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The Enumerative Significance of Elliptic Gromov-Witten Invariants in P

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The Enumerative Significance of Elliptic Gromov-Witten Invariants in P The Enumerative Significance of Elliptic Gromov-Witten Invariants in P3 Master Thesis Department of Mathematics Technische Universit¨atKaiserslautern Author: Supervisor: Felix R¨ohrle Prof. Dr. Andreas Gathmann May 2018 Selbstst¨andigkeitserkl¨arung Hiermit best¨atigeich, Felix R¨ohrle,dass ich die vorliegende Masterarbeit zum Thema The Enumerative Significance of Elliptic Gromov-Witten Invariants in " P3\ selbstst¨andigund nur unter Verwendung der angegebenen Quellen verfasst habe. Zitate wurden als solche kenntlich gemacht. Felix R¨ohrle Kaiserslautern, Mai 2018 Contents Introduction 1 1 The Moduli Space M g;n(X; β) 5 1.1 Stable Curves and Maps . .5 1.2 The Boundary . .9 1.3 Gromov-Witten Invariants . 11 1.4 Tautological Classes on M g;n and M g;n(X; β)........... 14 2 Dimension 17 3 The Virtual Fundamental Class 27 3.1 Reducing to DR [ DE;1 [···[ DE;n ................ 28 3.2 The Contribution of DR ....................... 31 3.3 Proof of the Main Theorem . 38 A The Ext Functor 41 B Spectral Sequences 43 Introduction Enumerative geometry is characterized by the question for the number of geo- metric objects satisfying a number of conditions. A great source for such prob- lems is given by the pattern \how many curves of a given type meet some given objects in some ambient space?". Here, one might choose to fix a combination of genus and degree of the embedding of the curve to determine its \type", and as \objects" one could choose points and lines. However, the number of objects is not free to choose { it should be determined in such a way that the answer to the question is a finite integer. For example one might ask [KV03]: How many plane rational curves of degree d pass through 3d − 1 points in general position? This question has successfully been answered and the solution is quite il- lustrative for how enumerative problems might be approached. The first step is to set up a moduli space parametrizing the objects of interest. By means of suitable morphisms the point incidence conditions are pulled back from P2 to this moduli space. The resulting subvarieties are then intersected using the calculus of intersection theory to yield a zero-dimensional Chow cycle including multiplicities. A simple degree evaluation then gives rise to integers which have been found to indeed answer the question asked above. The numbers obtained in the way just described can be defined in more gen- eral settings and are referred to as Gromov-Witten invariants. However, passing to positive genus introduces new phenomena causing a discrepancy between the invariants and the number sought by the enumerative question. Consider for example the problem: How many elliptic curves of degree d are there which meet a number of a given lines and b given points in P3 in general position, where 4d = a + 2b? Gromov-Witten invariants in the elliptic case can still be computed, but they are not answering the question. In fact, they are not even integers! In his 1996 paper [Get97, theorem 6.1] Getzler claims the answer to this question to be given by the formula 1 N (1) + (2d − 1)N (0); ab 12 ab (1) (0) where Nab and Nab are elliptic and rational Gromov-Witten invariants respec- tively. However, to the best of our knowledge no proof for the statement has 1 been made public. Therefore we made it the goal of this thesis to prove Getzler's claim. We now present an outline of our proceeding. In chapter 1 we will formally define the moduli space M g;n(X; d) of so-called stable maps of genus g with n marks and degree d (definition 1.9) { a notion due to Kontsevich. In this notation X is the ambient space which we will take to be P3 in the end. This moduli space naturally comes with so-called evaluation maps ev : M g;n(X; d) ! X (see definition 1.24) that enable us to translate incidence conditions in X into Chow cycles of the moduli space: ∗ V ⊆ X ! ev V ⊆ M g;n(X; d): With this technique at hand we may introduce Gromov-Witten invariants in definition 1.25 by intersection these cycles and finally we restate Getzler's claim as main result of this thesis in theorem 1.28. However, the definition of concepts like virtual dimension and virtual fundamental class even though important will be largely omitted. The discrepancy between Gromov-Witten invariants and enumerative num- bers arises from a phenomenon referred to as excessive dimension. This is caused by the compactification of the moduli space Mg;n(X; d) of smooth genus g curves that was defined in chapter 1. In chapter 2 we will explore the extent of the problem by determining all irreducible components of the moduli space of elliptic curves in P3 whose dimension exceeds the expected value (corollary 2.14). In order to do so we will prove some theorems constituting a dimension calculus for irreducible components (theorems 2.4 and 2.12). In the third and final chapter we determine for each of the components found in chapter 2 the impact they have on the Gromov-Witten invariants. We start by identifying most of the excessive dimensional components as enumeratively irrelevant in theorem 3.7. This is done by constructing a morphism to a space of sufficiently small dimension which still keeps all of the enumerative information. Then we utilize the projection formula from intersection theory to move the computation of Gromov-Witten invariants from the original moduli space to the condensed space, where the result will be trivial for dimensional reasons. Section 3.2 is dominated by the central computation of this thesis. We use the calculus of Chern classes to actually compute a virtual fundamental class of one of the excessive dimensional components. The result is expressed via tautological classes of the moduli spaces involved (theorem 3.21). In theorem 3.22 the coefficient claimed by Getzler shows up for the first time. Finally in section 3.3 we use the compatibility of virtual fundamental classes with pull-back along forgetful morphisms to prove the main theorem. Careful investigation of the central theorems in this thesis shows why we chose the special case of genus 1 curves in P3: In order for the Gromov-Witten invariants to have a chance of being enumerative we need dim Mg;n(X; β) = vdim M g;n(; β); otherwise the virtual fundamental class restricted to Mg;n(X; β) cannot be equal to the usual fundamental class. However our proof of theorem 2.12 works only for genus at most 1. As for the choice of an ambient space X we restricted ourselves to Pr because we needed to use smoothness and the Euler sequence at various occasions. The 2 choice of r = 3 is a bit more subtle however. Looking at the expression for the virtual dimension vdim M g;n(X; d) = −KX d + (dim X − 3)(1 − g) + n we see that it is independent of g only if dim X = 3. This independence is needed to state theorem 3.22, i.e. in order to be able to express the push- virt forward of [M 1;n(X; d)] along the map used in theorem 3.22 as a multiple of virt [M 0;n(X; d)] . This does of course not rule out the possibility of generaliza- tion, however if either of the constraints should be relaxed, different (and most likely more involved) tools will have to be employed. 3 Notation Throughout this thesis, whenever we say \scheme" we mean \k-scheme", where k is an algebraically closed, fixed base field of characteristic 0. Correspondingly all morphisms are morphisms of k-schemes. A variety is a separated scheme of finite type. Curly capitals will always denote sheaves, fraktur font indicates categories. For example Mod(R) is the category of modules over the ring R and Ab is the category of Abelian groups. x when the number of variables is clear from the context or irrele- vant we will write this instead of x1; : : : ; xN ()_ dual module, vector space, sheaf, etc. r r P r-dimensional projective space, P := Proj k[x0; : : : ; xr] OX structure sheaf of a scheme or variety X OX (D) sheaf corresponding to the divisor D 2 Div(X) r r r · r OP (n) Serre twisting sheaf; OP (n) ·= OP (n·H), where H ⊆ P denotes a hyperplane Hi(X; F) i-th cohomology k-module of the scheme X with coordinates in the sheaf of OX -modules F hi(X; F) ·= dim Hi(X; F) Ai(X) group of i-dimensional Chow cycles in X Ai(X) group of Chow cycles of codimension i in X A∗(X) Chow ring non-singular, quasi-projective variety X, ·= L1 i i=0 A (X) with ring structure given by intersection product V · W intersection product of V; W 2 A∗(X) ci(E) i-th Chern class of a vector bundle E · P1 ∗ c(E) total Chern class, c(E) ·= i=0 ci(E) 2 A (X) 1 g(C) arithmetic genus of the curve C, g(C) ·= h (C; OC ) ΩX cotangent sheaf of X _ TX tangent sheaf of X, TX ·= ΩX Vdim X !X canonical bundle of non-singular variety X, !X ·= ΩX KX canonical divisor of X, OX (KX ) = !X Γ(U; F) ·= F(U), sections of the sheaf F on U ⊆ X open Γ(F) ·= F(X), global sections of F · Fx ·=−! limx2U Γ(U; F) the stalk of the sheaf F at the point x 2 X 4 Chapter 1 The Moduli Space Mg;n(X; β) As very first step towards making any enumerative question accessible, one has to parametrize the objects of interest.
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