Algebraic Geometry Seminar

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.

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.

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.

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.

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.

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.

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.

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.