Spring 2024
February 13: Stephen Maguire (University of Illinois Urbana-Champaign)
Title: Cox rings, affine group actions, and the Weitzenböck conjecture
February 20: Nachiketa Adhikari (University of Illinois Urbana-Champaign)
Title: The moduli of sheaves on a Calabi-Yau fourfold as a local derived critical locus
Abstract: It is well known that the moduli space M of sheaves on a Calabi-Yau threefold has a symmetric obstruction theory. In the framework of derived algebraic geometry, this is the classical (i.e. non-derived) truncation or “shadow” of a richer construction known as a (-1)-shifted symplectic structure. It is also known that, locally, any derived scheme or stack with a (-1)-shifted symplectic structure is a derived critical locus. For example, M is locally the critical locus of a function on the space of representations of a quiver. In this talk, we will discuss an analog of this result: that any (-2)-shifted symplectic derived stack is locally a derived Lagrangian intesection. In particular, so is the moduli of sheaves on a Calabi-Yau fourfold.
Translation: A somewhat complicated object can always be (locally) understood as the result of performing a (slightly) less complicated operation on a (slightly) less complicated object.
This is based on joint work with Yun Shi.
No prior knowledge of derived geometry will be assumed in this talk
February 27: Ian Cavey (University of Illinois Urbana-Champaign)
Title: Equivariant Localization and Hilbert schemes of points
Abstract: The localization formula is a powerful tool in equivariant cohomology that, in the presence of a suitable group action, reduces cohomological computations on a variety to those on its fixed locus. Translating through the dictionary of toric geometry, one obtains Brion's formula expressing the generating function of integer points in a rational polytope as the sum of those in its vertex cones. In the first half of this talk, we will review both of these formulas with examples. In the second half, we will see how these formulas can be used to study line bundles on Hilbert schemes of points on smooth, projective, toric surfaces in terms of those on Hilbert schemes of points in the affine plane. In combination with earlier results in the affine case, we obtain a combinatorial indexing set for a basis of the space of global sections of any ample line bundle on the Hilbert schemes of points on any Hirzebruch surface.
March 5: Dan Bragg (University of Utah)
Title: Murphy's law for the stack of curves
Abstract: We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.
March 19: no seminar
March 26: Chris Dodd (University of Illinois Urbana-Champaign)
Title: Witt differential operators
April 2: Eric Jovinelly (University of Illinois Chicago)
Title: Geometric Manin's Conjecture for Fano threefolds
Abstract: A famous conjecture of Manin predicts an asymptotic formula for counting the number of rational points of bounded height on a Fano variety defined over a number field. In the 1990s, Batyrev developed a heuristic argument for a version of Manin's Conjecture over finite fields that assumes irreducibility of certain spaces of embedded rational curves. Though Batyrev's heuristics and Manin's initial conjecture are false in general, Geometric Manin’s Conjecture (GMC) translates Batyrev’s heuristic for Manin’s Conjecture to statements about free rational curves on Fano varieties. In this talk, I will first review this translation and motivate the framework of GMC with concrete examples. I will then describe a recent proof of GMC for smooth Fano threefolds over the complex numbers by appealing to relationships between this framework and the Mori structures of Fano threefolds.
April 9: Qianyu Chen (University of Michigan)
Title: The minimal exponent and singularities
Abstract: I will introduce the minimal exponent, an invariant of hypersurface/local complete intersection singularities, which refines the log canonical threshold. Many important features of the log canonical threshold were generalized to the minimal exponent. I will also discuss the characterization of higher Du Bois and higher rational singularities using the minimal exponent.
April 16: no seminar
April 23: no seminar
April 30: Deepam Patel (Purdue)
Title: Local monodromy of constructible sheaves
Abstract: Let X be a complex algebraic variety, and X -> D a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local Monodromy Theorem (due to Brieskorn-Grothendieck-Griffiths-Landsman) asserts that the eigenvalues of local monodromy are roots of unity. In this talk, we will discuss generalizations of this result to the setting of arbitrary morphisms between complex algebraic varieties, and with coefficients in arbitrary constructible sheaves. If there is time, I'll discuss applications to variation of monodromy in abelian covers, and applications to the monodromy of alexander modules.
This is based on joint work with Madhav Nori.