Algebraic Geometry Seminar
Fall 2016 — Tuesdays 3:30  4:30 PM, location: LCB 222 (Note room change starting 9/27)
Date  Speaker  Title — click for abstract (if available) 
August 30 Room: JFB 102 
Grigory Mikhalkin University of Geneva 
Quantum index of real plane curves and refined enumerative geometry
We note that under certain conditions, the area bounded by the
logarithmic image
of a real plane curve is a halfinteger multiple of pi square. The
halfinteger number
can be interpreted as the quantum index of the real curve and used to
refine enumerative invariants.

August 31 (special RTG seminar) time: 3:304:30pm room: LCB 219 
Jenny Wilson Stanford University 
Representation theory and higherorder stability in the configuration spaces of a manifold
Let F_k(M) denote the ordered kpoint configuration space of a
connected open manifold M. Work of Church and others shows that for a
given manifold, as k increases, this family of spaces exhibits a
phenomenon called homological "representation stability" with respect to
the natural symmetric group actions. In this talk I will explain what this
means, and describe a higherorder "secondary stability" phenomenon among
the unstable homology classes. The project is work in progress, joint
with Jeremy Miller.

September 6 
Chung Ching Lau University of Utah 
On subvarieties with nef normal bundle
The goal of this talk is to study the positivity of subvarieties
with nef normal bundle in terms of intersection theory. We first give a
generalization to the notion of subvarieties with nef normal bundle, via
the positivity of the exceptional divisor. We call these subvarieties nef.
Then we show that restriction of a pseudoeffective divisor to a nef
subvariety is pseudoeffective. We also show that nefness is a transitive
property. Next, we define the weakly movable cone as the cone generated by
the pushforward of cycle classes of nef subvarieties via proper surjective
maps, and show that this cone contains the movable cone and shares similar
intersectiontheoretic properties with it, using the aforementioned
properties of nef subvarieties. The main tool used in this work is the
theory of $q$ample divisors, as developed by Totaro.

September 13 
Natalie Hobson University of Georgia 
Identities between first Chern classes of vector bundles of conformal blocks
Given a simple Lie algebra $\mathfrak{g}$, a positive integer $\ell$,
and an $n$tuple $\vec{\lambda}$ of dominant integral weights for $\mathfrak{g}$ at level $\ell$,
one can define a vector bundle on $\overline{\operatorname{M}}_{g,n}$ known as a \textit{vector bundle of conformal blocks}.
These bundles are nef in genus $g=0$ and so this family provides potentially an infinite number of elements in the nef cone of
$\M_{0,n}$ to analyze. Result relating these divisors with different data is thus significant in understanding these objects.
In this talk, we use correspondences of these bundles with products in quantum cohomology in order to classify when a bundle with
$\sL_2$ or $\sP_{2\ell}$ is rank one.
We show this is also a necessary and sufficient condition for when these divisors are equivalent.

September 20 
Lei Song University of Kansas 
On normal generation of line bundles on a surface
Let $L$ be a line bundle on a smooth projective surface $X$. A conjecture attributed
to S. Mukai says if $L\simeq \omega_{X}\otimes A^{\otimes k}$ for some ample line
bundle $A$ and an integer $k\ge 4$, then $L$ is normally generated. In this talk, I
will discuss various methods and results in the curve and surface case. I will show
the conjecture holds for a double covering over an anticanonical rational surface,
which may be viewed as a two dimensional analogue of hyperelliptic curve, and
equivariant ample line bundle $A$. The work builds on an understanding of linear
systems on anticanonical rational surfaces, which is largely due to B. Harbourne.

September 27 Room change to LCB 222 starting today 
Aaron Bertram University of Utah 
Veronese secants, Gorenstein rings and stability
A hyperplane in the vector space k[x_0,...,x_n]_d
of homogeneous polynomials of degree d defines the socle of a Gorenstein
graded ring. A canonical Bridgeland stability condition allows us
to filter these rings (or more precisely their graded resolutions)
as objects of the derived category of coherent sheaves and we use this
to find some new invariants. In particular, this stratifies the space of
polynomials by generalized secant varieties to the Veronese.
This is joint work with Brooke Ullery.

October 4 
Chi Li Purdue University 
Minimizing normalized volumes of valuations
Motivated by the study of KahlerEinstein/SasakiEinstein metrics, I
will discuss an algebrogeometric problem of minimizing normalized volumes among all
real valuations centered at any QGorenstein klt singularity. It was conjectured
that there is always a unique minimizer, which should be quasimonomial. I will
discuss recent progresses on this problem, and explain how it is related to the
socalled Ksemistability and the deFernexEinMustata type inequalities. Part of
this work is based on joint works with Yuchen Liu and Chenyang Xu.

October 11 
Fall Break 

October 18 
Tom Alberts University of Utah 
The Geometry of the Last Passage Percolation Model
Last passage percolation is a wellstudied model in probability
theory that is simple to state but notoriously difficult to analyze. In
recent years it has been shown to be related to many seemingly unrelated
things: longest increasing subsequences in random permutations, eigenvalues
of random matrices, and longtime asymptotics of solutions to stochastic
partial differential equations. Much of the previous analysis of the last
passage model has been made possible through connections with
representation theory of the symmetric group that comes about for certain
exact choices of the random input into the last passage model. This has the
disadvantage that if the random inputs are modified even slightly then the
analysis falls apart. In an attempt to generalize beyond exact analysis,
recently my collaborator Eric Cator (Radboud University, Nijmegen) have
started using tools of tropical geometry to analyze the last passage model.
The tools we use to this point are purely geometric, but have the potential
advantage that they can be used for very general choices of random inputs.
I will describe the very pretty geometry of the last passage model, our
work in progress to use it to produce probabilistic information, and our
goal of eventually detropicalizing our approach and using it to analyze
the socalled directed polymer problem.

October 25 


November 1 
Calum Spicer UC San Diego 
Mori Theory for Foliations
Work by McQuillan and Brunella demonstrates the existence of a Mori theory for rank 1 foliations on surfaces. In this talk we will discuss an extension of some of these results to the case of rank 2 foliations on threefolds, as well as indicating how a complete Mori theory could be developed in this case.

November 8 
Thomas Goller University of Utah 
Finite quot schemes on the projective plane
Following ideas of Marian and Oprea, finite quot schemes can be
used to investigate Le Potier's strange duality conjecture for surfaces. I
will discuss recent work with Aaron Bertram and Drew Johnson in which we
prove the existence of a large class of finite quot schemes on the
projective plane. We use nice resolutions of general stable vector
bundles, which also yield an easy proof that these bundles are globally
generated whenever their Euler characteristic suggests that they should
be.

November 15 


November 16 Joint seminar 2:003:00PM LCB 222 
Rachel Pries Colorado State University 
Galois action on homology of Fermat curves
We prove a result about the Galois module structure of the
Fermat curve using commutative algebra, number theory, and algebraic
topology. Specifically, we extend work of Anderson about the action of the
absolute Galois group of a cyclotomic field on a relative homology group of
the Fermat curve. By finding explicit formulae for this action, we
determine the maps between several Galois cohomology groups which arise in
connection with obstructions for rational points on the generalized
Jacobian. Heisenberg extensions play a key role in the result. This is
joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

November 21 Special Time and Location 3:004:00PM LCB 225 
Giulia Saccà Stony Brook University 
Geometry of O'Grady's 6 dimensional example
There are not many known examples of compact irreducible hyperk?hler
manifolds. Two series of examples appear in dimension 2n, for every
n>1, and are related to the Hilbert scheme of points on a K3 or an
abelian surface; and in dimension 6 and 10 there is one extra, or
exceptional, deformation class, each of which was found by O'Grady.
While considerable work has been devoted to studying hyperk?hler
manifolds belonging to the first two deformation classes, not much is
known for the exceptional deformation classes. In this talk I will
present joint work with G. Mongardi and A. Rapagnetta, regarding the
geometry of O'Grady's six dimensional example. By realizing these
examples as "quotients" of another hyperk?hler manifold by a
birational involution, we are able to compute the Hodge numbers
and, in work in progress, also study properties of their moduli
spaces/deformation class.

November 29 
Jeff Achter Colorado State University 
Distinguished models of intermediate Jacobians
Consider a smooth projective variety over a number field. The image
of the associated (complex) AbelJacobi map inside the
(transcendental) intermediate Jacobian is a complex abelian variety.
We show that this abelian variety admits a distinguished model over
the original number field, and use it to address a problem of Mazur on
modeling the cohomology of an arbitrary smooth projective variety by
that of an abelian variety. (This is joint work with Sebastian
CasalainaMartin and Charles Vial.)

December 6 
Mihai Paun University of Illinois at Chicago 
Positivity of quotients of cotangent bundles
I will report on a joint work with F. Campana.
Consider a projective manifold whose
canonical bundle is pseudoeffective.
We show that the determinant of any quotient
of some tensor power of its cotangent bundle is
pseudoeffective as well.
This can be seen as a generalization of the important
generic semipositivity
result of Y. Miyaoka; some applications will equally
be discussed.

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