September 28 Michail Savvas (UCSD) |
Generalized Donaldson-Thomas Invariants via Kirwan Blowups |
Donaldson-Thomas (abbreviated as DT) theory is a sheaf theoretic technique of enumerating curves on a Calabi-Yau threefold. Classical DT invariants give a virtual count of Gieseker stable sheaves provided that no strictly semistable sheaves exist. This assumption was later lifted by the work of Joyce and Song who defined generalized DT invariants using Hall algebras and the Behrend function, their method being motivic in nature. In this talk, we will present a new approach towards generalized DT theory, obtaining an invariant as the degree of a virtual cycle inside a Deligne-Mumford stack. The main components are an adaptation of Kirwan’s partial desingularization procedure and recent results on the structure of moduli of sheaves on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem and Jun Li. |
October 5 Kristin DeVleming (UCSD) |
Moduli of surfaces in P^{3} |
For fixed degree d, one could ask for a meaningful compactification of the moduli space of smooth degree d surfaces in P^{3}. In other words, one could ask for a parameter space whose interior points correspond to [isomorphism classes of] smooth surfaces and whose boundary points correspond to degenerations of these surfaces. Motivated by Hacking's work for plane curves, I will discuss a KSBA compactification of this space by considering a surface S in P^{3} as a pair (P^{3}, S) satisfying certain properties. We will study an enlarged class of these pairs, including singular degenerations of both S and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree d is odd, we can give a rough classification of the objects on the boundary of the moduli space. |
October 12 Stefano Filipazzi (University of Utah) |
A generalized canonical bundle formula and applications |
Birkar and Zhang recently introduced the notion of generalized pair. These pairs are closely related to the canonical bundle formula and have been a fruitful tool for recent developments in birational geometry. In this talk, I will introduce a version of the canonical bundle formula for generalized pairs. This machinery allows us to develop a theory of adjunction and inversion thereof for generalized pairs. I will conclude by discussing some applications to a conjecture of Prokhorov and Shokurov. |
October 26 Joaquin Moraga (University of Utah) |
Minimal log discrepancies and Kollár components |
The minimal log discrepancy of an algebraic variety is an invariant which measures the singularites of the variety. For mild singularities the minimal log discrepancy is a non-negative real value; the closer to zero this value is, the more singular the variety. It is conjectured that in a fixed dimension, this invariant satisfies the ascending chain condition. In this talk we will show how boundedness of Fano varieties imply some local statements about the minimal log discrepancies of klt singularities. In particular, we will prove that the minimal log discrepancies of klt singularities which admit an ε-plt blow-up can take only finitely many possible values in a fixed dimension. This result gives a natural geometric stratification of the possible mld's on a fixed dimension by finite sets. As an application, we will prove the ascending chain condition for minimal log discrepancies of exceptional singularities in arbitrary dimension. |
November 16 Justin Lacini (UCSD) |
Log Del Pezzo surfaces in positive characteristic |
A log Del Pezzo surface is a normal log terminal surface with anti-ample canonical bundle. Over the complex numbers, Keel and McKernan have classified all but a bounded family of the simply connected log Del Pezzo surfaces of rank one. In this talk we extend their classification in positive characteristic, and in particular we prove that for p>5 every log Del Pezzo surface of rank one lifts to characteristic zero with smooth base. As a consequence, we see that Kawamata-Viehweg vanishing holds in this setting. Finally, we exhibit some counter-examples in characteristic two, three and five. |
November 17 (Saturday) |
The Southern California Algebraic Geometry Seminar takes place at UCLA. |
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. Contact Jonathan Conder at jconder@ucsd.edu about problems with the website or posters.