Algebraic Geometry Seminar

Spring 2011 — Tuesdays 3:30-4:30, LCB 323

Date Speaker Title — click for abstract (if available)
January 28
2-3pm, LCB 222
Chenyang Xu
Strong rational connectedness of open surfaces
We show that the smooth locus of a log del Pezzo surface is not only rationally connected but even strongly rationally connected. This gives an affirmative answer to a conjecture due to Hassestt-Tschinkel. It time permits, I will also talk about applications of this result, namely to prove some classes of del Pezzo surfaces over function fields of curves satisfy weak approximations.
February 1 Tyler Jarvis
Rescheduled: March 1
February 8 Daniel Erman
Sextic covers and Gale duality
The moduli space of n to 1 covers is well understood for n at most 5, and it turns that these moduli spaces are best understood in terms of a rather concrete question: when can the ideal of n points in projective space be generated by the minors of a matrix of linear forms? I will first explain some of what was previously known about the moduli of degree n covers, and then I will discuss some recent work on the case of sextic covers. In particular, by illustrating a connection with Gale duality, we identify local and global obstructions to extending previous structural theorems to the sextic case. This is joint work with Melanie Matchett Wood.
February 22 Alex Küronya
Arithmetic properties of volumes of divisors
The volume of a Cartier divisor on an irreducible projective variety describes the asymptotic rate of growth of the number of its global sections. As such, it is a non-negative real number, which happens to be rational whenever the section ring of the divisor in question is finitely generated.

In a joint work with Catriona Maclean and Victor Lozovanu we study the multiplicative semigroup of volumes of divisors. We prove that this set is countable on the one hand, on the other hand it contains transcendental elements.
March 1
3-4pm, JWB 335
Tyler Jarvis
Moduli of Curves with W-structure, Mirror Symmetry, and the Landau-Ginzburg/Calabi-Yau correspondence
The moduli of curves with W-structure and their corresponding cohomological field theories form an orbifolded Landau-Ginzburg A-model and are the subject of several beautiful mirror symmetry conjectures. Some of these conjectures have been proved, including the Witten ADE-integrable hierarchies conjecture, while others are still open. In this talk I will give an overview of the theory as well as discussing recent progress on some of the conjectures, including the Landau-Ginzburg/Calabi-Yau correspondence.
March 8 Mihai Fulger
Univ. of Michigan
Local volumes
Given a normal variety X of dimension at least two, we fix a point x on it and define a volume over x for any Cartier divisor on an arbitrary birational model dominating X. This volume function can be used to define and study a notion of volume for a normal isolated singularity that is a generalization of a volume introduced by J. Wahl for surface singularities. We also compare our volume of isolated singularities to a different recent generalization of Wahl's work, as introduced by S.Boucksom, T. de Fernex and C. Favre.
March 15 Dusty Ross
Colorado State
Open Gromov-Witten Theory and the Crepant Resolution Conjecture
The crepant resolution conjecture relates the GW theory of an orbifold to the GW theory of a crepant resolution. We suspect that the CRC for toric Calabi-Yau threefolds can be approached locally via "open" GW theory. In this talk, I will lay out the foundations for open GW theory of toric CY threefolds and describe the recent success of the local approach for the specific orbifold [O(-1)+O(-1)/Z_2] where the action is trivial on the base and nontrivial on the fibers. This is joint work with my advisor Renzo Cavalieri.
March 16
Wed, JWB 335
Martí Lahoz
Univ. de Barcelona
Bicanonical map of higher dimensional irregular varieties
I will give a numerical criterion for the birationality of the bicanonical map of a smooth irregular variety of arbitrary dimension. The criterion is given in terms of the generic vanishing of the canonical line bundle and related to the properties of the Albanese map of the variety.

The study of the pluricanonical maps of varieties of maximal Albense dimension was initiated by Chen and Hacon and has also been developed by Pareschi and Popa. Part of the work I will present has been done in collaboration with M.A. Barja, J.C. Naranjo and G. Pareschi.
March 29 Zach Teitler
Boise State
April 5 Stefan Kebekus
Extension properties of differentials and applications I
April 12 Stefan Kebekus
Extension properties of differentials and applications II
April 15
1-2pm, LCB 222
Stefan Kebekus
Extension properties of differentials and applications III
April 26 Jeff Achter
Colorado State
Arithmetic Torelli maps for cubic surfaces and threefolds
It has been known for some time that to a complex cubic surface or threefold one can canonically associate a principally polarized. I will explain a construction which works for cubics over an arithmetic base, and discuss what this tells us about the structure of the moduli space of cubic surfaces.

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