Algebra and Algebraic Geometry Seminar 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B131.

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Fall 2024 Schedule

date	speaker	title	host/link to talk
September 27	Joshua Mundinger (Madison)	Hochschild homology and the HKR spectral sequence	local
October 4	Dima Arinkin (Madison)	Derived category of the stacky compactified Jacobian	local
November 14 (2-3pm, Birge 348)	Yunfeng Jiang (Kansas)	Intro pre-talk in GAGS	Andrei/Ruobing
November 15	Yunfeng Jiang (Kansas)	Enumerative geometry for KSBA spaces	Andrei/Ruobing
December 13	Thomas Brazelton (Harvard)	Symmetry in classical enumerative geometry	Andrei/Josh
January 31	Jacob Koncki (Warsaw)	TBA	Laurentiu
April 4	Sam Grushevsky (Stony Brook)	TBA	Dima

Abstracts

Joshua Mundinger

Hochschild homology and the HKR spectral sequence

Hochschild homology of an algebraic variety carries the Hochschild-Konstant-Rosenberg (HKR) filtration. In characteristic zero, this filtration is split, yielding the HKR decomposition of Hochschild homology. In characteristic p, this filtration does not split, giving rise to the HKR spectral sequence. We describe the first nonzero differential of this spectral sequence. Our description is related to the Atiyah class.

Dima Arinkin

Derived category of the stacky compactified Jacobian

Abstract: The Jacobian of a smooth projective curve is an abelian variety which is identified with its own dual. This implies that its derived category carries a non-trivial auto-equivalence - the Fourier-Mukai transform. When the curve has planar singularities, the Jacobian is no longer compact (and, in particular, not an abelian variety), but it turns out that the Fourier-Mukai transform still exists, provided we compactify the Jacobian. The transform can be viewed as the `classical limit' of the geometric Langlands correspondence.

In this talk, I will explore what happens when the curve becomes reducible. From the point of view of the geometric Langlands conjecture, it is important to work with the compactified Jacobian viewed as a stack (rather than the corresponding moduli space). In my talk I will show that this also leads to certain issues, and in fact that the most general version of the statement is inconsistent, while more conservative versions are true.

Yunfeng Jiang

Enumerative geometry for KSBA spaces

Motivated by theoretical physics—string theory and gauge theory,  the curve counting invariants -- Gromov-Witten theory and Donaldson-Thomas theory have been a hot research  subject in recent decades. A key point in the development of Gromov-Witten theory is the Deligne-Mumford compactification of the moduli space \bar M_g of stable curves. Witten's conjecture and Kontsevich's theorem studied the tautological integral of tautological classes over this space \bar M_g. Gromov-Witten invariants are the curve counting invariants defined by the virtual fundamental class of the moduli space of stable maps to a target variety X. The role of the virtual fundamental class is crucial due to the fact that  the moduli space of stable maps is in general singular.

It is a long time question to generalize the above story to the counting of higher dimensions. The moduli space of general type surfaces (or log general type surfaces) was completed by Kollár-Shepherd-Barron-Alexeev (KSBA) called the KSBA compactification. Not like the moduli space of stable curves, where the worse singularities are nodal singularities which are locally complete intersection (lci) singularities,  the worst singularities in the boundary of the KSBA space are semi-log-canonical (slc) singularities. In these two talks I will first talk about the general background of the construction of the virtual fundamental class for the KSBA spaces. In the second talk I will outline my method to construct virtual fundamental class for KSBA spaces. This allows us to define tautological invariants on KSBA spaces and do the enumerative geometry on such spaces.

Thomas Brazelton

Symmetry in classical enumerative geometry In this talk we'll discuss an equivariant principle of conservation of number, proven using methods from equivariant homotopy theory. It roughly states that in the presence of symmetry, not only the number of solutions is conserved, but their symmetries are as well. For instance when a cubic surface is defined by a symmetric polynomial, its 27 lines always carry the same S4 action. We apply this idea in joint work with C. Bethea to compute bitangents to smooth plane quartics with nontrivial automorphism groups, where we see that homotopical techniques directly reveal patterns which are not obvious from a classical moduli perspective. We will also discuss work with S. Raman, in which we initiate a study of Galois groups of symmetric enumerative problems, leveraging tools from Hodge theory and computational numerical analysis.