Algebraic geometry seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Wednesday
September 06, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Rob Silversmith, SCGP
Gromov-Witten invariants of symmetric products of projective space

Through 3 general points and 6 general lines in $P^3$, there are exactly 190 twisted cubics; 190 is a (genus-zero) Gromov-Witten invariant of $P^3$. I will introduce Gromov-Witten invariants of a smooth complex projective variety X, and discuss how a torus action on X can help us compute its Gromov-Witten invariants. In the case when X is a toric variety, Kontsevich used this method to compute any Gromov-Witten invariant of X. Givental and Lian-Liu-Yau used Kontsevichís computation to prove a mirror theorem, which states that genus-zero Gromov-Witten invariants of X have an interesting rigid structure, which had been previously predicted by physicists. I will discuss the difficulties that arise when X is not toric. In particular, I will talk about the nontoric orbifold $X=Sym^d(P^r)$, the symmetric product of projective space. By studying the equivariant geometry of $Sym^d(P^r)$, I extended the strategies of Givental/Lian-Liu-Yau to prove a mirror theorem for $Sym^d(P^r)$.


Wednesday
September 13, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Francois Greer, Stony Brook University
Elliptic Fibrations and Noether-Lefschetz Theory

The mirror symmetry philosophy suggests that the Gromov-Witten series of an elliptic fibration should be a modular form. Using the theta correspondence from automorphic forms and the Hodge theory of surfaces, we give a proof of this statement for genus 0 and base degree 1, and compute the series explicitly in many examples. Time permitting, we will discuss some ideas on how to generalize to all genera and base degrees.


Wednesday
September 20, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Philip Engel, Harvard
Algebraic/symplectic K3 dictionary

Singular integral-affine structures on the sphere arise in connection to K3 surface in two distinct ways: Algebraically, as the dual complex of the central fiber of a degenerating family, and symplectically, as the base of a Lagrangian torus fibration. The Teichmuller space of such structures thus provides a dictionary between algebraic and symplectic moduli. Joint work with Simion Filip proves various structure results on the Teichmuller space of integral-affine structures on S^2---it is itself a separated 44-dimensional integral-affine manifold, with a natural 24 dimensional foliation, whose leaf space maps to the positive cone in R^{2,18}. Heuristically, any extension of the universal family of polarized K3 surfaces to a toroidal compactification would necessarily correspond to a section of the foliation over a certain hyperplane in R^{2,18}. Thus, the existence of such a section constructed by hyperKahler rotation evinces a program to extend the universal family.


Wednesday
October 04, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Luca Schaffler, University of Massachusetts, Amherst
The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces

In this talk we describe the moduli compactification by stable pairs (also known as KSBA compactification) of a 4-dimensional family of Enriques surfaces, which arise as the bidouble covers of the blow up of the projective plane at three general points branched along a configuration of three pairs of lines. The chosen divisor is an appropriate multiple of the ramification locus. Using the theory of stable toric pairs we are able to study the degenerations parametrized by the boundary and its stratification. We relate this compactification to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the boundary of this stable pairs compactification has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two. To conclude, we construct an explicit Looijenga semitoric compactification of this 4-dimensional family which we conjecture is isomorphic to the KSBA compactification studied.


Wednesday
October 11, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Xudong Zheng, J. Hopkins University
The Hilbert scheme of points on singular surfaces

The Hilbert scheme of points on a quasi-projective variety parametrizes its zero-dimensional subschemes. When the variety is a singular surface, the geometry of the Hilbert scheme should reflect the singularity of the underlying surface. I will present a sufficient condition for the Hilbert scheme to be irreducible in terms of the singularity of the surface, namely, the surface has only Kleinian singularities, via a purely algebraic approach. I will also report work in progress on some geometric consequences following their irreducibility.


Monday
October 16, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Daniele Agostini, Visiting Stony Brook University
Asymptotic syzygies and higher order embeddings.

The syzygies of an algebraic variety are the algebraic relations between its equations, and they often encode surprising geometric informations about the variety. For example, recently Ein and Lazarsfeld proved that one can read
the gonality of a curve off the syzygies of an embedding of very high degree. I will present a partial extension of this result in higher dimensions, especially in the case
of surfaces.


Wednesday
October 18, 2017

4:00 PM - 5:00 PM
Math Tower P-131
Harold Blum, U. Michigan, Ann Arbor
Valuations, Singularities, and K-stability

Let L be a line bundle on a projective variety X. We use valuations to measure the singularities of the linear system |mL| as m goes to infinity.
Specifically, we consider the global log canonical threshold, also known as Tianís alpha invariant, and the stability thresholds of L. The stability threshold generalizes an invariant recently introduced by Kento Fujita and Yuji Odaka. When X is a Fano variety, we show that the stability threshold detects the K-(semi)stability of X. This talk is based on joint work with Mattias Jonsson.


Wednesday
October 25, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Sebastian Casalaina-Martin, U. of Colorado,. Boulder
TBA

TBA


Wednesday
November 01, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Inna Zakharevich, Cornell
TBA


Wednesday
November 08, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Fedor Bogomolov, NYU
TBA

TBA


Wednesday
November 15, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Nicolas Addington, U. Oregon
TBA

TBA


Wednesday
November 22, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Happy TKSgiving!, Happy TKSgiving!
Happy TKSgiving!

Happy TKSgiving!


Wednesday
November 29, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Christian Schnell, Stony Brook University
TBA

TBA


Wednesday
December 06, 2017

4:00 PM - 5:30 PM
Math Tower P-131
Davesh Maulik, MIT
TBA

TBA


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars