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Dive into the research topics where Carel Faber is active.

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Journal of the European Mathematical Society | 2005

Relative maps and tautological classes

Carel Faber; Rahul Pandharipande

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 Ionels, Looijengas, and Getzlers vanishings, and applications to Gromov-Witten theory.


arXiv: Algebraic Geometry | 1999

A Conjectural Description of the Tautological Ring of the Moduli Space of Curves

Carel Faber

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.


arXiv: Algebraic Geometry | 1999

New Trends in Algebraic Geometry: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians

Carel Faber

We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.


Annales Scientifiques De L Ecole Normale Superieure | 2010

Tautological relations and the

Carel Faber; Sergey Shadrin; Dimitri Zvonkine

In [23, 24], Y.-P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.-P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.-P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.-P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it 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.


Progress in Math. | 1995

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Robbert Dijkgraaf; Carel Faber; Gerard B. M Geer

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. Wittens 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.


Comptes Rendus Mathematique | 2004

-spin Witten conjecture

Carel Faber; G.B.M. van der Geer

On the cohomology of local systems on the moduli spaces of curves of genus 2 and of Abelian surfaces, I. 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).  2004 Academie des sciences. Publie par Elsevier SAS. Tous droits reserves.


Archive | 1999

The moduli space of curves

Carel Faber; Eduard Looijenga

We discuss some aspects of the theory of the moduli space of curves as well as some recent research.


Manuscripta Mathematica | 1995

On the cohomology of local systems on the moduli spaces of curves of genus 2 and of Abelian surfaces, II

Paolo Aluffi; Carel Faber

The following elementary observation has proven useful in several enumerative geometry computations. Let X be any algebraic scheme over a eld, and let 2 K(X) be an element in the Grothendieck group of vector bundles over X. Then has a well-de ned rank rk , and Chern classes ck( ). Also, as tensor product makes K 0(X) a ring, we consider [L], where L is an arbitrary line bundle on X. Theorem. With notations as above, crk +1( ) = crk +1( [L]). (So this class is independent of L.) Proof. Write = [E] [F ], with E, F vector bundles of rankm and n = m r respectively. We have to show that the coe cient of t in the formal power series S(t) = ct(E L) ct(F L) is independent of ` = c1(L). By [1], Example 3.2.2, we have ct(E L) = (1+`t) c (E) with = t=(1+`t). Therefore S(t) = (1 + `t) c (E) c (F) = (1 + `t) 1 X


Mathematische Annalen | 2017

Remarks on Moduli of Curves

Fabien Cléry; Carel Faber; Gerard van der Geer

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.


Progress in Mathematics | 2016

A remark on the Chern class of a tensor product

Carel Faber; Gavril Farkas; Gerard van der Geer

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.

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Paolo Aluffi

Florida State University

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Gavril Farkas

Humboldt University of Berlin

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