This last theory makes possible an enumerative geometry with values in quadratic forms.
We define the tautological ring on moduli space of polarized hyperhahler manifolds, which is largely motivated from the work of Marian, Oprea and Pandharipande on moduli space of K3 surfaces. We will talk about the properties of the tautological ring and give some conjectural descriptions such as the tautological conjecture.
We will provide some evidence towards these conjectures. This is a joint work of Nicolas Bergeron. The moduli space of hyperelliptic K3's of degree 4 may be identified with an dimensional locally symmetric variety of Type IV, that we denote by F Starting from Looijenga's pioneering work on new compactifications defined by arrangements in Type IV BSD's, we have given a very precise conjectural decomposition of the period map into a composition of flips and divisorial contractions. Led by physics arguments Huang, Katz and Klemm conjectured that curve counting invariants of elliptic Calabi-Yau threefolds form Jacobi forms.
On the mathematics side the origin of the modularity is becoming clearer. Part of the modularity arises from sheaf theory.
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Fourier-Mukai transforms with respect to the Poincare line bundle of the fibration yield modular constraints. The other part has origins in Gromov-Witten theory. I will explain how this philosophy leads to a proof that the generating series of Gromov-Witten invariants of the product of a K3 surface and an elliptic curve with respect to primitive classes on the K3 is the reciprocal of the Igusa cusp form, a Siegel modular form.
I will discuss the K-trivial case of a conjecture of Lehn regarding the top Segre classes of tautological vector bundles over the Hilbert scheme of points. The method involves the study of the virtual geometry of certain Quot schemes via equivariant localization.
In addition, when the surface varies in moduli, the same method applied to the relative Quot schemes yields relations intertwining the k-classes over the moduli space of K3s and the Noether-Lefschetz loci. This was previously known only in genus 2. Kontsevich and V. Title: Torsionness for regulators of canonical extensions. Abstract: I will sketch a generalization of the results of Iyer and Simpson arXiv Title: Euler Characteristics of punctual quot schemes on threefolds.
Abstract: Let F be a homological dimension 1 torsion free sheaf on a nonsingular quasi-projective threefold.
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The first cohomology of the derived dual of F is a 1-dimension sheaf G supported on the singular locus of F. We prove a wall-crossing formula relating the generating series of the Euler characteristics of Quot F, n and Quot G,n , where Quot -,n denotes the quot scheme of length n quotients. We will use this relation in studying the Euler characteristics of the moduli spaces of stable torsion free sheaves on nonsingular projective threefolds. This is a joint work with Martijn Kool.
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The stability depends on a choice of an effective line bundle on the parameter space of weighted hypersurfaces and different choices pick out different birational model of the total space of the fibration. I will describe enumerative geometry that goes into understanding these stability conditions, and, if time permits, examples where this machinery can be used to produce birational models with good properties.
Joint work with Hamid Ahmadinezhad and Igor Krylov. Title: Rationality for geometrically rational threefolds. Abstract: We consider rationality questions for varieties over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics.
The combinatorial link is provided by the category of factorization modules over a certain factorization algebra, which in itself is a geometric device that concisely encodes the root data. Title: AGT relations in geometric representation theory. Title: Resolution in characteristic 0 using weighted blowing up. Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci — it goes into a loop.
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A similar result was discovered by G. Marzo and M. Title: On the base of a Lagrangian fibration for a compact hyperkahler manifold. Kemeny Duke Mathematical Journal , Singularities of the moduli space of level curves pdf with A. Arbarello, A. Bruno and G.
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The geometry of the moduli space of odd spin curves dvi pdf with A. Grushevsky, R. Salvati Manni and A. Chiodo, D.
Invited Talks and Conference Presentations
Eisenbud and F. Schreyer Inventiones Mathematicae , The universal theta divisor over the moduli space of curves dvi pdf with A. Verra Commentarii Mathematici Helvetici 88 , Brill-Noether with ramification at unassigned points dvi pdf Journal of Pure and Applied Algebra , Effective divisors on moduli spaces of curves and abelian varieties pdf with D.
Chen and I. Hassett, J.