Algebraic Geometry Seminar

Gross, Hacking, Keel, and Kontsevich recently constructed certain canonical bases for cluster algebras. The construction is combinatoric, but the bases are conjecturally controlled by the Gromov-Witten theory of the mirror cluster variety. I will discuss a new construction of these bases in terms of certain tropical curve counts which I hope to eventually show agree with the predicted Gromov-Witten counts. I will also discuss a refinement of the tropical counts which produces quantized versions of the canonical bases.

The space of arcs X_\infty of a singular variety X over a perfect field k has finiteness properties when we localize at its stable points. This allows to associate or recover invariants of X from its space of arcs. In the talk I will show some general properties of the stable points, pointing out our interest in computing the dimension of the complete local ring \^O_{X_\infty,P_E} when P_E is the stable point defined by a divisorial valuation v_E on X. I will also present our last result, together with H. Mourtada: assuming char k = 0, we have \embdim(\^O_{X_\infty,P_E}) = \^k_E + 1 where \^k_E is the Mather discrepancy of X with respect to ν_E. Expressed in terms of cylinders, stable points are precisely the generic points of the irreducible cylinders in X_\infty, and our result with H. Mourtada asserts that the embedding dimension of \^O_{X_\infty,P_E} is equal to the codimesion as cylinder of N_E, being N_E the closure of P_E in X_\infty. But in general we have \dim(\^O_{X_\infty,P_E}) < \embdim(\^O_{X_\infty,P_E}).

If X is a smooth variety embedded in projective space, we can form a new variety by looking at the closure of the union of all the lines through 2 points on X. This is called the secant variety of X. Similarly, the Hilbert scheme of 2 points on X parametrizes all length 2 zero-dimensional subschemes. I will talk about how these two constructions are related. More specifically, I will show how we can use certain tautological vector bundles on the Hilbert scheme to help us understand the geometry of the secant variety, leading to a proof that for sufficiently positive embeddings of X, the secant variety is a normal variety.

The aim of this talk is to present Popa and Schnell's result demonstrating that derived equivalent varieties over the complex numbers have isogenous Pic^0's and discuss the possibility of extending the result to varieties over algebraically closed fields of positive characteristic. We will give an application of such an extension to comparing zeta functions of derived equivalent varieties over finite fields. The talk will start with a short introduction to the information about algebraic groups required for Popa and Schnell's proof.

Given a smooth, embedded variety, Ulrich bundles are those bundles satisfying a certain natural extremal property. In my talk, I will survey the history of this class of bundles, and describe the connections with the representation theory of (generalized) Clifford algebras and Brill-Noether theory. Then I will present some recent work (joint with R. Kulkarni and Y. Mustopa) on the existence of Ulrich sheaves on general ACM surfaces.

The Waring rank of a complex homogeneous form is the least number of terms in an expression of the form as a sum of powers of linear forms. Waring rank, tensor rank, and various generalized ranks are interesting for a range of applications including secant varieties and geometric complexity theory, but they are difficult to compute. In particular we do not know the maximum Waring rank among forms in a given number of variables and a given degree; it is not even known whether forms of greater than generic rank exist. I will present upper and lower bounds for Waring rank and generalized ranks that narrow the possible ranges of maximum values of generalized ranks (joint work with Grigoriy Blekherman) and in some new cases show the existence of forms of above-generic Waring rank (joint work with Jaros{\l}aw Buczy\'nski).

Given a K3 surface S, a smooth curve C and a base-point free linear system A on C, there is an associated vector-bundle on S, depending on C and A, called the Lazarsfeld-Mukai vector bundle. For the case when A is a g^1_d or g^2_d for d small enough, we will show how we can use the vector-bundles to construct a rank-1 torsion free sheaf on S where the global sections cut out the divisors of |A|. We also show how these sheaves prove a special case of a conjecture posed by Donagi and Morrison.

Originally introduced by Eisenbud in the context of commutative algebra, matrix factorizations have since earned a prominent role in mathematical physics as the "D-branes of type B" in a Landau—Ginzburg model. In this spirit, Orlov proved that a certain category of matrix factorizations over affine space is equivalent to the derived category of a Calabi—Yau hypersurface.

In this talk I will describe preliminary work with Jérémy Guéré using matrix factorizations to give a new description of the virtual class in Gromov-Witten theory, FJRW theory, and so-called hybrid theories. We hope that this construction will help clarify the relationship between the virtual classes in these various theories.

There is a close relationship between the geometry of a variety in characteristic p and the action of Frobenius on its cohomology. This raises the question of how that action varies in a family of varieties of varying characteristic. I will describe our state of knowledge about the distribution of Frobenius conjugacy classes from this geometric perspective. My new results compute the exact density of a certain type of Frobenius action for varieties where we previously only knew lower bounds.

Landau-Ginzburg models appear in mirror symmetry and have connections to other important mathematical models, including those in Borcea-Voisin mirror symmetry. In this talk I will briefly review the history and construction of the A- and B-models in Landau-Ginzburg mirror symmetry and then discuss a recent Landau-Ginzburg theorem inspired by Borcea-Voisin mirror pairs (joint work with Nathan Priddis and Andrew Schaug).

We introduce the notion of a lim Cohen-Macaulay sequence of nonzero Noetherian modules {M_n}_n over a local ring R. The definition is phrased in terms of asymptotic length of higher Kozul homology of M_n with respect to one (equivalently, every) system of parameters. We prove that if lim Cohen-Macaulay sequences exist for the quotients of a regular local ring R by its prime ideals, then Serre's conjecture on positivity of intersection multiplicities holds for R. We also show that if R has such a sequence, one can use it to define a closure operation on ideals and submodules of finitely generated modules over R. In positive characteristic, lim Cohen-Macaulay sequences exist, and tight closure is a closure of operation that arises in this way. Quite generally, these closure operations enable one to construct big Cohen-Macaulay modules. Joint work with Bhargav Bhatt and Mel Hochster.

The set of k by n matrices with rank at most r naturally forms an algebraic variety. Its defining equations are given by determinants and it enjoys many beautiful properties. In this talk I'll discuss some recent work that describes how these varieties behave upon specialization with some applications to matroids and free resolutions. If time permits, I'll ask lots of questions about the role of coordinates in algebraic geometry.

How are the properties of a variety X related to those of an ample divisor D in X? Classical Lefschetz hyperplane theorems answer this question by comparing the cohomology or homotopy type of X to that of D. I'll describe new results, encapsulating some of these older Lefschetz theorems, which compare F(X) to F(D) where F is any functor which is representable in a suitable sense. For example, F can be \pi_1, or a moduli functor. While the main results are in characteristic zero, the method of proof passes through positive characteristic.

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture.