WAGS / Spring '04 Western Algebraic Geometry Seminar Abstracts University of Utah March 27 - 28, 2004

 Daniel Allcock Topology of Moduli Spaces and Discriminant Complements Several discriminant complements have contractible universal covers, modulo a conjecture concerning finite Coxter groups. These are the moduli spaces of K3 surfaces with polarization of given degree, minus their discriminants, and the discriminant complements in the versal deformations of the first few non-simple singularities in the Arnol'd hierarchy, including the singularity x2+y3+z7=0.

 Kalle Karu Toric Residue mirror conjecture The toric residue mirror conjecture of Batyrev and Materov states that a rational function, the toric residue, can be expanded in a Laurent series where the coefficients are integrals over certain moduli spaces of curves. In this talk I will explain the algebra behind the conjecture.

 Yuan-Pin Lee Invariance of tautological equations In this talk we will propose a conjecture about the tautological relations. The conjecture is closely related to the following "fact" that the only reasonable set of equations of tautological classes that holds for all GW theories is the set of (true) tautological equations. Assuming that conjecture, one can derive, using nothing more than calculus, all the known tautological equations and likely many new ones.

 Jun Li A mathematical theory of the topological vertex In "Topological vertex" (hep-th/0305132), Aganagic, Klemm, Marino and Vafa conjectured that certain open Gromov-Witten invariants of $\C^3$ can be defined for any three partitions $(\mu_1,\mu_2\mu_3)$ and three integers $(n_1,n_2,n_3)$, and that such invariants --which they call topological vertex-- can be used to derive a closed formula of the generating function of open Gromov-Witten invariants of toric Calabi-Yau threefold. This talk is to develop a mathematical theory of the topological vertex.

 Mircea Mustata Asymptotic intersection theory and base loci I will describe numerical invariants on the big cone of a smooth projective variety which can be used to describe asymptotic base loci of (small perturbations of) line bundles. This is joint work with L. Ein, R. Lazarsfeld, M. Nakamaye and M. Popa.

 Kristian Ranestad Rational and irregular surfaces in P4 Smooth rational and irregular surfaces are two kinds of surfaces that are quite rare in P4. Some finiteness results are known, but a classification is still far away. In this talk I will present old and new results focusing on different constructions, including the discovery of a new rational surface of degree 12 made this week with Hirotachi Abo

 Balázs Szendröi Calabi-Yau threefolds in weighted homogeneous varieties Let (X,D) be a Calabi--Yau threefold with quotient singularities, polarized by an ample Q-Cartier divisor. I prove a formula expressing the number of sections H^0(X, nD) in terms of global numerical invariants of the pair (X,D) and local invariants of D at the quotient singularities of X. Based on this formula, I construct several new families of Calabi--Yau threefolds in weighted homogeneous varieties, generalizations of weighted projective spaces introduced by Grojnowski, Corti and Reid. This is joint work with Anita Buckley.