I-70 Algebraic Geometry Symposium
Ana-Maria Castravet, Northeastern University
Derived Categories of Moduli Spaces of Stable Rational Curves
The Grothendieck-Knudsen moduli space M0,n parametrized stable, rational curves with n markings. It is a question of Orlov whether the derived category of M0,n admits a full, strong, exceptional collection that is invariant under the action of the symmetric group Sn. I will present several approaches towards answering this question; in particular, I will explain a construction of an invariant full exceptional collection on the Losev-Manin space. This is joint work with Jenia Tevelev.
Lawrence Ein, University of Illinois, Chicago
Hilbert Scheme of Points for Singular Surfaces
Let X be a singular surface and let Hilbn(X) be the Hilbert schemes of length n -closed subschemes of X. We would discuss joint work with Xudong Zheng on when Hilbn(X) is irreducible for all n.
David Eisenbud, Mathematical Sciences Research Institute
K3 Carpets and their Equations
Some time ago Gallego and Purnaprajna proved the remarkable fact that there is a unique double structure X on each smooth rational normal 2-dimensional scroll that “looks like” a K3 surface — that is, ΩX = OX and h1OX = 0, or, in commutative algebra language, arithmetically Gorenstein with a-invariant 0. Frank Schreyer and I have been studying the equations of these and a family of related degenerate K3 surfaces. I will begin by describing the lovely old classification of scrolls, and their equations, and then describe the K3 carpets.
Yifeng Liu, Northwestern University
Non-Archimedean Dolbeault Cohomology and Superforms
We will introduce a new cohomology theory for varieties over p-adic fields using analytic logarithmic differential forms, and study the corresponding cycle class map. We will see a close relation of this cohomology with superforms defined by Chambert-Loir and Ducros. If times permits, we will discuss how one can expect Hodge isomorphisms through so-called monodromy maps.
Hsian-Hua Tseng, Ohio State University
Gromov-Witten Theory of Hilbert Schemes of Points in the Plane.
The Hilbert scheme HilbnC2 of n points on C2 parametrize 0-dimensional length-n subschemes of C2. HilbnC2 admits a natural T=(C*)2 action arising from the standard (C*)2-action on C2. About a decade ago, the T-equivariant quantum cohomology of HilbnC2 was completely determined by Okounkov-Pandharipande. In this talk we discuss some recent progress on higher genus Gromov-Witten theory of HilbnC2. This is based on a joint work with R. Pandharipande.
Junwu Tu, University of Missouri, Columbia
Gromov-Witten Invariants of Calabi-Yau A-infinity Categories
Classical mirror symmetry relates Gromov-Witten invariants in symplectic geometry to Yukawa coupling invariants in algebraic geometry. Through non-commutative Hodge theory, one can define categorical Gromov-Witten invariants associated to (Calabi-Yau A-infinity) categories. Conjecturally, this construction should reproduce the Gromov-Witten invariants and Yukawa coupling invariants, when applied Fukaya categories and Derived categories, respectively. In this talk, I describe a first computation of categorical Gromov-Witten invariants at positive genus. This is a joint work with Andrei Caldararu.