Research Proposal Chitrabhanu Chaudhuri [email protected]

Research Proposal Chitrabhanu Chaudhuri Chitrabhanu@Gmail.Com

My area of research broadly falls under Complex Algebraic Geometry. Algebraic geometry is a branch of mathematics where one studies the geometry of zero sets of a collection of polynomials over a certain field, (C in my case), using algebraic techniques. These zero sets are called algebraic varieties. More specifically my research concerns Moduli of Algebraic curves. The Moduli Space of algebraic curves has been a classical object of study within Algebraic geometry and has connections with various other fields of mathematics, including but not limited to enumerative geometry [KM94], number theory [KM85], mathematical physics [GK98] and dynamical systems [EMM15]. I am currently pursuing several projects all revolving around the central topic of Moduli of Curves: 1. Arakelov Geometry of Modular curves, 2. Enumerative Geometry of Curves, 3. Topological study of the moduli of curves. I shall describe each of them in a separate section below.

1. Arakelov Geometry of Modular Curves This is ongoing joint work with Debargha Banerjee and Diganta Borah. We are interested in ﬁnding the minimal regular model and stable arithmetic self-intersection of certain modular curves with arithmetic applications in mind.

1.1. Arakelov Goemetry. Let K/Q be a number field and Ok be its ring of integers. An arithmetic surface X over OK is a regular scheme of dimension 2 with a proper map to X → Spec OK , defined in [Liu02]. The algebraic intersection theory doesn’t work very well in this situation since X is not proper. Arakelov [Ara75] introduced an intersection pairing suited to this situation for the special case K = Q. Deligne [Del87] later generalised this intersection theory to arbitrary number fields. Faltings [Fal84] studied this theory extensively, proving results like the Riemmann-Roch theorem, Hodge index theorem and Noether’s formula in this situation. Faltings’s motivation for studying Arakelov intersection pairing was to prove results about arithmetic surfaces (or number fields), that were known for algebraic surfaces (or function fields), in particular Mordell conjecture which is now known as Faltings’s theorem. Although Faltings’s proof of Mordell conjecture did not involve Arakelov theory, but Vojta [Voj91] gave a proof using Arakelov’s pairing. Faltings [Fal91]later proved a generalisation of Mordell conjecture using his results in Arakelov theory. In this section we shall use the notation h·, ·iAr for the Arakelov intersection pairing. 1.2. Modular curves. First let us discuss certain finite index subgroups of SL(2, Z), known as congruence subgroups. Fix a positive integer N called the level and let a b Γ (N) = ∈ SL(2, ) | c ≡ 0 (mod N) 0 c d Z a b Γ (N) = ∈ SL(2, ) | c ≡ 0 (mod N), a, d ≡ 0 (mod N) 1 c d Z 1 0 Γ(N) = A ∈ SL(2, ) | A ≡ (mod N) . Z 0 1 The modular group SL(2, Z) acts on the upper half plane H = {x + iy ∈ C | y > 0} through Möbiustransformations hence so does a subgroup Γ ⊂ SL(2, Z). The quotient Y (Γ) = Γ\H is a 1 2 non-compact Riemann surface. Y (Γ) can be compactified by adding finitely many points called cusps and the compactification is denoted by X(Γ). Being a compact Riemann surface, X(Γ) is an algebraic curve over C. In fact for congruence subgroups X(Γ) is an algebraic curve over Q. We abbreviate X0(N) = X(Γ0(N)), X1(N) = X(Γ1(N)) and X(N) = X(Γ(N)). In fact, X(Γ) are moduli spaces of generalised elliptic curves for congruence subgroups Γ and are closely connected to modular forms for the corresponding subgroups. A modern reference for modular curves and modular forms is [DS05].

1.3. Stable Arithmetic self-intersection. Let X be a smooth projective geometrically connected curve over a number field K. A regular model of X over OK is an arithmetic surface X → Spec OK with generic fiber isomorphic to X. If genus of X is atleast 1, there is a minimal (in the sense of birational geometry) regular model X which is unique upto isomorphism, but not functorial in X. Such a model is called semi-stable if all the scheme theoretic fibers of π : X → Spec OK are reduced and have only nodal singularities. There is a distinguished line bundle on X the relative dualizing sheaf ωX . This line bundle ωX when restricted to any fiber of π is the canonical line bundle of that curve. If X is semi-stable, the quantity

2 hωX , ωX iAr ωX = [K : Q] 0 0 is independent of K in the following sense: let K /K be a ﬁnite extension and X = X ×Spec OK Spec OK0 , then hω 0 , ω 0 i hω , ω i X X Ar = X X Ar . [K0 : Q] [K : Q] This quantity will henceforth be called the stable arithmetic self-intersection of X. The signiﬁcance of this invariant has been shown by Szpiro [Szp90], who showed that the strict positivity of this number is equivalent to Bogomolov’s conjecture.

1.4. Bogomolov’s conjecture. Let X be a smooth projective geometrically connected curve over a number ﬁeld K with genus g > 1. Let J be the Jacobian of X. Fix an embedding φ : X → J for some degree 1 divisor of X. Bogomolov’s conjecture, now a theorem due to Ullmo [Ull98] and Zhang [Zha93] says that there is an > 0 such that

{x ∈ X(K) | hNT(φ(x)) < } is ﬁnite; hNT denotes the Neron-Tate height on J. This conjecture is a generalisation of the Manin- Mumford conjecture. An eﬀective version of this conjecture is known to be true due to Zhang [Zha93], where the author demonstrates an explicit value for in terms of the stable arithmetic self-intersection of the curve X.

1.5. Survey of known results. The minimal regular models of modular curves have been computed in many cases. In the case of square free levels N, Deligne and Rapoport [DR73] constructed regular models of X(N),X1(N) and X0(N) as moduli spaces of generalised elliptic curves. Katz and Mazur [KM85] gave a diﬀerent moduli theoretic construction for Y (N),Y1(N) and Y0(N) using Drinfeld level structures and then compactiﬁed these by adding cusps. The encyclopaedic work of Katz and Mazur is now the most comprehensive reference for this material. It turns out that when N is not square, these models are not regular. Hence for non-square free level constructing a m minimal regular model is a major task, which has been undertaken by many people, for X0(p N) see Edixhoven [Edi90], and Weinstein [Wei16] for semi-stable models for X(pmN). The stable arithmetic self-intersection has also been studied extensively in the case of modular curves. In the case of X0(N), Abbes—Ullmo have given asymptotic upper and lower bounds for ω2 , when N is square-free and co-prime to 6. X0(N) 3

For the modular curves X1(N), Mayer has shown that when N is odd, square free and of the form N = N 0qr with q, r > 4 ω2 = 3g log N + o(g log N), X1(N) N N where gN is the genus of X1(N). In [BC18] using results from [BBC17], we have have shown that for N = p2, 2 2 2 ω 2 = 2g 2 log p + o(g 2 log p ). X0(p ) p p Using this and results of Zhang [Zha93], we are able to give an effective version of Bogomolov’s conjecture: for a sufficiently large prime p, the set 1 x ∈ X (p2)( ) | h (φ(x)) < − log(p2) 0 Q NT 2 is finite whereas 2 2 x ∈ X0(p )(Q) | hNT(φ(x)) ≤ (1 + ) log(p ) is infinite. 1.6. Future research. There are several future direction in this project. 3 Stable model for X0(p ) has been computed by Coleman and McMurdy in [CM06] and [CM10] along with a study of the special fiber. So using their work we would like to carry out the program 3 of [BC18] for X0(p ). More precisely: 3 Question 1. Find the stable arithmetic self-intersection of X0(p ). The stable model of Coleman—McMurdy is not regular so the first task is to compute a minimal 3 regular model for X0(p ). The geometry of the special fiber of the stable model is much more complicated in this case and it will be a much harder task to compute the algebraic (or finite) part of the self-intersection of the relative dualizing sheaf. However, computation of the analytic (or 2 infinite) part should be analogous in flavour to the case of X0(p ). n Finding stable models of X0(p ) for n > 3 is a well known open problem as well as finding regular semi-stable models. It would be a long term goal to study the geometry of regular minimal models n for X0(p ) over the ring of integers of finite extensions of Q over which the models are semi-simple. 2. Enumerative Geometry of Curves This is a joint project with Ritwik Mukherjee and Nilkantha Das. Our varieties are over C. An algebraic curve is a projective variety of dimension 1. There are two invariants associated to an algebraic curve, the geometric genus and the arithmetic genus. When the curve is smooth these are the same, but for singular curves the two numbers may be different. In this section genus will mean geometric genus. 2.1. Curve counting. A fundamental problem in enumerative algebraic geometry is:

(g) 2 Question 2. What is Ed , the number of genus g degree d curves in CP that pass through 3d−1+g generic points? 2 Here CP is the ambient variety and the requirement that the curves pass through certain points (g) is a constrain we put on the curves. Although the computation of Ed is a classical question, a complete solution to the above problem (even for genus zero) was unknown until the early 900s when (0) Ruan–Tian [RT95] and Kontsevich–Manin [KM94] obtained a formula for Ed . (g) The computation of Ed is now very well understood from several different perspectives. The (g) formula by Caporasso–Harris [CH98], computes Ed for all g and d. Since then, the computation of (g) Ed has been studied from many different perspectives; these include (among others), the algorithm 4 by Gathman [Gat03, Gat05] and the method of virtual localization by Graber and Pandharipande n ([GP99]) to compute the genus g Gromov-Witten invariants of CP (although for n > 2 and g > 0, (g) the Gromov-Witten invariants are not enumerative). More recently, the problem of computing Ed has been studied using the method of tropical geometry by Mikhalkin in [Mik05] (using the results (g) of that paper, one can in principle compute Ed for all g and d). Of course similar questions can be asked about other ambient varieties and the curves can be required to satisfy different constraints. 2.2. Gromov-Witten Invariants. A more general situation is as follows: let X be a smooth projective variety and β ∈ H2(X; Z) a given homology class. Given cohomology classes µ1, . . . , µk ∈ ∗ H (X, Q), the k-pointed genus g, Gromov-Witten invariant of X is defined to be Z (g) ∗ ∗ Vir (1) Nβ,X (µ1, . . . , µk) := ev1(µ1) ^ . . . ^ evk(µk) ^ M g,k(X, β) ,

M g,k(X,β) where M g,k(X, β) denotes the moduli space of genus g stable maps into X with k marked points th representing β and evi denotes the i evaluation map. For g = 0, this is a smooth, irreducible and proper Deligne-Mumford stack and has a fundamental class. However, for g > 0, M g,k(X, β) is not smooth or irreducible, hence it does not posses a fundamental class. Behrend, Behrend—Fantechi and Li-Tian, have however deﬁned the virtual fundamental class Vir 2Θ M g,k(X, β) ∈ H (M g,k(X, β)), Θ := c1(TX) · β + (dim X − 3)(1 − g) + k; which is used to deﬁne the Gromov-Witten invariants (see [Beh97],[BF97] and [LT98]). When all the µ1, . . . , µk represent the class Poincare dual to a point (and the degree of the cohomology class that is being paired in (1), is equal to the virtual dimension of the moduli space), then we abbreviate (g) (g) Nβ,X (µ1, . . . , µk) as Nβ . The number of genus g curves of degree β in X, that pass through (g) (g) c1(TX) · β + (dim X − 3)(1 − g) generic points is denoted by Eβ . In general, Eβ is not necessarily (g) equal to Nβ , i.e. the Gromov-Witten invariant is not necessarily enumerative (this happens for 3 example when X := CP and g = 1). An important class of surfaces for which the enumerative geometry is particularly important are Fano surfaces, which are also called del-Pezzo surfaces. When g = 0, it is proved in ([GP98], (0) (0) Theorem 4.1, Lemma 4.10) that for del-Pezzo surfaces Nβ = Eβ . In [Vak00], Vakil generalizes the approach of Caporasso-Harris in [CH98] to compute the numbers (g) Eβ for all g and β for del Pezzo surfaces. It is also shown in ([Vak00], Section 4.2) that all the (g) (g) genus g Gromov-Witten invariants of del Pezzo surfaces are enumerative (i.e. Nβ = Eβ ). 4 In [Get97], Getzler discovered a new relation in H (M 1,4, Q) and using this he gave a recursive 2 formula for the genus 1 Gromov-Witten invariants of CP . Later Pandharipande showed that Getzler’s formula is equivalent to a simpler formula predicted by the celebrated Virasoro conjecture. Using Getzler’s approach we give a recursive formula for the genus 1 Gromov-Witten invariants of del Pezzo surfaces in [CD19]. Since these invariants are enumerative, we get a formula for the number of degree d, genus 1 curves in a del Pezzo surface passing through the right number of (1) points, that is Eβ . There are many interesting question to be explored in this area. 2.3. Future research. Having tackled the genus 1 case a natural question to ask is can this method be used to count the number of genus two curves in a del Pezzo surface passing through the right number of points. More precisely: Question 3. How many genus 2 curves are there in a del Pezzo surface X, representing the homology class β that pass through c1(TX) · β + 1 points? 5

We want to emphasize that our objective here is to answer Question3 using the method of [CD19]. The numbers have already been calculated using diﬀerent techniques by Vakil [Vak00], Shustin—Shoval [SS13], Abramovich—Bertram [AB01] and others. 4 In [BP00], Pandharipande and Belorousski have found a new relation in H (M 2,3, Q). In fact this 2 relation is an algebraic relation in the Chow ring A (M 2,3). Using this new relation, the authors 2 are able to calculate the genus 2 Gromov-Witten invariants for CP . We believe that this method can be extended to calculate the genus 2 Gromov-Witten invariants for del Pezzo surfaces. The same question can of course be asked for genus g > 2, however the moduli spaces M g,n become more and more complicated and the homology groups are less well understood as the genus grows. Homological relations have been found for genus 3 and 4 in [KL06] and [Wan18] respectively. We hope that these relations can be used to calculate the Gromov-Witten invariants of del Pezzo surfaces for the corresponding genera. 3 P ,Planar In a diﬀerent direction, in [MPS18] the authors give are recursive formula for Nd (r, s) the 3 number of degree d genus 0 curves in CP lying inside a hyperplane, intersecting r generic lines and passing through s generic points. This can be viewed as a family version of Question2 of 2 counting rational curves in CP . Similar problems related to the enumerative geometry of singular 3 rational curves moving in a family inside CP have been studied by Kleiman—Pein [KP04] and Laarakker [Laa18]. Together with Ritwik Mukherjee and Nilkantha Das we are investigating the genus 1 version of this problem. 3 Question 4. How many genus 1 degree d curves are there in CP that lie on a hyperplane, intersect r generic lines and pass through s generic points. This seems to be a much deeper question than its genus 0 analogue due to the much richer 3 3 geometry of M 1,k(P , d) as compared to M 0,k(P , d).

3. Topological Study of Moduli of Curves In [Cha15] I study a certain topological invariant called cohomological excess of the moduli of curves. This originated from a conjecture by Looijenga.

Conjecture 1. Mg can be covered by g − 1 open affine subvarieties, when g ≥ 2. 3.1. Certain topological invariants for algebraic varieties. Roth and Vakil introduced the related concept of affine stratifications (see [RV04]) and rephrased the conjecture in terms of the affine stratification number (asn). Looijenga later a nicer invariant, the cohomological excess (ce), which has proven to be easier to work with (see [Cha13, §4.2]). For a complex algebraic variety X, these invariants are related in the following way, ce(X) ≤ asn(X) ≤ acn(X). The cohomological excess gives a bound on the topological complexity of the variety. In particular X is homotopic to a real CW complex of dimension at most

dimC X + ce(X). 3.2. Conjectures and known results. Roth and Vakil [RV04] generalised Looijenga’s conjecture 6k in several directions. Let M g,n denote the locus of curves inside M g,n which have at most k rational components. The following increasing ﬁltration, often comes up in the literature e.g. Graber and Vakil [GV05] 60 62g−2+n M g,n ⊂ ... ⊂ M g,n = M g,n. One of the conjectures by Roth and Vakil is the following.

6k Conjecture 2. asn M g,n ≤ g − 1 + k. 6

Fontanari and Pascolutti [FP12] have proved Conjecture1 of Looijenga for genus 2 ≤ g ≤ 5. However, Conjecture2 of Roth and Vakil, as of now is still open, and nothing is known about the sharpness of the bounds. Let Hg be the hyperelliptic locus inside M g. Hyperelliptic curves are those which admit a degree 1 2 map to P . Hg inherits the same ﬁltration as M g. We prove the following. 6k Theorem 1 (Corollary 4.18 [Cha13]). asn Hg ≤ g − 1 + k. This provides some evidence towards the correctness of Conjecture2. Furthermore we show that in certain cases the upper bound is sharp, provided that it is correct. 60 60 Theorem 2 (Theorem 1.6 [Cha15]). ce(Hg ) ≥ g − 1 and hence asn M g ≥ g − 1. 60 The proof of Theorem2 is constructive. We demonstrate a constructible sheaf L on Hg , which has non-vanishing cohomology in the correct degree, thus proving the lower bound on the cohomological excess. The main tools are mixed hodge structure and modular operads (see [GK98]). 3.3. Future research. The vast majority of the cases are undecided yet and are the subject of my current and future research. The easier and more tractable problems concern the hyper-elliptic locus. 6k Question 5. What is ce(Hg,n? We hope to answer this question using techniques of [Cha13, Cha15]. Our long term goal in this direction would be to say something about Conjecture2.

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