Algebraic Geometry Seminar Spring 2014

The seminar meets on Fridays at 2:25 pm in Van Vleck B231.

The schedule for the previous semester is here.

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Spring 2014 Schedule

date	speaker	title	host(s)
January 31	Marci Hablicsek (local)	Twisted derived intersections and twisted de Rham complexes
February 21	Rares Rasdeaconu (Vanderbilt)	Counting real rational curves on K3 surfaces	Maxim
April 4	Kevin Tucker (University of Illinois at Chicago)	Comparing multiplier ideals to test ideals on numerically QQ-Gorenstein varieties	Daniel
April 18	Manuel Gonzalez Villa (CIMAT, Mexico)	Recursions for motivic iterated vanishing cycles for quasi-ordinary surface singularities	Maxim
April 25	Charles Doran (University of Alberta)	TBA	Song

Abstracts

Marci Hablcisek

Given a smooth variety and two smooth closed subvarieties, derived algebraic geometry assigns to this data a differential graded scheme, the derived intersection. Equipping the ambient space with an Azumaya algebra, we obtain the notion of twisted derived intersections. In order to compare the twisted and the "untwisted" derived intersections, we assume that the Azumaya algebra is split along the two subvarieties. For such twisted intersection problem, we associate a natural line bundle on the derived intersection, which measures the difference between the two derived intersections. We give a criterion for the triviality of this line bundle. As an application, we prove a special case of the Barannikov-Kontsevich theorem, and we give a decomposition theorem for the hypercohomology spaces of the twisted de Rham complexes. The work is joint with Dima Arinkin and Andrei Caldararu.

Rares Rasdeaconu

Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. I will discuss the case of counting real rational curves on K3 surfaces equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface, the real version of a result due to Gottsche.

The talk is based on a joint work with V. Kharlamov.

Kevin Tucker

In this talk, I will focus on the connection between two important measures of singularities: multiplier ideals in characteristic zero and test ideals in positive characteristic. While their relationship is well understood in many cases (e.g. hypersurface or finite quotient singularities), it remains conjectural for non-QQ-Gorenstein varieties (such as the cone over the Segre embedding of PP^1 x PP^2 in PP^5). I will discuss positive recent progress on this conjecture for so-called numerically QQ-Gorenstein varieties (which include all normal surface singularities). This is joint work with T. de Fernex, R. Do Campo, and S. Takagi.