Algebraic Geometry Seminar
Spring 2017 — Tuesdays 3:30  4:30 PM, location LCB 219
Date  Speaker  Title — click for abstract (if available) 
January 17 


January 24 


January 31 


February 7 
Travis Mandel University of Utah 
Descendant log GromovWitten theory and tropical curves
GromovWitten (GW) theory is concerned with virtual counts of
algebraic curves which satisfy various conditions. I will motivate the log
GW theory of GrossSiebert and AbramovichChen by explaining how log
structures result in a theory which is better behaved than ordinary GW
theory (e.g., less superabundance, betterbehaved psiclasses, easily
imposed tangency conditons, and invariance in logsmooth families). I will
then explain a correspondence between certain descendant log GW invariants
and certain counts of tropical curves (from the perspective of
NishinousSiebert, but allowing for psiclasses and arbitrary genus). This
is based on joint work with H. Ruddat.

February 14 


February 21 Room Change: JWB 333 
Donghai Pan Stanford University 
Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces
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.

February 28 LCB 219 from now on 
Fei Xie UCLA 
Toric varieties over nonclosed fields
Toric varieties over nonclosed fields can be viewed as "noncommutative"
algebraic varieties. More precisely, in MerkurjevPanin 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
nonclosed fields, and study some examples in the motivic category. Time permits, I
will briefly discuss the relation between the motivic category and noncommutative
motives.

March 7 
Jakub Witaszek Imperial College London 
Images of toric varieties and liftability of the Frobenius morphism
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
(DemaillyHwangMokPeternell) or toric varieties (OcchettaWi?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.

March 14 
Spring Break 

March 21 
Wenliang Zhang University of Illinois at Chicago 
A duality theorem for de Rham cohomology of graded Dmodules
I will discuss a duality result for de Rham cohomology of graded Dmodules
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
Dmodules will be assumed.) This is a joint work with Nicholas Switala.

March 28 
Roi Docampo University of Oklahoma 
Differentials on the arc space
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.

April 4 
David H. Yang Massachusetts Institute of Technology 
Curves on Fano Varieties and Manin's Conjecture
Following work of LehmannTanimoto, 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 BatyrevManin conjecture. Then we will discuss generalizations to higher genus and give some concrete applications of this framework.

April 11 
Tyler Kelly University of Cambridge 
A Toric Orlov Theorem via LandauGinzburg Models
Orlov's theorem defines and proves that any smooth Fano (resp.
general type) hypersurface in projective nspace has a subcategory (resp.
supercategory) in its bounded derived category of coherent sheaves that is
a Fractional CalabiYau category. We prove this is the case for a certain
class of toric complete intersections. The method to find this is by
studying a LandauGinzburg 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).

April 18 
Kalina Mincheva Yale University 
Prime congruences and Krull dimension for additively idempotent semirings
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.

April 25 
Colleen Robles Duke University 
Generalizing the SatakeBailyBorel compactification
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 MumfordTate domain. The later are
generalizations of period domains and I will also discuss the connections with Hodge
theory and moduli.

This web page is maintained by Tiankai Liu and YuChao Tu.