Algebraic geometry and moduli seminar

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Autumn Semester 2011

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Date / Time Speaker Title Location
30 September 2011
Prof. Dr. Carel Faber
KTH Stockholm
Modular forms and the cohomology of moduli spaces  HG G 43 
Abstract: This will be an overview over the cohomology classes on moduli spaces of curves or abelian varieties (of low genus respectively dimension) corresponding to modular forms. Elliptic cusp forms give classes for curves of genus 1, and their generalizations, Siegel cusp forms, give classes for abelian varieties. One obtains a formula for moduli of pointed curves of genus 2 in these terms. For curves of higher genus, Teichmüller modular forms enter the picture. The related cohomology classes are still quite mysterious. Joint work with Jonas Bergström and Gerard van der Geer.
7 October 2011
Dr. Oscar Randal-Williams
Cohomology of the universal Picard varieties, and tautological relations  HG G 43 
Abstract: I will discuss a recent calculation (with J. Ebert) of the stable cohomology of the universal Picard variety over the moduli space of curves, and of the gerbe over it which classifies Riemann surfaces equipped with a holomorphic line bundle. The cohomology of this gerbe has a useful description in terms of "generalised Mumford-Morita-Miller classes", and using this description I will show how to reinterpret and extend the construction of Morita in the 80's of a relation among tautological classes. Up to genus 9 this construction produces all top-dimensional relations among tautological classes.
14 October 2011
Dr. Emanuel Scheidegger
Pencils of Cubic Fourfolds  HG G 43 
Abstract: We will discuss some Hodge-theoretic aspects of families of cubic fourfolds. Our focus will lie on so-called special cubic fourfolds, which contain an algebraic surface not homologous to a plane. We will show that there is a modular form that counts the special members of a pencil of cubic fourfolds. We will go on with some speculations on the derived category of special cubic fourfolds and homological mirror symmetry.
21 October 2011
Prof. Dr. Gavril Farkas
Humboldt University, Berlin
Green's conjecture for ALL curves on K3 surfaces  HG G 43 
Abstract: Formulated in 1984 and still open in its full generality, Green's Conjecture on syzygies is a deceptively simple statement which predicts that the intrinsic geometry of the curve (in the form of linear series) can be recovered in a precise way from the extrinsic geometry of the canonical embedding (in the form of syzygies). I will discuss how one can use Voisin's solution to Green's Conjecture for GENERAL curves together with the geometry of the moduli space of curves, in order to prove Green's Conjecture for ARBITRARY curves lying on K3 surfaces. This is joint work with M. Aprodu.
28 October 2011
Prof. Dr. Andrew Kresch
Univerisity of Zürich
Maps from curves with ramification conditions HG G 43 
11 November 2011
Prof. Dr. Lothar Goettsche
ICTP / Bonn
Refined curve counting on algebraic surfaces  HG G 43 
Abstract: Most of the results discussed in this talk are conjectural. Let L be ample line bundle on an a projective algebraic surface S. Let g be the genus of a smooth curve in the linear system |L|. If L is sufficiently ample with respect to d, the number of n_{L,d} of d-nodal curves in a general d-dimensional sublinear system of |L| will be finite. Kool-Shende-Thomas use relative Hilbert schemes of points of the universal curve over |L| to de fine the numbers n_{L,d} as BPS invariants and prove a conjecture of mine about their generating function. We use the generating function of the chi_y-genera of these relative Hilbert schemes to defi ne and study refi ned curve counting invariants N_{L,g}(y), which are now polynomials in y, with N_{L,d}(1)=n_{L,d}. If S is a K3 surface we relate these invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas. In the case of real toric surfaces we see that the refi ned invariants interpolate between the Gromov-Witten invariants (at y = 1) and the Welschinger invariants (which count real curves) at y = -1. - Show quoted text -
18 November 2011
Prof. Dr. Barbara Fantechi
SISSA / Bonn
Moduli stack of degenerations and twisted degenerations  HG G 43 
Abstract: In order to define relative Gromov-Witten invariants in algebraic geometry one needs a stack of "half accordeons". To prove a degeneration formula one needs a similar stack of accordeons. We will describe these stacks and their twisted versions, showing some geometric properties. We will hint at the relationship between the twisted and the log geometry language.
2 December 2011
Prof. Dr. Albrecht Klemm
Generalized holomorphic anomaly and Motivic DT invariants HG G 43 
9 December 2011
Dr. Tamas Hausel
Symmetries of SL(n) Hitchin fibers  HG G 43 
Abstract: In this talk we show how the computation of the group of components of Prym varieties of spectral covers leads to cohomological results on the moduli space of stable bundles originally due to Harder-Narasimhan. This is joint work with Christian Pauly.
16 December 2011
Prof. Dr. Marcos Marino
Torus knots and mirror symmetry HG G 43 

Organizers: Rahul Pandharipande 

Archive: SS 17  AS 16  SS 16  AS 15  SS 15  AS 14  SS 14  AS 13  SS 13  AS 12  SS 12  AS 11 

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