January 10: David Stapleton (UCSD)
Title: Hypersurfaces which are far from being rational Abstract: Rational varieties are some of the simplest examples of varieties, e.g. most of their points can be parametrized by affine space. It is natural to ask (1) How can we determine when a variety is rational? and (2) If a variety is not rational, can we measure how far it is from being rational? A famous particular case of this problem is when the variety is a smooth hypersurface in projective space. This problem has attracted a great deal of attention both classically and recently. The interesting case is when the degree of the hypersurface is at most the dimension of the projective space (the "Fano" range), these hypersurfaces share many of the properties of projective space. In this talk, we present recent work with Nathan Chen which says that smooth Fano hypersurfaces of large dimension can have arbitrarily large degrees of irrationality, i.e. they can be arbitrarily far from being rational. |
January 17: Ziquan Zhuang (MIT)
Title: Positivity of CM line bundle on the K-moduli space Abstract: Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties. It is conjectured that this moduli space is projective and the polarization is given by a natural line bundle, the Chow-Mumford (CM) line bundle. In this talk, I will present a recent joint work with Chenyang Xu where we show the CM line bundle is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties, which conjecturally should be the entire K-moduli. As an application, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. |
January 24: Kristin DeVleming (UCSD)
Title: Wall crossing for K-moduli spaces of plane curves Abstract: I will discuss compactifications of the moduli space of smooth plane curves of degree d at least 4. We will regard a plane curve as a log Fano pair (P^{2}, aC), where a is a rational number, and study the compactifications coming from K stability for general log Fano pairs. We establish a wall crossing framework to study these spaces as a varies and show that, when a is small, the moduli space coming from K stability is isomorphic to the GIT moduli space. We describe all wall crossings for degree 4, 5, and 6 plane curves and discuss the picture for general Q-Gorenstein smoothable log Fano pairs. This is joint work with Kenneth Ascher and Yuchen Liu. |
January 31: Kiran Kedlaya (UCSD)
Title: The tame Belyi theorem in positive characteristic Abstract: Belyi's theorem says that on one hand, a curve over a field of characteristic 0 that admits a finite map to P^{1} ramified over at most three points must descend to a subfield algebraic over Q, and on the other hand any curve over such a subfield does indeed admit such a morphism (without any further base extension). One might ask whether a similar statement holds over a field of characteristic p, replacing Q with F_{p}. For general morphisms this is false, but it becomes true if we restrict to tamely ramified morphisms to P^{1}. Such a statement was originally given by Saidi, in which the "other hand" assertion was made conditional on the existence of some tamely ramified morphism from the given curve to P^{1}. In the pre-talk, we will discuss how to establish existence of a tamely ramified morphism in characteristic p>2. This is "classical" over an infinite algebraic extension of F_{p}; to do it over a fixed finite field requires a density statement in the style of Poonen's finite field Bertini theorem. In the talk proper, we will discuss work of Sugiyama-Yasuda that establishes the existence of a tamely ramified morphism when the base field is algebraically closed of characteristic 2. The case where the base field is finite of characteristic 2 requires a further geometric reinterpretation of the key construction of Sugiyama-Yasuda; this is joint work with Daniel Litt and Jakub Witaszek. |
Enumerative geometry seminar (AP&M5829)
February 20: Sanyanta Mandal (UIC) Title: Betti numbers of the moduli space of sheaves on projective plane Abstract: We will discuss stabilization of the Betti numbers of the moduli space of sheaves on surfaces, and in the special case of the projective plane we will produce lower bounds on the second Chern class of the sheaves such that the Betti numbers of their moduli space stabilizes. |
February 28: Ming Zhang (UBC)
Title: K-theoretic quasimap wall-crossing for GIT quotients Abstract: When X is a Grassmannian, Marian-Oprea-Pandharipande and Toda constructed alternate compactifications of spaces of maps from curves to X. The construction has been generalized to a large class of GIT quotients X=W//G by Ciocan-Fontanine-Kim-Maulik and many others. It is called the theory of epsilon-stable quasimaps. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants and prove their wall-crossing formulae for all targets in all genera. The wall-crossing formulae generalize Givental's K-theoretic toric mirror theorem in genus zero. In physics literature, these K-theoretic invariants are related to the 3d N = 2 supersymmetric gauge theories studied by Jockers-Mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the UV and the IR phases of the 3d gauge theory. It is based on joint work with Yang Zhou. |
February 29: Southern California Algebraic Geometry Seminar, at UCSD. |
March 6: Woonam Lim (UCSD)
Title: Rationality of the virtual invariant series of Quot schemes on surfaces Abstract: Oprea and Pandharipande studied the virtual Euler characteristic of Quot schemes on surfaces. Based on the calculations in several cases, they conjectured the rationality of the generating series of the virtual invariants. In this talk, I will explain the virtual χ_{-y}-genera of Quot schemes, thus refining the work of [OP]. The main result expresses the Quot scheme invariants universally in terms of Seiberg-Witten invariants of Dürr, Kabanov, and Okonek. Based on the calculations in several cases and the blow up formula, we resolve the χ_{-y}-genus analogue of the rationality conjecture of [OP], for all surfaces with p_{g}. In addition, the reduced Quot scheme invariants for K3 surfaces will be discussed with connections to the Kawai-Yoshioka formula. |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019, Fall 2019.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Iacopo Brivio at ibrivio@ucsd.edu about problems with the website or posters.