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Tautological Rings of Moduli Spaces of Curves Tautological Rings of Moduli Spaces of Curves MEHDI TAVAKOL Doctoral Thesis Stockholm, Sweden 2011 TRITA-MAT-11-MA-03 KTH ISSN 1401-2278 Institutionen för Matematik ISRN KTH/MAT/DA 11/02-SE 100 44 Stockholm ISBN 978-91-7501-043-4 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik Onsdag den 15 juni 2011 kl 13.00 i sal F3, Kungl Tekniska högskolan, Lindstedts- vägen 26, Stockholm. c Mehdi Tavakol, 2011 Tryck: Universitetsservice US AB iii Abstract The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mum- ford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincaré duality. The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra. In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result. In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well. Contents Contents v Acknowledgements vii Introduction 1 Introduction 1 1.1 Intersection theory of moduli spaces of curves . 2 1.2 Tautological rings of moduli spaces of curves . 9 1.3 Cohomology of moduli spaces of curves . 22 1.4 Integral computations on the moduli stack . 29 References 37 Papers Paper I ct The tautological ring of M1,n Paper II The Chow ring of the moduli space of curves of genus zero Paper III rt The tautological ring of the moduli space M2,n v Acknowledgements My special thanks go to my scientific advisor, Carel Faber. I would like to thank all members of the math department at KTH who helped me during my studies. vii Chapter 1 Introduction In this introduction we are going to present a short review of well-known facts and conjectures about the moduli spaces of curves and their invariants. Our main focus is on the study of the tautological classes on these moduli spaces and their images in the homology and cohomology groups of the spaces. Let us first explain the words we have used in this paragraph. The moduli space of a certain class of geometric objects parameterizes the iso- morphism classes of these objects. Moduli spaces occur naturally in classification problems. There are many examples of moduli spaces in algebraic geometry. Among them is the moduli space Mg parameterizing smooth curves of genus g for a natural number g ≥ 2 and the moduli space Ag of principally polarized abelian varieties of dimension g. Another important example is the moduli space of vector bundles with some specified properties on a given algebraic variety. To develop intersection theory with nice properties it is more convenient to work with compact spaces. The moduli space Mg is not a compact space as smooth curves can degenerate into singular ones. Deligne and Mumford in [17] introduced a com- pactification of this space by means of stable curves. Stable curves may be singular but only nodal singularities are allowed. Such a curve is stable when it is reduced, connected and each smooth rational component meets the other components of the curve in at least 3 points. This guarantees the finiteness of the group of auto- morphisms of the curve. The moduli space of stable curves of genus g is denoted by M g. It is phrased in [17] that the moduli space M g is just the “underlying coarse variety” of a more fundamental object, the moduli stack Mg. We do not attempt to explain this word and refer the interested reader to the original paper [17]. Other introductory references are [19] and [40]. We only mention that basic properties of algebraic varieties (schemes) and morphisms between them generalize for algebraic stacks in a natural way. It is proven in [17] that Mg is a separated, proper and smooth algebraic stack of finite type over Spec(Z) and the complement Mg\Mg is a divisor with normal crossings relative to Spec(Z). They prove that for an algebraically closed field k the moduli space M g over k is irreducible. 1 2 CHAPTER 1. INTRODUCTION A bigger class of moduli spaces of curves deals with pointed curves. Let g, n be integers satisfying 2g−2+n > 0. The moduli space Mg,n classifies the isomorphism classes of the objects of the form (C; x1, . , xn), where C is a smooth curve of genus g and the xi’s are n distinct points on the curve. The compactification of this space by means of stable n-pointed curves is studied by Knudsen in [59]. The notion of an stable n-pointed curve is defined as above with slight modifications: First of all we assume that all the markings on the curve are distinct smooth points. For a nodal curve of arithmetic genus g with n marked points the nodes and markings are called special points. Such a pointed curve is said to be stable if a smooth rational component has at least 3 special points. The Deligne-Mumford compactification M g,n of Mg,n parameterizes stable n-pointed curves of arithmetic genus g. The space Mg,n sits inside M g,n as an open dense subset and the boundary M g,n\Mg,n is a divisor with normal crossings. There are partial compactifications of Mg,n inside M g,n which we recall: The ct moduli space of stable n-pointed curves of compact type, denoted by Mg,n, param- rt eterizes stable curves whose Jacobian is an abelian variety. The moduli space Mg,n of stable n-pointed curves with rational tails is the inverse image of Mg under the rt ct natural morphism M g,n → M g, when g ≥ 2. The space M1,n is defined to be M1,n rt and M0,n = M 0,n. 1.1 Intersection theory of moduli spaces of curves It is an important question to determine the whole intersection ring of the moduli spaces of curves. In general it is a difficult task to compute the Chow ring of a given variety. On the other hand the moduli spaces of curves are equipped with natural vector bundles. The theory of Chern classes gives natural algebraic cycles on these spaces. Enumerative Geometry of the moduli space of curves In [75] Mumford started the program of studying the enumerative geometry of the set of all curves of arbitrary genus g. He suggests to take as a model for this the enumerative geometry of the Grassmannians. In this case there is a universal bundle on the variety whose Chern classes generate both the cohomology ring and the Chow ring of the space. Moreover, there are tautological relations between these classes which give a complete set of relations for the cohomology and Chow rings. The first technical difficulty is to define an intersection product in the Chow group of M g. The variety M g is singular due to curves with automorphisms. Mumford solves this problem by observing that M g has the structure of a Q-variety. A quasi-projective variety X is said to be a Q-variety if in the étale topology it is locally the quotient of a smooth variety by a finite group. This is equivalent to saying that there exists an étale covering of the quasi-projective variety X by a 1.1. INTERSECTION THEORY OF MODULI SPACES OF CURVES 3 finite number of varieties each of which is a quotient of a quasi-projective variety by a faithful action of a finite group. For a Q-variety X it is possible to find a global covering p : Xe → X with a group G acting faithfully on Xe and X = X/Ge . In this situation there is a covering of Xe by a finite number of open subsets Xeα and finite groups Hα such that the collection Xα = Xeα/Hα gives an étale covering of X. The charts should be compatible in the sense that for all α and β the projections from the normalization Xαβ of the fibered product Xα ×X Xβ to Xα and Xβ should be étale. An important notion related to a Q-variety X is the concept of a Q-sheaf. By a coherent Q-sheaf F on X we mean a family of coherent sheaves Fα on the local charts Xα, plus isomorphisms F ⊗ O =∼ F ⊗ O . α OXα Xαβ β OXβ Xαβ Equivalently, a Q-sheaf is given by a family of coherent sheaves on the local charts, such that the pull-backs to Xe glue together to one coherent sheaf Fe on Xe on which G acts. Mumford proves the following fact about Q-sheaves on varieties with a global Cohen-Macaulay cover: Proposition 1.1. If Xe is Cohen-Macaulay, then for any coherent sheaf F on the Q-variety X, Fe has a finite projective resolution.
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