Geometry of Moduli Spaces

Title: Stratifying Symmetric Tensors by Instability

Abstract: To a symmetric d-tensor (= hyperplane in the space of homogeneous polynomials) one can associate a minimal free resolution of a Gorenstein ring. Using a canonical stability condition on the derived category of vector bundles on projective space, we can then stratify the space of symmetric d-tensors according to the instability of the free resolution. This gives an interesting refinement of the usual stratifications of symmetric tensors, and it "explains" the defective secant varieties to Veronese embeddings. This is joint work with Brooke Ullery.

Title: Every rational Hodge isometry between two K3 surfaces is algebraic

Abstract: We present a proof of the fact that given a Hodge isometry Psi between the rational second cohomology of two Kahler K3 surfaces S_1 and S_2, we can find a finite sequence of K3 surfaces interpolating between S_1 and S_2, and analytic (2, 2)-classes supported on successive products, such that the isometry Psi is the convolution of these classes. The proof of this fact implies that for projective K3 surfaces S_1, S_2 the class of Psi is algebraic. This proves a conjecture of I. Shafarevich.

Title: Geometry of moduli spaces of differentials

Abstract: Moduli spaces of differentials with a given type of zeros and poles on Riemann surfaces provide a stratification of the Hodge bundle, whose study has broad connections to flat geometry and billiard dynamics. In this talk we will introduce this topic from the perspective of an algebraic geometer, with a focus on recent results and open problems.

Title: Beyond geometric invariant theory

Abstract: Geometric invariant theory is an essential tool for constructing moduli spaces in algebraic geometry. Its advantage, that the construction is very concrete and direct, is also in some sense a drawback, because for a given moduli problem it is often intractable to explicitly describe GIT semistable objects in an intrinsic and simple way. Recently a theory has emerged which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will discuss some new progress in this program: joint with Jarod Alper and Jochen Heinloth, we give a simple necessary and sufficient criterion for an algebraic stack to have a good moduli space. This leads to the construction of good moduli spaces in many new examples, such as the moduli of Bridgeland semistable objects in derived categories.

Title: Properties of general sheaves on Hirzebruch surfaces

Abstract: Let X be a Hirzebruch surface. Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter. Therefore it makes sense to ask about the properties of a general sheaf. We consider two main questions of this sort. First, the weak Brill-Noether problem seeks to compute the cohomology of a general sheaf, and in particular determine whether sheaves have the "expected" cohomology that one would naively guess from the sign of the Euler characteristic. Next, we use our solution to the weak Brill-Noether problem to determine when a general sheaf is globally generated. A key technical ingredient is to consider the notion of prioritary sheaves, which are a slight relaxation of the notion of semistable sheaves which still gives an irreducible stack. Our results extend analogous results on the projective plane by Gottsche-Hirschowitz and Bertram-Goller-Johnson to the case of Hirzerbruch surfaces. This is joint work with Izzet Coskun.

Title: A conjectural formula for Witten's r-spin class

Abstract: Witten's r-spin class is a cycle in the moduli space of r-spin curves, which plays the role of the virtual class in the quantum singularity theory of the A_{r-1}-singularity. One motivation to study Witten's class is that in the case r=5, its fifth power is connected via the conjectural LG/CY correspondence to the Gromov-Witten theory of the quintic threefold. There are several (technical) constructions of Witten's class, and an explicit formula for the image of Witten's class on the moduli space of curves. In my talk, I want to discuss a conjectural formula for Witten's class on the moduli space of r-spin curves.

Title: Schreyer's Conjecture and Hurwitz Spaces

Abstract: Green's famous conjecture predicts a close relationship between the Brill-Noether theory of a curve and the algebraic invariants associated to its homogeneous coordinate ring. Schreyer's Conjecture goes beyond this by proposing the following geometric explanation for Green's conjecture: all "extremal" linear syzygies should come from rational normal scrolls associated to minimal pencils. I will discuss a proof of Schreyer's Conjecture for generic k-gonal curves. The approach taken involves a study of a certain determinantal divisor on a partial compactification of Hurwitz space. This is joint work with G. Farkas.

Title: MMP for moduli of sheaves on Enriques and bielliptic surfaces via Bridgeland wall-crossing

Abstract: Since Bridgeland introduced his mathematical formulation of Douglas’pi-stability, Bridgeland stability conditions have become a powerful tool for answering many questions in the study of coherent sheaves on varieties, especially with regard to the birational geometry of their moduli. In this talk, I will report on the application of this perspective to the study of stable sheaves on Enriques and bielliptic surfaces. In joint work with K. Yoshioka, we prove that any two moduli spaces of Bridgeland stable objects of Mukai vector v with respect to two generic stability conditions are birational. We achieve this by completely classifying the geometric behavior induced by crossing any given wall W. We further conjecture that all minimal models of these (often singular) moduli spaces arise as Bridgeland moduli. In sole authored work, I obtain a similar classification for bielliptic surfaces, proving on the way many heretofore unknown fundamental results about moduli of sheaves and Bridgeland stable objects on bielliptic surfaces (such as the existence of coarse projective Bridgeland moduli spaces and criteria for their nonemptiness).

Title: Hilbert schemes of points and multiple q-zeta values

Abstract: This talk is about a conjecture of Okounkov regarding connections between Hilbert schemes of points on algebraic surfaces and multiple q-zeta values.

Title: A tale of four theories

Abstract: Around a decade ago the following four (C^*)^2-equivariant theories are proven to be equivalent:

• Gromov-Witten theory of P^1 \times C^2 relative to three fibers;
• Donaldson-Thomas theory of P^1\times C^2 relative to three fibers;
• Quantum cohomology of Hilbert schemes of points on C^2;
• Quantum cohomology of symmetric product stacks of C^2.

Title: Enumeration of singular subvarieties with tangency conditions

Abstract: In this talk we will discuss the enumeration of subvarieties cut by sections of vector bundles. These subvarieties can be in a smooth variety of any dimension with very general given singularity type and tangency conditions. The motivation is the enumeration of nodal curves on surfaces (Gottsche'ss conjecture) and Caporaso-Harris' recursive formula for nodal plane curves satisfying given tangency conditions with a line. We construct a new type of algebraic cobordism theory and show part of enumeration of subvarieties are invariants of them, and in general additional series are needed. This gives us the structure and some explicit terms of the enumeration, and in particular a new relation of Caporaso-Harris invariants.

Title: Canonical points on K3 surfaces and hyper-Kahler varieties

Abstract: The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one hand, the Chow group of 0-cycles is known to be huge. On the other hand, the 0-cycles arising from intersections of divisors and the second Chern class of the tangent bundle all lie in a one dimensional subgroup. A conjecture of Beauville and Voisin gives a generalization of this property to hyper-Kähler varieties. In my talk, I will recall these beautiful stories, and explain a conjectural connection between the K3 surface case and the hyper-Kähler case. If time permits, I will also explain how to extend this connection to Fano varieties of lines on some cubic fourfolds. This is based on a joint work with Junliang Shen and Qizheng Yin.