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We prove that the moduli space of stable n-pointed curves of genus one and the projector associated to the alternating representation of the symmetric group on n letters define (for n>1) the Chow motive corresponding to cusp forms of weight n+1 for SL(2,Z). This provides an alternative (in level one) to the construction of Scholl., 18 pages. To appear in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, 2006

Introduction.- Samuel Boissiere, Andrea Cattaneo, Marc Nieper-Wisskirchen, and Alessandra Sarti: The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface.- Igor Dolgachev: Orbital counting of curves on algebraic surfaces and sphere packings.- V. Gritsenko and K. Hulek: Moduli of polarized Enriques surfaces.- Brendan Hassett and Yuri Tschinkel: Extremal rays and automorphisms of holomorphic symplectic varieties.- Gert Heckman and Sander Rieken: An odd presentation for W(E_6).- S. Katz, A. Klemm, and R. Pandharipande, with an appendix by R. P. Thomas: On the motivic stable pairs invariants of K3 surfaces.- Shigeyuki Kondo: The Igusa quartic and Borcherds products.- Christian Liedtke: Lectures on supersingular K3 surfaces and the crystalline Torelli theorem.- Daisuke Matsushita: On deformations of Lagrangian fibrations.- G. Oberdieck and R. Pandharipande: Curve counting on K3 x E, the Igusa cusp form X_10, and descendent integration.- Keiji Oguiso: Simple abelian varieties and primitive automorphisms of null entropy of surfaces.- Ichiro Shimada: The automorphism groups of certain singular K3 surfaces and an Enriques surface.- Alessandro Verra: Geometry of genus 8 Nikulin surfaces and rationality of their moduli.- Claire Voisin: Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kahler varieties.

Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In "Limits of PGL(3)-translates of plane curves, I" we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C. The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in "Linear orbits of arbitrary plane curves" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance., Comment: 22 pages. Minor revision. Final version

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the $P^8$ of 3x3 matrices to the $P^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in our earlier paper "Linear orbits of arbitrary plane curves"., 28 pages. Minor revision. Final version

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations., 46 pages, 7 figures, A discussion of the analyticity of Gromov-Witten potentials and a more careful description of Givental's group action added in Section 5; minor changes elsewhere

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Resume Nous etudions la cohomologie des systemes locaux sur les espaces M 2 de modules des courbes de genre 2 et A 2 de modules des surfaces abeliennes. Nous donnons une formule explicite pour la cohomologie d'Eisenstein et une formule conjecturale pour la contribution endoscopique. Notre calcul des courbes sur des corps finis donne des renseignements precis sur les formes modulaires de Siegel. Pour citer cet article : C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required.

Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper, we compute the class of the bielliptic locus in the moduli space \overline{M}_3 of stable curves of genus three in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in \overline{M}_3 .

After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, we prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true., Comment: 6 pages. To appear in Annales de l'Institut Fourier (Grenoble)

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological push-forwards to the moduli space of curves. In particular, the fundamental classes of all moduli spaces of admissible covers push-forward to tautological classes. Consequences for the tautological rings of the moduli spaces of curves include methods for generating new relations, uniform derivations of the socle and vanishing statements of the Gorenstein conjectures for the complete, compact type, and rational tail cases, tautological boundary terms for Ionel's, Looijenga's, and Getzler's vanishings, and applications to Gromov-Witten theory.

0.1. Overview. Let Mg be the moduli space of Deligne–Mumford stable curves of genus g ≥ 2. The study of the Chow ring of the moduli space of curves was initiated by Mumford in [Mu]. In the past two decades, many remarkable properties of these intersection rings have been discovered. Our first goal in this paper is to describe a new perspective on the intersection theory of the moduli space of curves that encompasses advances from both classical degeneracy studies and topological gravity. This approach is developed in Sections 0.2–0.7. The main new results of the paper are computations of basic Hodge integral series in A∗(Mg) encoding the canonical evaluations of κg−2−iλi . The motivation for the study of these tautological elements and the series results are given in Section 0.8. The body of the paper contains the Hodge integral derivations.

The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work., Comment: 33 pages, AmS-TeX 2.1

The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of curves supported on unions of lines. Together with the results of math.AG/9805020, this encompasses the enumerative geometry of all plane curves with small linear orbit. This information will serve elsewhere as an ingredient in the computation of the degree of the orbit closure of an arbitrary plane curve., 16 pages, one figure. Proof of main result expanded, references added

We formulate a number of conjectures giving a rather complete description of the tautological ring of M g and we discuss the evidence for these conjectures.

Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these integrals from the standard descendent potential (for any target X). We use virtual localization and classical degeneracy calculations to find trigonometric closed form solutions for special Hodge integrals over the moduli space of pointed curves. These formulas are applied to two computations in Gromov-Witten theory for Calabi-Yau 3-folds. The genus g, degree d multiple cover contribution of a rational curve is found to be simply proportional to the Euler characteristic of M_g. The genus g, degree 0 Gromov-Witten invariant is calculated (in agreement with recent string theoretic calculations of Gopakumar-Vafa and Marino-Moore). Finally, with Zagier's help, our Hodge integral formulas imply a general genus prediction of the punctual Virasoro constraints applied to the projective line., 24 pages, LaTeX2e

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary plane curves. Linear orbits of smooth plane curves are studied in [A-F1]., Comment: 34 pages, 4 figures, AmS-TeX 2.1, requires xy-pic and epsf

Developments in theoretical physics, in particular in conformal field theory, have led to a surprising connection to algebraic geometry, and especially to the fundamental concept of the moduli space Mg of curves of genus g, which is the variety that parametrizes all curves of genus g. Experts in the field explore in this volume both the structure of the moduli space of curves and its relationship with physics through quantum cohomology. Witten's conjecture in 1990 describing the intersection behaviour of tautological classes in the cohomology of Mg arose directly from string theory. Shortly thereafter an interesting proof was provided by Kontsevich who, in this volume, describes his solution to the problem of counting rational curves on certain algebraic varieties and includes suggestions for further development. The same problem is given treatment in a paper by Manin. There follows a number of contributions to the geometry, cohomology and arithmetic of the moduli spaces of curves. In addition, several contributors address quantum cohomology and conformal field theory.

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the ��tale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures., Added a section on Harder's conjectural congruences. Some minor changes. 16 pages

ring of the moduli space of stable curves of genus 4. These results are not complete. We find generators for the Chow ring of 4 and for the Chow groups in codimension 1 and 2 of -W4. For A2(G'4) we find fourteen generators. Using test surfaces we prove that the dimension of A 2(4/'4) is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist. Finally, we have two proofs that this relation does indeed hold, so that the dimension of A2( 4/4) equals 13. This enables us to determine the Chow ring of ,4'4. Our original proof is based on a rather delicate argument; the second proof uses a result of Ran (see [R]) and is much simpler.

International audience; This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.