Harvard/MIT Algebraic Geometry Seminar

Fall 2011
Tuesdays at 3 pm

The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT (2-146) and Harvard (Science Center 507).

• Sep 132011 SC 507Harvard Vivek Shende
  On Severi degrees abstract±   (pdf) I will explain how Euler characteristics of Hilbert schemes of points encode the number of nodal curves present in a family. This allows results on the smoothness of relative Hilbert schemes to be translated into enumerative information. In particular, the multiplicities of the Severi varieties in the versal deformation of an integral planar curve are encoded by the Euler characteristics of the Hilbert schemes of the central fibre alone. In the global setting, the number of nodal curves in a general slice of a sufficiently ample linear system on a surface can be determined from the Euler characteristics of the relative Hilbert schemes. These in turn may be computed by a tautological integral over the Hilbert scheme of points; as a consequence we learn that these Severi degrees are given by universal polynomials in the Chern classes of surface and bundle.
• Sep 202011 2-146MIT Jack Huizenga
  Hilbert schemes of points, vector bundles, polynomial multiplication, and the golden ratio abstract±   (pdf) The Hilbert scheme of n points in the projective plane is a smooth projective variety of dimension 2n parameterizing zero-dimensional subschemes of length n. An interesting question is to determine the various birational models of this space. A first step in this program is to determine the cone of effective divisors.
  It turns out that this cone is closely related to the geometry of vector bundles on the plane. For many values of n, we will construct extremal effective divisors on the Hilbert scheme by studying Steiner Bundles, which are particularly nice vector bundles generalizing the tangent bundle. In particular, we will investigate the stability and splitting properties of these vector bundles. This will lead us to consider a basic problem about multiplication of polynomials on the projective line, whose answer involves a surprising occurrence of the golden ratio and its continued fraction expansion.
  Note: the speaker will give a pre-talk at the Baby Algebraic Geometry Seminar on September 19th.
• Sep 272011 SC 507Harvard Elham Izadi
  Some limits of the Scorza correspondence abstract±   (pdf) The Scorza correspondence is defined on the second Cartesian power of a smooth curve for an even theta-characteristic with no global sections (this is a special case of the Szego Kernel). We compute some limits of this correspondence, for instance when the theta characteristic acquires two sections or for some simple degenerations of the underlying curve. These are useful for understanding the map between the moduli spaces of curves given by the Scorza correspondence. This is joint work with Gavril Farkas.
• Oct 42011 2-146MIT Benjamin Bakker
  Lagrangian Hyperplanes in Holomorphic Symplectic Varieties abstract±   (pdf) It is well known that the extremal rays in the cone of effective curve classes on a K3 surface are generated by rational curves C for which (C,C)=-2; a natural question to ask is whether there is a similar characterization for a higher-dimensional holomorphic symplectic variety X. The intersection form is no longer a quadratic form on curve classes, but the Beauville-Bogomolov form on X induces a canonical nondegenerate form ( , ) on H₂(X;R) which coincides with the intersection form if X is a K3 surface. We therefore might hope that extremal rays of effective curves in X are generated by rational curves C with (C,C)=-c for some positive rational number c. In particular, if X contains a Lagrangian hyperplane Pⁿ ⊂ X, the class of the line l ⊂ Pⁿ is extremal. For X deformation equivalent to the Hilbert scheme of n points on a K3 surface, Hassett and Tschinkel conjecture that (l,l)=-(n+3)/2; this has been verified for n<4. In joint work with Andrei Jorza, we prove the conjecture for n=4, and discuss some general properties of the ring of Hodge classes on X.
• Oct 112011 SC 507Harvard Dawei Chen
  Geometry of Teichmueller Curves II: Quadratic Differentials abstract±   (pdf) Teichmueller curves are generated by quadratic differentials. A year ago in this seminar I talked about the case when the generator is a global square of an abelian differential. Now I would like to address the other case when it is not a global square. The situation is similar in the interior of the moduli space, but more intriguing at the boundary. Using 4-branched covers of the Riemann sphere as an example, I will highlight a beautiful interplay between billiard dynamics, flat geometry, Hurwitz counting problem, and the intersection theory on moduli space. A non-varying numerical property for such Teichmueller curves will be explained. This is a joint work in progress with Martin Moeller.
• Oct 182011 2-146MIT Anand Deopurkar
  Alternate compactifications of Hurwitz spaces abstract±   (pdf) Moduli spaces of geometrically interesting objects are usually not compact. They need to be compactified by allowing certain carefully chosen degenerations. Often, this can be done in several ways, leading to different birational models that are related in interesting ways. I will describe a range of compactifications of the Hurwitz space H^d_g, which parametrizes d sheeted, simply branched, genus g covers of the projective line. These compactifications are constructed by allowing degenererations where the branch points can collide in a prescribed way, recovering as a particular case the standard compactification by admissible covers.
  After the general construction, I will focus on the case of d = 3. In this case, the above construction gives a sequence of compactifications which contract the boundary divisors in the admissible cover compactification. I will construct a sequence of yet more compactifications that modify the interior, featuring the classical Maroni invariant of trigonal curves.
• Oct 252011
  3:00-5:15 2-132MIT Sean Keel
  Mirror symmetry for open Calabi-Yau manifolds abstract±   (pdf) I'll explain my conjecture (and theorem in dim 2), joint with Hacking and Gross, which constructs the mirror to an affine CY as the spectrum of an explicit algebra (which we conjecture is symplectic cohomology of the CY) with vector space basis a Mori theoretic generalisation of Thurston's integer laminations, and with multiplication rule governed by counts of rational curves. Then I'll indicate some of the many implications for classical algebraic geometry, e.g. that the complement to a nodal anti-canonical curve on a del Pezzo surface has canonical "theta functions" -- a canonical vector space basis of regular functions.
  
  In the second hour I will reformulate the theory of cluster algebras in terms of birational geometry of CYs, obtaining a simple conceptual proof of the Laurant phenomenon, and then indicate how the Fock-Goncharov conjecture on dual bases is a special case of our general mirror conjecture.
  Note the special time and location!
• Nov 12011 SC 507Harvard Zsolt Patakfalvi
  Moduli spaces of higher dimensional varieties abstract±   (pdf) The moduli spaces of smooth curves of genus at least two and its compactification, the space of stable curves, are one of the most investigated objects of algebraic geometry. In the past two decades, natural higher dimensional generalizations, the moduli space of canonically polarized manifolds and of stable schemes have been constructed. After giving a short introduction to the above spaces, I will talk about their global geometry, in particular about the hyperbolicity properties of the moduli space of canonically polarized manifolds.
• Nov 82011 SC 507Harvard Andrew Snowden
  Finiteness of K3 surfaces and the Tate conjecture abstract±   (pdf) Given a class of varieties, it is natural to ask if it has only finitely many members defined over each finite field. I will discuss this question for K3 surfaces and explain the following result: one has such a finiteness statement for K3 surfaces if and only if the Tate conjecture holds for K3 surfaces. This is joint work with Max Lieblich and Davesh Maulik.
• Nov 152011 2-146MIT Jie Wang
  Generic vanishing results on certain Koszul cohomology groups abstract±   (pdf) A central problem in curve theory is to describe algebraic curves in projective spaces with fixed genus and degree. One wants to know the extrinsic geometry of the curve, i.e information on the equations defining the curve. Koszul cohomology groups in some sense carry 'everything one wants to know' about the extrinsic geometry of curves in projective space: the number of equations of each degree needed to define the curve, the relations between the equations, etc. In this talk, I will present a new method using deformation theory to study Koszul cohomology of general curves. Using this method, I will describe a way to determine number of defining equations of a general curve in some special degree range (but for any genus).
• Nov 222011 SC 507Harvard Artan Sheshmani
  Higher rank stable pairs and virtual localization over local Calabi Yau threefolds and K3 surfaces abstract±   (pdf) We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X^r(-n) → F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformation- theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local P¹ using the Graber-Pandharipande virtual localization technique. Moreover, if time permits we will talk about similar computations over K3 surfaces which provide us with a higher rank analog of the Kawai-Yoshioka formula.
• Nov 292011 2-146MIT Dima Arinkin
  Compactified Jacobians of reducible singular curves abstract±   (pdf) Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In particular, the Fourier-Mukai transform gives an auto-equivalence of the coherent derived category D(J) of J.
  In this talk, I discuss extensions of this fact to singular curves C with plane singularities. In these settings, the Jacobian J has to be replaced by its compactification J'. If C is irreducible, the category D(J') still carries a Fourier-Mukai auto-equivalence. This 'categorical autoduality' appeared in my talk in this seminar this March; I will review the relevant results.
  If C is reducible, the situation is more interesting, because the 'compactification' J' actually has infinitely many irreducible components. Sometimes one can use a stability condition to select finitely many components and obtain an auto-equivalence. Alternatively, one can consider the entire J', which leads to new unexpected phenomena.
  In my talk, I will discuss the advantages and drawbacks of these two approaches. (This project is joint with T.Pantev.)
• Dec 62011 SC 507Harvard Patrick Brosnan
  Degenerations of mixed Hodge structure and applications abstract±   (pdf) I'll talk about joint work with Greg Pearlstein showing that the zero locus of an admissible normal function is algebraic. Appealing to joint work with Pearlstein and Schnell, I'll explain why this generalizes the theorem of Cattani, Deligne and Kaplan. Then I'll explain the Tanakian Galois group of a category of certain well-behaved degenerations of mixed Hodge structure (and why this relates to the zero locus).
• Dec 132011 2-146MIT Maxim Arap
  Classification of smooth weak Fano threefolds of Picard number two abstract±   (pdf) Smooth Fano threefolds have been classified through the works of Fano, Iskovskikh, Shokurov, Mori and Mukai. Threefolds whose anti-canonical class is nef and big, but not ample, are called weak Fano (or almost Fano). The aim of this seminar is to briefly review the classification of smooth Fano threefolds and to report the recent results on the classification of smooth weak Fano threefolds of Picard number two.

This seminar is being organized by Joe Harris (Harvard), James McKernan (MIT), David Smyth (Harvard), and Vivek Shende (MIT). Abstracts for previous semesters' seminars are located here. 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.