Algebraic Geometry Seminar
Department of Mathematics
Spring 2016 Lectures
Regular meeting time: Tuesdays
4:30-5:30 (Tea served at 3:30)
Place: Fine 322
|February 9||Eric Larson
|Interpolation for normal bundles of general curves
This talk will address the following question: When does there exist a curve of given degree d and genus g, passing through n general points p_1, p_2,..., p_n in P^r?
|February 16||No Algebraic Geometry Seminar|
|February 23||Benjamin Schmidt
Ohio State University
|Ample divisors on Hilbert schemes of points on surfaces
The Hilbert scheme of n points on a smooth projective surface X is a smooth projective variety due to a classical result of Fogarty. A natural question about these spaces is to determine their ample divisors. Using techniques from derived categories developed by Bayer and Macrì, we describe the nef cones if X has Picard rank 1, irregularity 0 and n is large. Moreover, we compute the nef cone if X is the blow-up of the projective plane at 8 general points for any n. This is joint work with Bolognese, Huizenga, Lin, Riedl, Woolf, and Zhao originating from the boot camp of the 2015 Algebraic Geometry Summer Institute in Utah.
|March 1||Daniel Litt
|Automorphisms of Blowups
We use p-adic analytic methods to analyze automorphisms of smooth projective varieties. We prove a version of the dynamical Mordel-Lang conjecture for arbitrary subschemes of a variety. We apply this result to (1) classify automorphisms of X for which there exists a divisor D whose intersection with its iterates are not dense in D, and (2) show that various properties of Aut(X) (for example finiteness of its component group) are not altered by blowups in high codimension. This is joint work with John Lesieutre.
|March 8||Chris Skalit
University of Illinois at Chicago
|Intersection multiplicity over a two-dimensional base
Let X be smooth over a regular, two-dimensional base scheme Y. We show that properly-meeting cycles on X intersect with positive multiplicity (in the sense of Serre's Tor-formula). When Y is one-dimensional - say, the ring of integers of a number field - we use these techniques to investigate the extent to which intersection multiplicities can detect transversality.
|March 15 (spring break) Special time: 2 pm, same room||Roberto Svaldi
|Adjoint dimension of foliations
The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.
|March 22||Kevin Tucker
University of Illinois at Chicago
|Test Ideals in F-regular rings |
In this talk, I will discuss an answer to a question of Mustata-Yoshida -- showing that every ideal in a strongly F-regular ring can be realized as a test ideal. I will also parallel the story in characteristic zero for multiplier ideals, and highlight some open questions in this direction.
University of Utah
|Ulrich bundles and variants on ACM surfaces
A sheaf F on a polarized variety (X,O_X(1)) is ACM if F(k) has no intermediate cohomology for any integer k. The variety X is ACM if O_X is ACM. One important class of ACM bundles is the class of Ulrich bundles. An Ulrich bundle is a globally generated ACM bundle which has the largest possible space of global sections in a certain sense. Spinor bundles on quadrics are a familiar example of Ulrich bundles. It is an important problem to understand whether or not a smooth projective variety admits an Ulrich bundle, even in dimension two. In this talk, I will describe striking links between Ulrich bundles and problems in commutative algebra and representation theory. I will also discuss some partial progress on the existence problem with R. Kulkarni and Y. Mustopa where we construct vector bundles on every ACM surface that are Ulrich along a general hyperplane section.
|March 29||Brooke Ullery
University of Utah
|Measures of irrationality for hypersurfaces of large degree
The gonality of a smooth projective curve is the smalles degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. I will discuss some of these notions, and present joint work with L. Ein and R. Lazarsfeld. Our main rezult is thet if X is an n-dimensional hypersurface of degree d at least (5/2)n, then any dominant rational map from X to P^n must have degree at least d-1.
|April 5||Li Li
|Frobenius semisimplicity for convolution morphisms
In the joint work with M. de Cataldo and T. Haines, we study the conjecture that, under a proper morphism of varieties over a finite field, the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple. We prove the conjecture for the (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. This implies that a stronger form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid for the convolution morphisms.
|April 12||Botong Wang
University of Wisconsin - Madison
|Cohomology jump loci and examples of NonKahler manifolds
Cohomology jump loci are homotopy invariants associated to topological spaces of finite homotopy type. They are generalizations of usual cohomology groups. I will give a survey on the theory of cohomology jump loci of projective, quasi-projective and compact Kahler manifolds, due to Carlos Simpson, Nero Budur and myself. In the second part of the talk, I will introduce some concrete examples of 6-dimensional symplectic-complex Calabi-Yau manifold, which satisfies all the known topological criterions of compact Kahler manifolds such as Hodge theory and Hard Lefschetz theorem, but fail the cohomology jump locus property of compact Kahler manifolds. The second part is joint work with Lizhen Qin.
|April 19||Zhiwei Yun
|Intersection numbers and higher derivatives of L-functions for functions fields, I
In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under the assumption that the relevant objects are everywhere unramified. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2). In this first talk, I will explain the geometric construction behind the formula.
Fine Hall 214
|Tensors and their eigenvectors
Note the time and place: 4:30pm on Monday, in Fine 224.
|April 26||Mihnea Popa
| Hodge ideals
I will present joint work with M. Mustata, in which we study a sequence of ideals arising naturally from M. Saito's Hodge filtration on the localization along a hypersurface. The multiplier ideal of the hypersurface appears as the first step in this sequence, which as a whole provides a more refined measure of singularities. We give applications to the comparison between the Hodge filtration and the pole order filtration, adjunction, and the singularities of hypersurfaces in projective space and theta divisors on abelian varieties.
( 2:30 - 3:30, Fine 322 )
University of Illinois at Urbana-Champaign
|Mirror symmetry, elliptic fibrations, and Jacobi forms
We conjecture, with evidence, that the all-genus Gromov-Witten generating function of an elliptically fibered Calabi-Yau threefold is expressed as a quotient of weak Jacobi forms with a universal denominator. For the Calabi-Yau Weierstrass fibration over the projective plane, the conjecture allows the GW invariants for any curve class to be computed algorithmically up to genus 189, while the GW invariants for curve classes which project to a plane curve of degree at most 20 can be computed algorithmically to arbitrarily high genus. This talk is based on joint work with Min-xin Huang and Albrecht Klemm.
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