Algebraic Geometry Seminar

Gromov-Witten (GW) theory is concerned with virtual counts of algebraic curves which satisfy various conditions. I will motivate the log GW theory of Gross-Siebert and Abramovich-Chen by explaining how log structures result in a theory which is better behaved than ordinary GW theory (e.g., less superabundance, better-behaved psi-classes, easily imposed tangency conditons, and invariance in log-smooth families). I will then explain a correspondence between certain descendant log GW invariants and certain counts of tropical curves (from the perspective of Nishinous-Siebert, but allowing for psi-classes and arbitrary genus). This is based on joint work with H. Ruddat.

Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there's a correspondence between hyperelliptic curves and pencil of quadrics. I'll give a brief review of Reid's work and then describe a higher degree generalization of the correspondence.

Toric varieties over non-closed fields can be viewed as "noncommutative" algebraic varieties. More precisely, in Merkurjev-Panin category of motives (a full subcategory of Tabuada's category of noncommutative motives), a smooth projective toric variety subject to certain conditions is an "affine object", i.e, isomorphic to a single (noncommutative) algebra. In particular, any smooth projective toric surface is an "affine object" in this sense. I will introduce toric varieties over non-closed fields, and study some examples in the motivic category. Time permits, I will briefly discuss the relation between the motivic category and noncommutative motives.

The celebrated proof of the Hartshorne conjecture by Shigefumi Mori allowed for the study of the geometry of higher dimensional varieties through the analysis of deformations of rational curves. One of the many applications of Mori's results was Lazarsfeld's positive answer to the conjecture of Remmert and Van de Ven which states that the only smooth variety that the projective space can map surjectively onto, is the projective space itself. Motivated by this result, a similar problem has been considered for other kinds of manifolds such as abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties (Occhetta-Wi?niewski). In my talk, I would like to present a completely new perspective on the problem coming from the study of Frobenius lifts in positive characteristic. This is based on a joint project with Piotr Achinger and Maciej Zdanowicz.

I will discuss a duality result for de Rham cohomology of graded D-modules and its applications to understanding local cohomology modules associated with embedding of complex smooth projective varieties into projective spaces. (No background on de Rham cohomology or D-modules will be assumed.) This is a joint work with Nicholas Switala.

We study the sheaf of K?hler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, and a new proof of the curve selection lemma for arc spaces. This is joint work with Tommaso de Fernex.

Following work of Lehmann-Tanimoto, we will discuss what structure can be expected in the moduli space of rational curves of fixed degree in an arbitrary Fano variety X. Among other things, we propose a heuristic that should recover the classical Batyrev-Manin conjecture. Then we will discuss generalizations to higher genus and give some concrete applications of this framework.

Orlov's theorem defines and proves that any smooth Fano (resp. general type) hypersurface in projective n-space has a subcategory (resp. supercategory) in its bounded derived category of coherent sheaves that is a Fractional Calabi-Yau category. We prove this is the case for a certain class of toric complete intersections. The method to find this is by studying a Landau-Ginzburg model associated to the toric complete intersection and then using geometric invariant theory. We will try to focus on studying a few motivating examples from previous literature. This work is joint with David Favero (Alberta).

We propose a definition for prime congruences which allows us to define Krull dimension of a semiring as the length of the longest chain of prime congruences. We give a complete description of prime congruences in the polynomial and Laurent polynomial semirings over the tropical semifield $\mathbb{R}_{\max}$, the semifield $\mathbb{Z}_{\max}$ and the Boolean semifield $\mathbb{B}$. We show that the dimension of the polynomial and Laurent polynomial semiring over these idempotent semifields is equal to the number of variables plus the dimension of the ground semifield. We extend this result to all additively idempotent semirings. We relate this notion of Krull dimension to dimension of tropical varieties.

The SBB compactification realizes a locally Hermitian symmetric space as an open, dense subset of a projective algebraic variety. I will discuss a generalization of this construction to obtain a "horizontal completion" (not a compactification) of an arithmetic quotient of a Mumford-Tate domain. The later are generalizations of period domains and I will also discuss the connections with Hodge theory and moduli.