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Algebraic Geometry Seminar 20/21 Term 1

The algebraic geometry seminar in Term 1 2020/21 will meet on Zoom. The usual time is Wednesdays at 3pm, though we will sometimes change to allow speakers from other time zones.

Wednesday 10 November, 2020. Brendan Hassett (Brown/ICERM).

Title: Group actions, symbols, and modular forms

Abstract: Consider a finite group acting faithfully on a variety X. We seek to classify such actions up to birational equivalence, i.e., that coincide after removing subvarieties of X. Analyzing the fixed points, we can construct symbol invariants that may be used to distinguish actions. The vector spaces where these take values have rich structure in their own right; for cyclic groups acting on surfaces they are related to modular forms. For more general groups, they remain mysterious and we discuss some open questions. (joint with Kresch and Tschinkel, building on work of Kontsevich and Pestun).

Wednesday 18 November, 2020. 4pm Isabel Vogt (Washington).

Title: Brill--Noether theory over the Hurwitz space

Abstract: Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. ?However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. ?The simplest case is when C is general among curves of fixed gonality. ?Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Hannah Larson that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. ?In the course of our degenerative argument, we'll exploit a close relationship with the combinatorics of the affine symmetric group.

Wednesday 25 November, 2020. Matthias Paulsen (Hannover).

Title: The construction problem for Hodge numbers

Abstract: The Hodge numbers of smooth projective varieties in a given dimension are subject to Serre duality, and in characteristic zero also to Hodge symmetry. Since unexpected further restrictions on the Hodge numbers exist, a complete classification of possible Hodge diamonds seems to be out of reach. However, the construction problem for Hodge numbers turns out to be fully solvable from an arithmetic perspective, i.e. when considering the Hodge numbers modulo an arbitrary positive integer m. In joint work with Stefan Schreieder (in characteristic zero) and Remy van Dobben de Bruyn (in positive characteristic), we show that any collection of numbers in Z/mZ which satisfy Serre duality and in characteristic zero also Hodge symmetry, can be realized as the Hodge diamond of a smooth projective variety. As an application, this solves a question of János Kollár on polynomial relations between Hodge numbers (in Z, not in Z/mZ).

Wednesday 2 December, 2020. Lie Fu (Université Lyon 1 and Radboud University)

Title: Derived category of flips and cubic hypersurfaces

Abstract: Given a standard flip between two algebraic varieties, Bondal and Orlov established a fully faithful embedding between their derived categories. We complete their result by identifying the complement, resulting a semi-orthogonal decomposition. The main application is a lifting to the level of derived categories of the so-called Galkin-Shinder relation between a cubic hypersurface, its Fano variety of lines and its Hilbert square. The 4-dimensional case provides the first example of hyper-Kähler Fano visitors, in the sense of Bondal. This is based on a joint work with Pieter Belmans and Theo Raedschelders.

Wednesday 9 December, 2020. Alex Perry (Michigan)

Title: Kuznetsov's Fano threefold conjecture via K3 categories

Abstract: Kuznetsov conjectured the existence of a correspondence between different types of Fano threefolds which identifies a distinguished semiorthogonal component of the derived category on each side. I will explain joint work with Arend Bayer which resolves one of the outstanding cases of this conjecture. This relies on the study of the Hodge theory of certain K3 categories associated to the semiorthogonal components.