# Algebraic Geometry Seminar

The algebraic geometry seminar in Term 2 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.

See the talks from Term 1 here.

Wednesday 3 February 2021, 3pm. Speaker: Nick Addington (Oregon)

Title: A categorical sl_2 action on some moduli spaces of sheaves

Abstract: We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman, Yoshioka, and Nakajima. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. I'll spend most of my time on a nice example. This is joint with my student Ryan Takahashi.

Wednesday 10 February 2021, 3pm. Speaker: Dan Corey (Wisconsin).

Title: The Ceresa class: tropical, topological, and algebraic

Abstract: The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve. It is homologically trivial but not algebraically equivalent to zero for a very general curve. In this sense, it is one of the simplest algebraic cycles that goes ``beyond homology.'' The image of the Ceresa cycle under a certain cycle class map produces a class in étale homology called the Ceresa class. We define the Ceresa class for a tropical curve and for a product of commuting Dehn twists on a topological surface. We relate these to the Ceresa class of a smooth algebraic curve over C((t)). Our main result is that the Ceresa class in each of these settings is torsion. Nevertheless, this class is readily computable, frequently nonzero, and implies nontriviality of the Ceresa cycle when nonzero. This is joint work with Jordan Ellenberg and Wanlin Li.

Wednesday 17 February 2021, 3pm. Speaker: Anne Lonjou (Paris-Saclay)

Title: Actions of Cremona groups on CAT(0) cube complexes

Abstract: A key tool to study the plane Cremona group is its action on a hyperbolic space. Sadly, in higher rank such an action is not available. Recently in geometric group theory, actions on CAT(0) cube complexes turned out to be a powerful tool to study a large class of groups. In this talk, based on a common work with Christian Urech, we will construct such complexes on which Cremona groups of rank n act. We will then see which kind of results on these groups we can obtain.

Wednesday 24 February 2021, 3pm. Speaker: Asher Auel (Dartmouth)

Title: Brill-Noether theory for cubic fourfolds

Abstract: It is well-known that the study of special cubic fourfolds leads to beautiful connections between algebraic cycles, the rationality problem, K3 surfaces, and derived categories. In this talk, I'll explain a connection to the theory of algebraic curves, via a notion of Brill-Noether theory for cubic fourfolds. Specifically, I'll discuss the open problem of establishing lower bounds on the Clifford index of smooth Brill-Noether special cubic fourfolds.

Wednesday 3 March 20201, 3pm. Speaker: Marcello Bernardara

Title: Fanos of K3 Type: isomorphisms and classification of Hodge structures and K3 categories

Abstract: Fano varieties of (derived) K3 type are Fano varieties whose Hodge structure (derived category) contains a K3-type sub-Hodge structure (subcategory). Many examples of such varieties are known, arising as zeroes of homogeneous bundles on Grassmannians, in dimensions that grow up to 19. In this talk, I will first present joint work with Fatighenti and Manivel showing that many of these examples can be related by geometric correspondences and have actually the same K3-type Hodge structure. I will also present an ongoing project with Fatighenti, Manivel and Tanturri, whose aim is to show that in the case of Fano fourfolds, the only possible K3-type structures which are not actual K3 can arise from Gushel-Mukai, cubics and Küchle c5 fourfolds.