Abstracts - Spring 2020

Speaker:  Fabio Bernasconi

Title: On del Pezzo fibrations in positive characteristic

Abstract: Del Pezzo fibrations appear naturally as one of the possible outcomes of the Minimal Model Program for threefolds. In this talk, I will discuss some 'pathologies' that can appear in characteristic p and how it is possible to bound the bad behavior of such fibrations depending on the prime p. This is joint work with H. Tanaka.

Speaker:  Nathaniel Bottman

Title: Functoriality for the Fukaya category and a compactified moduli space of pointed vertical lines in C^2

Abstract: A Lagrangian correspondence between symplectic manifolds induces a functor between their respective Fukaya categories. I will begin by introducing this construction, along with a family of abstract polytopes called 2-associahedra (introduced in math/1709.00119), which control the coherences among this collection of functors. Next, I will describe joint work with Alexei Oblomkov (math/1910.02037), in which we construct a compactification of the moduli space of configurations of pointed vertical lines in C^2 modulo affine transformations (x,y) -> (ax+b,ay+c). These spaces are proper complex varieties with toric lci singularities, which are equipped with forgetful maps to \overline{M}_{0,r}. Our work yields a smooth structure on the 2-associahedra, thus completing one of the last remaining steps toward a complete functoriality structure for the Fukaya category.

Speaker:  David Stapleton

Title: Rationality & Irrationality - Specialization & Obstruction

Abstract: A theorem of Matsusaka states that ruledness behaves well under specialization. Kollár used this theorem and a specialization to characteristic p argument to show that for many degrees d, a very general Fano hypersurface of degree d is not rational. Since then there has been a great deal of work in developing new specialization results (and obstructions for the various central fibers) in order to prove the nonrationality of Fano varieties. In this talk we survey these results, and present joint work with Nathan Chen, that is a new specialization result for studying the degree of irrationality in higher dimensions. A consequence of this result, is that Fano hypersurfaces can have arbitrarily large degrees of irrationality, which is proved via specialization to characteristic p.

Speaker:  Maria Yakerson

Title: The motive of the Hilbert scheme of infinite affine space

Abstract: The Hilbert schemes (of points) of surfaces have many nice properties, however they are hard to study for higher dimensional schemes. In particular the Hilbert scheme of the two-dimensional affine space has a pure Tate motive, but we don’t know the motives of Hilbert schemes of A^n for higher n. Surprisingly, we observe that the ind-scheme Hilb(A^\infty) has a pure Tate motive, at least rationally. Moreover, if we let both the dimension of affine space and the degree of points go to infinity, we get that colim_d Hilb_d(A^\infty) is A^1-homotopy equivalent to the infinite Grassmanian, which has a well-known pure Tate motive thanks to its cellular structure. Time permitting, we will discuss applications of this result to algebraic K-theory and its version for hermitian K-theory. This is joint work with Marc Hoyois, Joachim Jelisiejew, and Denis Nardin.