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October 8: Johannes Schmitt (University of Zürich)
SPECIAL TIME: 10:00am pretalk 10:30-11:30am main talk Title: Strata of k-differentials and double ramification cycles Abstract: The moduli space of stable curves parameterizes tuples (C,p1,...,pn) of a compact, complex curve C together with distinct marked points p1, ..., pn. Inside this moduli space, there are natural subsets, called the strata of k-differentials, defined by the condition that there exists a meromorphic k-differential on C with zeros and poles of some fixed multiplicities at the points pi. I will discuss basic properties of these strata and explain a conjecture relating their fundamental class to the so-called double ramification cycles on the moduli space. I explain the idea of the proof of this conjecture and some ongoing work with Costantini and Sauvaget on how to use this relation to compute intersection numbers of the strata with psi-classes on the moduli of curves. |
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October 15: Nikolas Kuhn (Institut Mittag-Leffler)
SPECIAL TIME:10:30am pretalk 11:00am-12:00pm main talk Title: Blowup formulas for virtual sheaf-theoretic invariants on projective surfaces Abstract:For a smooth projective surface X, natural objects of study are its moduli spaces of (semi-) stable coherent sheaves. In rank one, their structural invariants are well-understood, starting with Göttsche's famous formula for the Betti numbers of the Hilbert schemes of points of X in terms of the Betti numbers of X itself. Even for rank two, however, little is known. There are results for particular choices of X by Yoshioka and others, and a blowup formula for virtual Hodge numbers due to Li-Qin. In general, the moduli spaces are non-smooth and one often studies virtual analogues of invariants, which are better behaved and have connections to physics. For example, there is an elegant conjectural formula for the virtual Euler characteristics of rank 2 moduli spaces due to Göttsche and Kool. I will present joint work with Y. Tanaka on a blowup formula for virtual invariants of moduli spaces of sheaves on a surface, which presents a step towards proving an analogue of Li-Qin's blowup formula for the virtual Euler characteristic. |
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October 22: Ming Zhang (UCSD)
Title: Equivariant Verlinde algebra and quantum K-theory of the moduli space of vortices Abstract: In studying complex Chern-Simons theory on a Seifert manifold, Gukov-Pei proposed an equivariant Verlinde formula, a one-parameter deformation of the celebrated Verlinde formula. It computes, among many things, the graded dimension of the space of holomorphic sections of (powers of) a natural determinant line bundle over the Hitchin moduli space. Gukov-Pei conjectured that the equivariant Verlinde numbers are equal to the equivariant quantum K-invariants of a non-compact (Kahler) quotient space studied by Hanany-Tong. In this talk, I will explain the setup of this conjecture and its proof via wall-crossing of moduli spaces of (parabolic) Bradlow-Higgs triples. It is based on work in progress with Wei Gu and Du Pei. |
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October 29: Justin Lacini (University of Kansas)
Title: Logarithmic bounds on Fujita's conjecture Abstract: A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then K_X+mL is basepoint free for m>=n+1. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for n>=2 the conjecture holds for m larger than n(loglog(n)+3). This is joint work with L. Ghidelli. |
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SPECIAL ENUMERATIVE GEOMETRY SEMINAR
November 2: Naoki Koseki (University of Edinburgh) Time: 2-3:30pm Title:Cohomological chi-independence for Gopakumar-Vafa invariants Abstract: Maulik and Toda recently proposed a mathematical definition of Gopakumar-Vafa(GV) invariants, which count one-dimensional torsion sheaves on a Calabi-Yau 3-fold (CY3), and is conjecturally equivalent to other curve counting invariants such as Gromov-Witten invariants (GV/GW correspondence conjecture). One of the mysterious features of the GV invariants is the so-called cohomological chi-independence, which is expected from the GV/GW correspondence. In this talk, I will explain backgrounds and recent developments on the GV theory, including my work in progress, joint with Tasuki Kinjo (Tokyo), on the cohomological chi- independence for certain non-compact CY3s. |
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November 5: Yi Hu (University of Arizona)
Title: Local Resolution of Singularities Abstract: Mnev's universality theorem asserts that every singularity type over the ring of integers appears in some thin Schubert cell of the Grassmannian Gr(3,E) for some vector space E. We construct sequential blowups of Gr(3,E) such that certain induced birational transforms of all thin Schubert cells become smooth over prime fields. This implies that every singular variety X defined over a prime field admits local resolutions. For a singular variety X over a general perfect field k, we spread it out and deduce that X/k admits local resolution as well. |
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November 19: Valery Alexeev (University of Georgia)
Title: Compact moduli spaces of K3 surfaces Abstract:I will explain recent results on modular, geometrically meaningful compactifications of moduli spaces of K3 surfaces, most of which are joint with Philip Engel. A key notion is that of a recognizable divisor: a canonical choice of a divisor in a multiple of the polarization that can be canonically extended to any Kulikov degeneration. For a moduli of lattice-polarized K3s with a recognizable divisor we construct a canonical stable slc pair (KSBA) compactification and prove that it is semi toroidal. We prove that the rational curve divisor is recognizable, and give many other examples. |
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SPECIAL ENUMERATIVE GEOMETRY SEMINAR
December 7: Nicola Tarasca (Virginia Commonwealth University) Time: 2:15-3:30pm Title: Incident varieties of algebraic curves and canonical divisors Abstract:The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by the recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. The strata are in fact the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a graph formula for the class of the restriction of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings of decorated stable graphs. I will conclude with some applications. Joint work with Iulia Gheorghita. |
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, Winter 2020, Spring 2020, Fall 2020, Winter 2021, Spring 2021.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Samir Canning at srcannin@ucsd.edu about problems with the website or posters.