Harvard/MIT Algebraic Geometry Seminar
Fall 2011
Tuesdays at 3 pm
The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT
(2146) and Harvard (Science Center 507).

Sept 132016
SC 507Harvard
Brooke Ullery
Measures of irrationality for hypersurfaces of large degree
abstract±
The gonality of a smooth projective curve is the smallest 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 definitions, and present joint work with Lawrence Ein and Rob Lazarsfeld. Our main result is that if X is an ndimensional hypersurface of degree d at least (5/2)n, then any dominant rational map from X to P^{n} must have degree at least d1.

Sept 202016
4153MIT
Simion Filip
Hodge theory and its applications in Teichmuller dynamics
abstract±
The moduli space of Riemann surfaces equipped with a holomophic 1form carries an interesting action of the group SL(2,R) which shares some features with locally homogeneous spaces. Understanding this action provides insight into understanding dynamics on individual surfaces. Hodge theory, in particular techniques from variations of Hodge structures, play a role in understanding the dynamics in moduli space.
I will introduce the basic objects in the story and explain how concepts such as real multiplication or torsion points on Jacobians come into play. Time permitting, I will discuss questions in Hodge theory motivated by dynamics, in particular the concept of Lyapunov exponents associated to a variation of Hodge structures.

Sep 272011
SC 507Harvard

Oct 42016
4153MIT
Brian Lehmann
Exceptional sets and Manin's Conjecture
abstract±
Manin's conjecture predicts the growth rate of points of bounded height after removing an exceptional set. We study the geometric consistency of the conjecture by analyzing the exceptional sets. This is joint work with Sho Tanimoto.

Oct 112016
SC 507Harvard
Robert Friedman
Deformations of cusp singularities
abstract±
Cusp singularities are a class of normal surface singularities with a rich geometry and deformation theory. In particular their smoothing components are very closely connected to the moduli of certain rational surfaces. In 1981, Looijenga gave a necessary condition for a cusp singularity to be smoothable and conjectured that this condition was also sufficient, a conjecture recently proved by GrossHackingKeel and P. Engel. This talk describes recent joint work with Engel. We define an invariant λ for every semistable Ktrivial model of a one parameter smoothing of a cusp singularity and show that all possible values of the invariant arise. Using this result, we characterize those rational double point singularities which are adjacent to cusp singularities.

Oct 182016
4153MIT
Claudiu Raicu
Cohomology of determinantal thickenings
abstract±
I will explain how to determine the cohomology groups of arbitrary equivariant thickenings of generic determinantal ideals, as well as the ranks of the maps in cohomology induced by inclusions of such thickenings. One application of this work is a concrete description of the linear functions that describe the CastelnuovoMumford regularity of sufficiently large powers and symbolic powers of determinantal ideals.

Oct 252016
SC 507Harvard
Ian Shipman
Ulrich bundles and a generalization
abstract±
Let (X,L) be a smooth, polarized variety. A vector bundle E on X is
called an Ulrich bundle if all cohomology of the bundles E(L),...,E( dim(X) L
) vanishes. I will give an introduction to Ulrich bundles, highlighting their
connection to several elementary problems of interest. Then I will describe
some of my work (joint with Y. Mustopa & R. Kulkarni) on a generalization of
Ulrich bundles and on the connection between Ulrich bundles and (higher rank)
BrillNoether theory.

Nov 12016
4153MIT
Emanuele Macri
Derived categories of cubic fourfolds and K3 surfaces
abstract±
The derived category of coherent sheaves on a cubic fourfold has a subcategory which can be thought as the derived category of a noncommutative K3 surface.
This subcategory was studied recently in the work of Kuznetsov and AddingtonThomas, among others.
In this talk, I will present joint work in progress with Bayer, Lahoz, and Stellari on how to construct Bridgeland stability conditions on this subcategory.
This proves a conjecture by Huybrechts, and it allows to start developing the moduli theory of semistable objects in these categories, in an analogue way as for the classical Mukai theory for (commutative) K3 surfaces.
I will also discuss a few applications of this result.

Nov 82016
SC 507Harvard
Francois Greer
NoetherLefschetz Theory and Elliptic CalabiYau Threefolds
abstract±
The Hodge theory of surfaces provides a link between enumerative geometry and modular forms. I will introduce a general approach to studying the GromovWitten theory of elliptically fibered CY3's using the cohomological theta correspondence. As an application, we will examine the Weierstrass model over P^{2}, proving part of the modularity statement predicted by HuangKatzKlemm using the topological string partition function.

Nov 152016
4153MIT
No Seminar
No Seminar

Nov 222016
SC 507Harvard
Kenny Ascher
Uniformity of integral points and the LangVojta conjecture
abstract±
Caporaso, Harris, and Mazur showed, assuming Lang's conjecture, that the number of rational points on a smooth curve of genus greater than 1 is uniformly bounded by an integer depending solely on the genus and number field the curve is defined over. This theorem was proven by means of a purely algebrogeometric theorem known as a "fibered power theorem". We discuss how this uniformity result follows from the fibered power theorem, and discuss recent extensions (joint with A. Turchet) aiming to, assuming the LangVojta conjecture, study uniformity results for integral points on curves and surfaces of log general type.

Nov 292016
4153MIT
Chenyang Xu
TBD

Dec 62016
SC507Harvard
Gabriele di Cerbo
Log birational boundedness of CalabiYau pairs
abstract±
I will discuss a joint work with Roberto Svaldi on
boundedness of CalabiYau pairs. Recent works in the minimal model
program suggest that pairs with trivial log canonical class should
satisfy some boundedness properties. I will show that CalabiYau pairs
which are not birational to a product are indeed log birationally
bounded. This result is the first step towards boundedness of
elliptically fibered CalabiYau manifolds.

Dec 132016
4153MIT
Mark Mclean
TBD
This seminar is organized by Joe Harris (Harvard), Bjorn Poonen (MIT), Davesh Maulik (MIT), Sam Raskin (MIT), Maksym Fedorchuk (BC), Dawei Chen (BC), Yaim Cooper
(Harvard), Anand Patel (BC), Aaron Pixton (Harvard). This seminar is supported in part by
grants from the NSF. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s)
and do not necessarily reflect the views of the National Science
Foundation.