R → (R ⊗k R)/I the map induced by the projection q. In other m words, dR f is the class of the element 1⊗f. For each element f ∈ R and i = 0, . . . , m, we i i m m denote by d f the coeﬃcient of (dt) in the expression of dR f in the basis 1, dt, . . . , (dt) . Note that for any f ∈ R

1 ⊗ f = f ⊗ 1 + (1 ⊗ f − f ⊗ 1) and so, since 1 ⊗ f − f ⊗ 1 ∈ I, we have d0f = f. Furthermore,

1 ⊗ t = t ⊗ 1 + (1 ⊗ t − t ⊗ 1) = t + dt and so for any h ≥ 0, h h X h dmth = (dmt) = th−i(dt)i (2.11) R R i i=0 i h h h−i In particular, we have that d t = i t . Note that this is true for any i = 0, . . . , m, even when i > h.

Lemma 2.4.1 ([Lak81, Lemma 8]). Let f ∈ R, h ≥ 0, and 0 ≤ i ≤ m, and suppose

h h+1 i h h−i h−i+1 f ≡ at mod t , with a ∈ k. Then d f ≡ a i t mod t . Proof. We have f = ath + th+1g for some g ∈ R, and so h dif = a th−i + di th+1g i i i h+1 h−i+1 by (2.11) and the k-linearity of d . It suﬃces to show that d t g = t g1 for some g1 ∈ R. Note that m i m h+1 m h+1 m X X i−j h+1 j i dR t g = dR t (dR g) = d t d g (dt) i=0 j=0 where the second equality follows from the fact that (dt)i = 0 for i > m. We thus have i i X X h + 1 di th+1g = di−jth+1 djg = th−j+1 · djg i − j j=0 j=0 i X h + 1 = th−i+1 ti−j · djg i − j j=0 Chapter 2. Laksov’s Weierstrass points on curves 42

∼ Fix an isomorphism OC (D)x −→OC,x, and let f0, f1, . . . , fr ∈ OC,x be the image 0 0 ∼ of a basis e0, . . . , er of H (C,D) under the map H (C,D) → OC (D)x −→OC,x. Let

hj = ordxfj. The ej can be chosen s.t. h0 < h1 < . . . < hr.

Deﬁnition 2.4.2. We call the local invariants h0, h1, . . . , hr the Hermite invariants of D

at x. For our convenience, we deﬁne hr+1 = ∞.

With respect to the latter choice of basis of H0(C,D) and the basis 1, dt, . . . , (dt)m of

m m 0 m P (D)x, the map vx : H (C,D) ⊗k R → P (D)x is represented by the (m + 1) × (r + 1) th i matrix whose (i, j) entry is d fj. (Note that this matrix has a zeroth row and zeroth column.)

hj For j = 0, . . . , r, we have fj ≡ ajt for some non-zero aj ∈ k. Hence, by Lemma 2.4.1, hj dif ≡ a thj −i mod thj −i+1 (2.12) j j i for i = 0, . . . , m.

Now let jm be the unique integer satisfying hjm ≤ m < hjm+1. Then, by 2.12, at t = 0 i the matrix (d fj) takes the form             a0       ∗     .   .       ∗ a1    (2.13)    ∗ ∗     . .   . .       ∗ ∗ . . . ajm       ∗ ∗ ... ∗     . . .. .   . . . .    ∗ ∗ ... ∗

th where the empty space represents zeroes and aj is in the hj row for j = 0, 1, . . . , jm. 43 2.5. The classical situation

Theorem 2.4.3 ([Lak84, Thm. 10]). Fix a point x on C and let h0 < h1 < . . . < hr be the Hermite invariants of D at x. The following assertions hold: (i) for all integers j = 0, 1, . . . , r, we have dim(im vm(x)) = j + 1 for all integers m

satisfying hj ≤ m < hj+1; m m−1 m (ii) the inclusion ker(P (D)(x) → P (D)(x)) ⊆ imv (x) holds if and only if m = hj for some j = 0, 1, . . . , r;

(iii) gi+1(x) = hi + 1 for i = 0, 1, . . . , r;

(iv) for all but ﬁnitely many points x on C, we have bi = hi for i = 0, 1, . . . , r.

Proof. With respect to the chosen bases for H0(C,D) and Pm(D)(x), the map vm(x) is represented by the matrix 2.13, from which assertions (i) and (ii) follow immediately. Indeed, with respect to the chosen bases, the map Pm(D)(x) → Pm−1(D)(x) is simply the projection onto the ﬁrst m − 1 coordinates. We obtain assertion (iii) from (ii) by Prop. 2.2.1iii, and (iv) follows from (iii) by Theorem 2.3.5.

2.5 The classical situation

In this section, we show that if the characteristic of the ground ﬁeld k is equal to 0 or larger than the degree d of the divisor D, then the generic gap sequence of D is the classical generic gap sequence 0, 1, . . . , r. Thus, in this context, if D is a canonical divisor then the D-Weierstrass points on C are precisely the na¨ıve Weierstrass points deﬁned in the previous chapter.

Lemma 2.5.1 ([Lak81, Lemma 9]). Let M be the (r + 1) × (r + 1) matrix with entries in

th xj k[x0, x1, . . . , xr] whose (i, j) entry is i . (Note that this matrix has a zeroth row and zeroth column.) We have

r ! Y Y (i!) det M = (xi − xj) i=0 0≤j

Proof. Note that the expression on the left is the determinant of the matrix M˜ obtained

xj ˜ from M by scaling each entry i so that it is monic in xj. Now, det M is clearly an alternating polynomial and as such it is a multiple of the Vandermonde polynomial Chapter 2. Laksov’s Weierstrass points on curves 44

Q r+1 0≤j

Corollary 2.5.2 ([Lak81, Thm. 10 (i)]). With the notation used in Section 2.4, for all

r points x on C the determinant of the map vx satisﬁes

r ! r ! Y r Y Y h h+1 (i!) det vx ≡ ai (hi − hj) t mod t i=0 i=0 0≤j

Pr where h = i=0(hi − i).

Proof. This follows directly from 2.12, Lemma 2.5.1, and the observation that for any Pr Pr permutation σ of 0, 1, . . . , r, we have i=0(hσ(i) − i) = i=0(hi − i) = h.

Theorem 2.5.3 ([Lak81, Thm. 11 (i)]). Let p = char(k) and assume that p = 0 or p > d.

Then bi = i for i = 0, 1, . . . , r.

Proof. As noted in Remark 2.2.4, for all points x on C we have gi(x) ≤ d + 1 for i =

1, . . . , r + 1, and so it follows from Theorem 2.4.3 that hi ≤ d for i = 0, . . . , r, where

h0, . . . , hr are the Hermite invariants of D at x. Moreover, from the Riemann-Roch Theorem, we have g − h0(C,K − D) = d − r (2.14)

where K is a canonical divisor. Since g = h0(C,K) and D is an eﬀective divisor, the

r diﬀerence on the left is nonnegative, and hence r ≤ d. Thus, by Corollary 2.5.2, det vx is r non-zero and so rk B (D) = r. This is only possible if bi = i for i = 0, . . . , r. Chapter 3

Weierstrass points in families

In the previous chapter, we presented Laksov’s theory of Weierstrass points and gap sequences associated to a divisor on a single smooth curve over an algebraically closed ﬁeld. In Section 3.1, we survey Laksov and Thorup’s generalization of this theory to smooth families of curves over arbitrary base schemes. While the availability of this theory seems to suggest that Weierstrass points behave somewhat “regularly” in families—for example, that Weierstrass points reduce to Weierstrass points mod p—we show in Section 3.2 that this hypothesis may fail rather dramatically. Indeed, we show that, while the Weierstrass points of the Klein quartic reduce injectively mod 3, none of the points thus obtained is a Weierstrass point of the reduction.

3.1 Weierstrass points on schemes

Fix a noetherian scheme X. If F is an OX -module and x is a point of X, denote by κ(x) the residue ﬁeld at x and, as in the previous chapter, denote by F(x) the ﬁber

F ⊗OX,x κ(x) of F at x.

If v : G → F is a map of OX -modules, denote by v(x): G(x) → F(x) the induced κ(x)-linear map on the ﬁbers. For the rest of this chapter, unless otherwise stated, a module is assumed to be a coherent OX -module and a locally free module is assumed to be of ﬁnite rank.

45 Chapter 3. Weierstrass points in families 46

Deﬁnition 3.1.1. The rank of a map v : W → Q of modules is the largest integer r s.t. Vr v 6= 0.

If Q is locally free and X is a smooth projective curve, then the image of v is locally free and the rank of v is equal to the rank of its image. In any case, rk v is bounded above by the largest integer t such that Vt W 6= 0. In the case that W is locally free, t is simply the rank of W. The following two lemmas relate the rank of a map v of modules to the rank of the κ(x)-linear map induced by v on the ﬁbers at a point x of X.

Lemma 3.1.2 ([LT94, Lemma 1.4]). Let v : W → Q be a map of modules, x a point of X. If Q is locally free, then rk v ≥ rk v(x) (3.1)

and equality holds if and only if in a neighbourhood of x, the cokernel of v is free and and the image of v is free of rank equal to rk v(x).

Lemma 3.1.3 ([LT94, Lemma 1.5]). Suppose we have a commutative diagram

v u q Q P

where q is a surjection of locally free modules whose kernel is an invertible module. Then the following inequalities hold

rk v ≥ rk u ≥ rk v − 1 (3.2)

rk v(x) ≥ rk u(x) ≥ rk v(x) − 1 (3.3) where x is any point of X. Moreover, the following are equivalent: (i) the ﬁrst inequality of 3.3 is strict; (ii) equality holds in the second inequality of 3.3; (iii) the map of ﬁbers Coker v(x) → Coker u(x) at x induced by the diagram is an isomorphism. 47 3.1. Weierstrass points on schemes

Now assume that X is a smooth family of curves over a base scheme S, that is, assume we have a proper smooth morphism f : X → S whose geometric ﬁbers are curves. Fix

∗ an invertible OX -module L, and let W = f f∗L.

Deﬁnition 3.1.4. Denote by Pm(L) the sheaf of ith order principal parts of L, deﬁned

∗ as p∗(O∆m ⊗ q L) where p and q are the two projections X ×S X → X and ∆m is the th subscheme of X ×S X deﬁned by the (m + 1) power of the ideal sheaf I of the diagonal.

The short exact sequence

m+1 0 → I → OX×S X → O∆m → 0 (3.4)

∗ of OX×S X -modules remains exact after tensoring with q L. Applying p∗ gives us an exact sequence

∗ m 1 m+1 ∗ 1 ∗ p∗q L → P (L) → R p∗(I ⊗ q L) → R p∗q L → 0 (3.5)

that is exact to the right because the restriction of p to the closed subscheme ∆m is a topological homeomorphism.

∗ Apply p∗(· ⊗ q L) to the exact sequence

m m+1 0 → I /I → O∆m → O∆m−1 → 0 (3.6)

to obtain the exact sequence

1 ⊗m m m−1 0 → (ΩS/X ) ⊗ L → P (L) → P (L) → 0 (3.7)

As P0(F) = F, it follows by induction that Pm(L) is locally free of rank m + 1. (See Section 2.1 or [Per95, pp. 3219-3220] for clariﬁcation on some of the above claims.) We thus obtain for every m a commutative diagram

W vm vm+1 (3.8) Pm(L) Pm−1(L)

∗ as in Lemma 3.1.3, where vm is the composition of the base change map W = f f∗L → ∗ p∗q L with the ﬁrst map in 3.5. Note that the base change map is an isomorphism since we are assuming f is ﬂat ([Har77, III 9.3]). Chapter 3. Weierstrass points in families 48

By Lemma 3.1.3, we have the following inequalities

rk vm+1 ≥ rk vm ≥ rk vm+1 − 1 (3.9)

rk vm+1(x) ≥ rk vm(x) ≥ rk vm+1(x) − 1 (3.10)

for any point x of X, and the following are equivalent: (i) the ﬁrst inequality of 3.10 is strict; (ii) equality holds in the second equality of 3.10;

(iii) the map Em+1(L)(x) → Em(L)(x) induced by the commutative diagram 3.8 is an isomorphism,

where Em(L) is the cokernel of the map vm.

Deﬁnition 3.1.5. An integer m + 1 ≥ 1 is called a gap of L at x if the above three

th equivalent conditions hold. The s gap at x is denoted gs(x). An integer m + 1 ≥ 1 is th called a generic gap of L if rk vm+1 > rk vm. Denote by gs the s generic gap.

The following two propositions relate the number of generic gaps to the number of gaps at a point.

Proposition 3.1.6 ([LT94, Prop. 2.4]). The number of generic gaps is finite and equal to the rank of vm for all sufficiently large m. Analogously, for any point x, the number of gaps at x is finite and equal to rk vm(x) for all sufficiently large m. The number of gaps at x is at most equal to the number of generic gaps, and for any s ≥ 1, we have

gs(x) ≥ gs

Proposition 3.1.7 ([LT94, Prop. 2.5]). Let x be a point of X. The following are equivalent: (i) The number of gaps at x is equal to the number of generic gaps.

(ii) For all suﬃciently big m, there is a neighbourhood of x over which the cokernel Em

is free and the image of vm is free of rank equal to rk vm. Let U be the set of points x at which the above two conditions are satisﬁed. Then U is an open subset of X, and for suﬃciently large m, the cokernel Em is locally free and the image of vm is locally free of rank equal to rk vm. 49 3.1. Weierstrass points on schemes

If, moreover, W is locally free of rank r, then the following are equivalent: (iii) The number of gaps at x is equal to r.

(iv) The map vm(x) is injective for some m. Furthermore, if there exists a point in X satisfying (ii) and (iii), then the four conditions are equivalent and they are satisﬁed at every point in X.

Finally, suppose W is locally free. If the map vm is injective for some m, then the above four conditions are equivalent and they hold for every associated point of X. Conversely,

if condition (iv) holds for every associated point of X, then the map vm is injective for m suﬃciently large.

Remark 3.1.8. Recall that a point x is an associated point of X if every element of mx

is a zero divisor, where mx is the maximal ideal of the stalk OX,x. Note that, since we are taking X to be noetherian, X certainly contains associated points. Indeed, the generic point of each irreducible component of X is an associated point of X in this setting.

We are now in a position to associate Weierstrass points to the module L. As we did with curves, we will deﬁne a point on X to be Weierstrass if the gap sequence at that point is “strange”.

Deﬁnition 3.1.9. A point x of X is called an L-Weierstrass point if the sequence of gaps at x is diﬀerent from the sequence of generic gaps.

Remark 3.1.10. Note that the gap sequences, and hence the Weierstrass points, associated to the invertible module L depend only on the isomorphism class of L.

The following theorem tells us that we can interpret L-Weierstrass points as the zeroes of a global wronskian, in complete analogy to Section 2.3.

Theorem 3.1.11 ([LT94, Thm. 3.1]). There is a canonical map

r ^ 1 ⊗(g1+g2+...+gr−r) ⊗r w : W → ΩX/S ⊗ L (3.11)

where r is the number of generic gaps, with the property that a point x of X belongs to the zero scheme Z(w) of w if and only if x is an L-Weierstrass point. Chapter 3. Weierstrass points in families 50

Deﬁnition 3.1.12. The map w is called the wronskian of L.

Remark 3.1.13. The above discussion indeed generalizes the theory of Weierstrass points on curves over algebraically closed fields. If we take S = Spec(k) for some algebraically closed field k and let D be a divisor on X, we find that Pm(O(D)), as defined in 3.1.4,

m 0 is simply P (D) as deﬁned in 2.1.1. In this case, W is the free module H (C,D)C and m 0 m vm+1 is the map v : H (C,D)C → P (D) discussed in the previous chapter. Setting 0 r = h (C,D), we associated a sequence of r generic gaps b0, b1, . . . , br−1 to X in Prop.

2.1.3. For i = 0, 1, . . . , r − 1, we have gi+1 = bi + 1. This chapter’s definition of the gap sequence g1(x), . . . , gr(x) at a point x is consistent with the previous definition, although the previous notion was only defined for closed points x of X. Finally, the wronskian w of O(D) defined above is the wronskian of D in Theorem 2.3.3.

3.2 Reductions of the Klein quartic

In this section, we re-examine the so-called Klein quartic introduced at the beginning of the last chapter. It is the genus 3 plane curve C given by the equation

X3Y + Y 3Z + Z3X = 0 (3.12)

We assume that this curve is nonsingular over the ground ﬁeld k (this is certainly true for k = C, and when char(k) = 2, 3, as we will see). Over C, the curve C is isomorphic to the modular curve X(7) and the automorphism

group G of C is isomorphic to PSL2(F7); it is the simple group of order 168 ([Elk99]). G is generated by the following transformations of the projective plane:       ζ − ζ6 ζ2 − ζ5 ζ4 − ζ3 0 1 0 ζ4 0 0        2 5 4 3 6     2  T2 = ζ − ζ ζ − ζ ζ − ζ  ,T3 = 0 0 1 ,T7 =  0 ζ 0 (3.13)       ζ4 − ζ3 ζ − ζ6 ζ2 − ζ5 1 0 0 0 0 ζ

2πi/7 where ζ = e . Note that the orders of T2,T3,T7 are 2, 3, 7, respectively. For the remainder of this section, let P, Q, R be the points (0 : 0 : 1), (0 : 1 : 0), (1 : 0 : P2 0) in the projective plane k, respectively, and let D be the canonical divisor 3[Q] + [R]. 51 3.2. Reductions of the Klein quartic

P2 We note that D is indeed a canonical divisor on C since C is canonically embedded in k and the line Z = 0 intersects C only at Q and R with multiplicities 3 and 1, respectively.

Y X 0 Moreover, the functions 1, Z , Z constitute a basis of H (C,D). Indeed, we have Y div = [P ] − 3[Q] + 2[R] Z X div = 3[P ] − 2[Q] − [R] Z In Example 2.0.1, we showed that if the characteristic of the ground ﬁeld k is 3, then every point on C is a Weierstrass point in the na¨ıve sense. We make this more precise with Prop. 3.2.4, but ﬁrst we examine the case char(k) = 2.

Lemma 3.2.1. The Klein quartic C (given by equation 3.12) is nonsingular in characteristic 2.

Proof. Setting f(X,Y,Z) = X3Y + Y 3Z + Z3X, we have ∂f ∂f ∂f = X2Y + Z3, = Y 2Z + X3, = Z2X + Y 3 (3.14) ∂X ∂Y ∂Z since char(k) = 2. We would like to show that these partials all vanish at a point (X,Y,Z) satisfying f(X,Y,Z) = 0 if and only if X = Y = Z = 0. If, say, X = 0, then the condition that the partials all vanish implies that Y = Z = 0, and similarly for the other two variables. Thus, we may assume XYZ 6= 0. In this case, we obtain from 3.14 and f(X,Y,Z) = 0 that ∂f Y 3Z ∂f Z3X ∂f X3Y = − , = − , = − ∂X X ∂Y Y ∂Z Z and so none of the partials vanish.

Proposition 3.2.2. If char(k) = 2, then the generic gap sequence (as deﬁned in Def. 2.1.3) of D is 0, 1, 2.

Proof. Let a, b ∈ k such that x := (a : b : 1) is a point on C. We would like to show that, unless a, b satisfy some stringent conditions, the Hermite invariants of D at x are 0, 1, 2. ˜ X ˜ Y ˜ ˜ Let X = Z − a, Y = Z − b. From equation 3.12, we have that X and Y satisfy

X˜ 3Y˜ + bX˜ 3 + aX˜ 2Y˜ + Y˜ 3 + abX˜ 2 + a2X˜Y˜ + bY˜ 2 + (a2b + 1)X˜ + (a3 + b2)Y˜ = 0 (3.15) Chapter 3. Weierstrass points in families 52 and so

X˜(X˜ 2Y˜ + bX˜ 2 + aX˜Y˜ + abX˜ + a2Y˜ + a2b + 1) = Y˜ (Y˜ 2 + bY˜ + a3 + b2) (3.16)

Thus, if we assume a2b 6= 1 and a3 6= b2, then X˜ and Y˜ are related by a unit. Since X,˜ Y˜ ˜ certainly generate the maximal ideal of the local ring at x, we conclude that ordxX = ˜ 2 ˜ 3 2 ˜ ˜ 0 ordxY = 1. Now let f = (a b + 1)X + (a + b )Y . Then 1, X, f form a basis of H (C,D). Furthermore, from 3.15, we have

f = X˜ 3Y˜ + bX˜ 3 + aX˜ 2Y˜ + Y˜ 3 + abX˜ 2 + a2X˜Y˜ + bY˜ 2 (3.17)

and so certainly ordxf ≥ 2. Thus, if we can show that ordxf is almost always exactly 2, then we are done. Now, from (3.16), we have

u2 · a2X˜Y˜ = X˜ 2(a7b + a4b3 + a5 + a2b2 + H.O.T.)

u2 · bY˜ 2 = X˜ 2(a4b3 + b + H.O.T.) where u = Y˜ 2 + bY˜ + a3 + b2, and certainly

u · abX˜ 2 = X˜ 2(a7b + ab5 + H.O.T.)

Hence, u2 abX˜ 2 + a2X˜Y˜ + bY˜ 2 = X˜ 2(ab5 + a5 + a2b2 + b + H.O.T.) (3.18)

Since we are assuming u is a unit, it follows that unless ab5 + a5 + a2b2 + b = 0, we have

˜ 2 2 ˜ ˜ ˜ 2 ˜ 2 ordxf = ordx abX + a XY + bY = ordx(X ) = 2

Note that it follows from the above proposition that P = (0 : 0 : 1) is a Weierstrass

Y X point in characteristic 2. Indeed, by examining the orders of 1, Z , Z at P , we see that the Hermite invariants of D at P are 0, 1, 3. This fact leads to the following proposition.

Proposition 3.2.3. In characteristic 2, the Weierstrass points of C are its F8-rational points. 53 3.2. Reductions of the Klein quartic

Proof. It follows from the splitting behaviour of 2 in Q(ζ) that all the automorphisms

3.13 in G are deﬁned over F8. Since P is F8-rational and Weierstrass, any point in its orbit

under G will also have these two properties. As it is known that C has 24 F8-rational points ([Elk99, p. 77]), and since C has at most 24 Weierstrass points by Theorem 2.3.5 and Prop. 3.2.2, it suﬃces to show that P has 24 points in its G-orbit. To that end, we

show that the stabilizer H of P in G is the order-7 cyclic subgroup generated by T7.

We certainly have hT7i ⊆ H. Now, H 6= G since for example T3P 6= P , and so H is contained in a maximal subgroup of G whose order is a multiple of 7. There is only one such maximal subgroup M, and its order is 21 ([DF04, 6.2 Prop. 14]). Now suppose

2 H = M. Then H contains an element h of order 3 that is conjugate to one of T3,T3 . −1 Suppose wlog that T3 = γhγ for some γ ∈ G. Then γP is a ﬁxed point of T3, and so

γP = (1 : 1 : 1), but (1 : 1 : 1) is not on the curve C. Thus, H 6= M, and so H = hT7i.

We now show that the situation in characteristic 3 is considerably diﬀerent. Note that the model of the Klein quartic given by Equation 3.12 remains nonsingular in characteristic 3.

Proposition 3.2.4. If char(k) = 3, then the generic gap sequence of D on C is 0, 1, 3.

Proof. Let a, b ∈ k be nonzero, and let x be the point (a : b : 1). We set out to ﬁnd the Hermite invariants of D at x. ˜ X ˜ Y ˜ ˜ Let X = Z − a, Y = Z − b. By equation 3.12, we have that X and Y satisfy

X˜ 3Y˜ + Y˜ 3 + X˜ + bX˜ 3 + a3Y˜ = 0 (3.19)

and so X˜ 2Y˜ + bX˜ 2 + 1 Y˜ = − X˜ (3.20) Y˜ 2 + a3 Since neither Y˜ 2 +a3 nor X˜ 2Y˜ +bX˜ 2 +1 vanish at x, this shows that X˜ and Y˜ are related by a unit. Since the maximal ideal of the local ring at x is certainly generated by X˜ and ˜ ˜ ˜ Y , we conclude that ordxX = ordxY = 1. Now, from equation 3.19, we have

X˜ + a3Y˜ = −(X˜ 3Y˜ + Y˜ 3 + bX˜ 3) (3.21) Chapter 3. Weierstrass points in families 54

˜ 3 ˜ ˜ ˜ 3 ˜ 0 and so certainly ordx(X + a Y ) ≥ 3. Since 1, X, X + a Y form a basis of H (C,D) by previous considerations, it follows that the Hermite invariants of D at the point x are ˜ 3 ˜ ˜ 3 ˜ ˜ 3 ˜ 3 0, 1, h, where h = ordx(X + a Y ) = ordx(X Y + Y + bX ). Now, from 3.20, we have

X˜ 3(X˜ 2Y˜ + bX˜ 2 + 1)3 X˜ 3(X˜ 6Y˜ 3 + b3X˜ 6 + 1) Y˜ 3 = − = − (Y˜ 2 + a3)3 Y˜ 6 + a9 and so

(Y˜ 6 + a9)(Y˜ 3 + bX˜ 3) = −X˜ 3(X˜ 6Y˜ 3 + b3X˜ 6 + 1) + bX˜ 3(Y˜ 6 + a9)

= −X˜ 9Y˜ 3 − b3X˜ 9 + bX˜ 3Y˜ 6 + (a9b − 1)X˜ 3

˜ 6 9 ˜ 9 ˜ 3 3 ˜ 9 ˜ 3 ˜ 6 Since Y + a is a unit and ordx(−X Y − b X + bX Y ) ≥ 9, it follows from the above that h = 3 unless a9b = 1, in which case h = 4.

Remark 3.2.5. Note that Prop. 3.2.4, together with Theorem 2.3.5, implies that the sum of the weights of the Weierstrass points of C is 28. Moreover, from the proof of Prop. 3.2.4, we see that the Hermite invariants of D at x := (1 : 1 : 1), which is indeed a point on C if char(k) = 3, are 0, 1, 4, and so by Theorem 2.4.3, the point x is Weierstrass in characteristic 3.

Remark 3.2.6. Over C, the Klein quartic C maps to the projective curve given by

X4 + Y 4 + Z4 + 3α(X2Y 2 + X2Z2 + Y 2Z2) = 0 (3.22)

2 under the transformation of PC with matrix   1 1 + ζα ζ2 + ζ6    2 6  1 + ζα ζ + ζ 1  (3.23)   ζ2 + ζ6 1 1 + ζα

√ 2 4 −1+ −7 where α = ζ + ζ + ζ = 2 . Under the change of basis 3.23, the transformations 0 0 0 T2,T3,T7 of the projective plane (see 3.13) correspond to T2,T3,T7, respectively, where       1 0 0 0 1 0 −1 1α ¯       0   0   0   T2 = 0 0 1 ,T3 = 0 0 1 ,T7 =  α α 0  (3.24)       0 1 0 1 0 0 −1 1 −α¯ 55 3.2. Reductions of the Klein quartic

(See [Elk99, pp. 55-56] for details.) Observe that equation 3.22 reduces mod 3 to the curve F given by X4 + Y 4 + Z4 = 0. This is the so-called Fermat quartic, and by the above, it is isomorphic to the Klein quartic in characteristic 3. Furthermore, the transformation 3.23 of the projective plane ﬁxes x = (1 : 1 : 1), and so, by Remark 3.2.5, the point x (considered now as a point on F ) is Weierstrass. This leads us to the following result.

Proposition 3.2.7. In characteristic 3, the Weierstrass points of the Fermat quartic F

are its F9-rational points.

Proof. Since 3 is inert in Q(α), it follows that all the automorphisms 3.24 of F in G are deﬁned over F9. Since the point x := (1 : 1 : 1) is F9-rational and Weierstrass, any point in its orbit under G will also have these two properties. As it is known that F has 28

F9-rational points, and since F has at most 28 Weierstrass points (see Remark 3.2.5), it is enough to show that x has 28 points in its G-orbit. ∼ 0 0 Let H = StabG(x). Certainly, we have S3 = hT2,T3i ⊆ H, and we know H 6= G since 0 T7x 6= x. This shows H is contained in a maximal subgroup of G whose order is a multiple of 6. There is only one such maximal subgroup, and its order is 24 ([DF04, 6.2 Prop. 14]).

Thus, to show that #OrbG(x) = 28, it suﬃces to show that #OrbG(x) > 168/24 = 7. Now, we have   0 1 0   : 02 0 0−2   A = T7 T2T7 = 1 0 0    0 0 −1 and so any point obtained by performing a signed permutation of the homogeneous coordinates of x is in OrbG(x). This provides us with four points (1 : 1 : 1), (1 : −1 : 1), (1 : −1 : −1). Moreover, we have

0 2 T7x = (¯α : 2α : −α¯) = (1 : α : −1) where we abuse notation here and use α, α¯ to denote the reductions of α, α¯ ∈ Z[α] mod

2 3, respectively; that is, α, α¯ ∈ F9 are the roots of the polynomial t + t + 2 over F3 (in 2 particular, αα¯ = 2). Thus, the four points (1 : α : 1) are also in OrbG(x), and so we indeed have #OrbG(x) ≥ 8. Chapter 3. Weierstrass points in families 56

Remark 3.2.8. It is important to note that, while the Weierstrass points of C over C reduce injectively mod 3 ([Elk99, p. 80]), the 24-point set thus obtained (this is precisely the set of cusps of C once we identify it with the modular curve X(7)) is disjoint from the set of C’s Weierstrass points in characteristic 3. Indeed, the group G acts transitively on both of these sets, but we just showed that they are not of the same size.

Proposition 3.2.9. Suppose char(k) = 3, so we may identify the curve C with the modular curve X(7) over k. Then the 28 Weierstrass points of C correspond to the 28 classes of full level 7 structures on the unique elliptic curve E over k with #Aut(E) = 12.

Proof. By the considerations in Remark 3.2.8, a Weierstrass point is not a cusp of C, and so it corresponds to an equivalence class of a pair (E, ξ), where E is an elliptic curve over k and ξ is a full level 7 structure on E; that is, ξ is a basis (T1,T2) of the F7-vector space

E[7] of 7-torsion points on E, satisfying w(T1,T2) = ζ, where w is the Weil pairing and ζ th 0 0 0 is some ﬁxed primitive 7 root of unity in k. Two pairs (E, (T1,T2)) and (E , (T1,T2)) are 0 0 equivalent if and only if there is an isomorphism ϕ : E → E over k such that ϕ(Ti) = Ti . ∼ An element γ ∈ G = PSL(E[7]) acts on a class [(E, (T1,T2))] by sending it to 0 0 [(E, (γT1, γT2))], and so #OrbG([(E, ξ)]) is the number of classes [(E, ξ )], where ξ is a full level 7 structure on E. Thus,  #SL(E[7]) 336 168 #Aut(E) = 2 #Orb ([(E, ξ)]) = = = G #Aut(E) #Aut(E) 28 #Aut(E) = 12

where the ﬁrst equality holds because an automorphism that acts trivially on E[7] must be trivial (indeed, X(7) is a ﬁne moduli space — see, for example, [Zha01, p. 183]). Since the orbit of a Weierstrass point of C has size 28 by the proof of Prop. 3.2.7, a Weierstrass point must correspond to a class [(E, ξ)] with #Aut(E) = 12. There is one such elliptic curve E up to isomorphism in characteristic 3 ([Sil09, III Thm. 10.1]). Conclusion

In Chapter 1, we defined a point on a curve of genus g to be Weierstrass if its gap sequence differed from the classical gap sequence 1, . . . , g. This definition proved to be somewhat crude in low (positive) characteristic, as demonstrated by the example of the Klein quartic. We showed that in characteristic 3, every point of this genus 3 curve has a non-classical gap sequence (Example 2.0.1). However, all but finitely many of its points have the gap sequence 1, 2, 4 (Prop. 3.2.4), and in general, almost all points on a curve share the same gap sequence (Thm. 2.3.5).

We note that, in light of Laksov’s theory, if a curve has infinitely many Weierstrass points in the na¨ıve sense — that is, infinitely many points with non-classical gap sequences — then all of its points are Weierstrass in that sense (so, in retrospect, Example 2.0.1 is not as surprising). Indeed, if a curve C has infinitely many points with non-classical gap sequences, then by Theorem 2.3.5 the generic gap sequence of C cannot be classical. Since by Prop. 2.2.7 the gap sequence at a point must be “greater” than the generic gap sequence, this implies that there is no point of C whose gap sequence is classical.

Laksov distinguishes between a point on a curve whose gap sequence is generic with respect to that curve and a point with a non-generic gap sequence, the latter of which he calls a Weierstrass point. However, if we adopt Laksov’s definition, Weierstrass points do not behave well under reduction, as exemplified by the reduction of the Klein quartic mod 3 (Remark 3.2.8). It would be useful to determine under what conditions Weierstrass points reduce to Weierstrass points modulo primes. This appears to be related to how the wronskian map defined in Theorem 3.1.11 behaves when pulled back to fibers.

Another topic of interest to the author is the action of the automorphism group of

57 58 a curve on that curve’s Weierstrass points, particularly in low characteristic. In Section 1.3, we showed that if char(k) = 0 or char(k) > 2g − 2, where g > 1 is the genus of a curve C over k, then an automorphism that fixes the Weierstrass points of C is either trivial or, in the case that C is hyperelliptic, equal to the hyperelliptic involution of C. This comes from the fact that, under the above conditions on char(k), there is a lower bound for the number of distinct Weierstrass points of C. The situation when 0 < char(k) ≤ 2g − 2 appears to be more complex. In Section 3.2, we showed that the Klein quartic in characteristics 2 and 3 has 24 and 28 distinct Weierstrass points, respectively. Thus, by Lemma 1.3.4, an automorphism of the Klein quartic that fixes its Weierstrass points is indeed trivial if char(k) = 2, 3. However, as discussed in [Lak81, p. 246], in characteristic 2 there are hyperelliptic curves of arbitrarily high genus carrying a single Weierstrass point. Finally, suppose a curve C of genus g > 1 is defined over a number field K. If we accept the analogy between Weierstrass points and N-torsion points on elliptic curves, it might be interesting to examine the smallest extension K(W ) of K over which all the Weierstrass points of C are defined. The field K(W ) is the analogue of the N th division field K(E[N]) of an elliptic curve E over K. It is known, for example, that if p is a prime of K of good reduction for E and whose norm is prime to N, then p is unramified in K(E[N]) ([DT02, Thm. 2.1]). It would be interesting to find a criterion for when primes are unramified in K(W ). Bibliography

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