Algebraic Geometry Seminar
Spring 2012 — Tuesdays 3:304:30, LCB 222
Date  Speaker  Title — click for abstract (if available) 
January 10 
Qile Chen Columbia Univeristy 
Stable log maps
The theory of stable log maps was developed recently by
AbramovichChenGrossSiebert to generalize the theory of
relative stable maps. To illustrate the idea of using log
geometry, I will focus on the basic situation when the target
is given by a variety with a Cartier divisor. Stable log maps
relative to more complicated boundary, for example simple
normal crossings or even toric, can be constructed in a
similar manner. This is based on a joint work with Dan
Abramovich.

January 17 
Noah Giansiracusa Universität Zürich 
GIT compactifications of \(M_{0,n}\) and flips
Despite intensive investigation over the years and its innocuously
classical appearance, there remain some tantalizing open questions
regarding the birational geometry of the moduli space of marked
rational curves. One such question, dating from a 2000 paper of
Hu and Keel, is to determine whether \(\bar{M}_{0,n}\) is a Mori dream
space. Roughly speaking, this would say that its Moritheoretic
information is completely determined by variational GIT in a
natural way. In this talk I will discuss joint work with Dave
Jensen and HanBom Moon in which we construct a wide range of
birational models using a GIT construction inspired by two
constructions of Kapranov from 1993. These models include
\(\bar{M}_{0,n}\) itself as well as all the Hassett weighted models.
A consequence is that we exhibit explicit flips between models and
give a glimpse of the Mori dream behavior envision by Hu and Keel.

January 24 
Yi Zhu Stony Brook 
Families of Homogeneous Spaces over Curves
A basic question in arithmetic geometry is whether a given
variety defined over a nonclosed field admits a rational
point. When the base field is of geometric nature, i.e.,
function fields of varieties, one naturally hopes to solve the
problem via purely geometric methods. In this talk, I will
discuss the geometry of the moduli space of sections of a
projective homogeneous space fibration over an algebraic curve
and its MRC quotient. By the results of Esnault and
GraberHarrisStarr, it leads to answers for the existence of
rational points on projective homogeneous spaces defined either
over a global function field or over a function field of an
algebraic surface.

January 27 Friday, 3:30pm LCB 225 
Morgan Brown University of California, Berkeley 
Singularities of Cox Rings of Fano Varieties
The Cox Ring of an algebraic variety is a generalization of the
homogeneous coordinate ring of a projective variety. I will
give an introduction to Cox Rings and how they are used in
birational geometry, as well as some ideas from the minimal
model program, with the aim of showing that the Cox ring of a
Fano variety over the complex numbers is Gorenstein.

February 14 
Chenyang Xu University of Utah 
Special test configurations and Kstability
The famous YauTianDonaldson conjecture says that a polarized
variety (X,L) admits a constant scalar curvature metric if and
only if it is Kpolystable. The last notion is a completely
algebraic notion which I will concentrate on in this talk. More
precisely, we will study the Kstability question of Fano
varieties. Our new point is that we will put in the machinery
of the Minimal Model Program to modify the test configuration
and show that the DFinvariants are decreasing. In particular,
we answer a conjecture of Tian under the assumption that the
Picard number is 1. This is a joint work with Chi Li.

February 21 
ChingJui Lai University of Utah 
Bounding volumes of singular Fano three folds
Mildly singular Fano varieties of Picard number one are
important objects to study from the minimal model program point
of view. For the set of three dimensional \(\epsilon\)klt log
\(\mathbb Q\)Fano pairs \((X,\Delta)\) of Picard number one, we show
that there is a volume bound \((K_X+\Delta)^3\leq
M(3,\epsilon)\) depending only \(\epsilon\). This result is
related to the BorisovAlexeevBorisov conjecture which asserts
boundedness of the set of ndimensional \(\epsilon\)klt log
\(\mathbb Q\)Fano varieties.

February 28 
Dung Nguyen Colorado State University 
Characteristic numbers of elliptic space curves
Counting curves in projective spaces that pass through various
linear subspaces and that are tangent to various hyperplanes
(or hypersurfaces) is a classical theme in algebraic geometry.
An example is there are 3264 conics tangent to 5 general conics
in the projective plane. Several fundamental problems in this
area remained unsolved until the advent of Kontsevich moduli
space of stable maps. In this talk, I will discuss how to use
this tool to count genus one space curves.

March 6 
Xiaodong Jiang University of Utah 
Effective Iitaka fibrations
In this talk, we are going to prove a uniformity result for the
Iitaka fibration f from X to Y, provided that the generic fiber
has a good minimal model and the variation of f is zero or that
the Kodaira dimension of X is equal to the dimension of X minus
1.

March 27 
Yuchen Zhang University of Utah 
Pluricanonical map in positive characteristic
For a nonsingular projective variety X of general type, it's
known that mK_X induces a birational map for any m
sufficiently large. It's an important problem to bound this
integer m. In this talk, we will show that, in positive
characteristic, 4K_X is birational providing that X has
maximal Albanese dimension.

April 3 
Yi Hu University of Arizona 
Derived and modular resolutions of the Stable map moduli and
applications
In this talk, I will present a derived version of the
resolutions of the moduli spaces of stable maps. This
resolution can be used to rigorously define the socalled
reduced GW numbers of CY threefolds (i.e., the GW numbers
associated to the main components of the stable map moduli).
The derived resolutions are singular (in the usual sense). For further applications, a resolution (in the usual sense) is desirable; I will describe how to achieve this by a sequence of modular blowups in the case of genera 1 and 2. 
April 10 
Melissa Liu Columbia University 
Moduli spaces of real and quaternionic vector bundles over a real
algebraic curve
Moduli spaces of semistable real and quaternionic vector bundles
of fixed topological type over a smooth real algebraic curve can
be expressed as Lagrangian quotients and embedded into the
symplectic quotient corresponding to the moduli variety of
semistable algebraic vector bundles of fixed rank and degree on
the complexified curve. When the rank and degree are coprime,
these Lagrangian quotients are connected components of the real
locus of the complex moduli variety endowed with the real
structure induced from the real structure of the complex curve.
The presentation as a quotient enables us to generalize the
methods of Atiyah and Bott to a setting with involutions, and
compute the mod 2 Poincare polynomials of these moduli spaces of
real and quaternionic vector bundles in the coprime case. This is
based on joint work with Florent Schaffhauser.

April 17 
Alberto Chiecchio University of Washington 
TBA 
April 24 
David Steinberg University of British Columbia 
Tilted pairs and the DonaldsonThomas crepant resoultion conjecture
DonaldsonThomas theory provides a virtual count of curves on a
smooth CalabiYau threefold X. When X is singular,
DonaldsonThomas theory is not defined. However, when X is the
coarse moduli space of an orbifold, there are two candidates
for producing virtual counts related to X: virtual counts on
the orbifold itself, and virtual counts on a crepant resolution
of X. The DonaldsonThomas crepant resolution conjecture states
that these two approaches are equivalent. In this talk, I will
present progress in proving this conjecture by introducing the
intermediate counting theory of tilted pairs.

Fall 2011 — Tuesdays 3:304:30, LCB 222
Date  Speaker  Title — click for abstract (if available) 
September 6 
Jie Wang University of Utah 
Generic vanishing results on certain Koszul cohomology groups
A central problem in curve theory is to describe algebraic
curves in a given projective space with fixed genus and degree.
One wants to know the extrinsic geometry of the curve, i.e
information on the equations defining the curve. Koszul
cohomology groups in some sense carry 'everything one wants to
know' about the extrinsic geometry of curves in projective
space: the number of equations of each degree needed to define
the curve, the relations between the equations, etc. In this
talk, I will present a new method using deformation theory to
study Koszul cohomology of general curves. Using this method, I
will describe a way to determine number of defining equations
of a general curve in some special degree range (but for any
genus).

September 13 
Steffen Marcus University of Utah 
A comparison theorem for double Hurwitz classes and Jacobian classes.
Consider the locus L of curves inside the moduli space of
smooth curves admitting a map to the projective line with
prescribed ramification profile over two points. This geometric
condition can be expressed in two equivalent ways, either as a
Hurwitz space, or by intersecting sections of the universal
Jacobian. Each gives rise to a Chow class that corresponds to
some closure of L inside some partial compactification of the
moduli space of curves. In this talk, I will discuss how these
classes compare, how they may be expressed in the tautological
ring (thanks to recent work of Hain, and GrushevskyZakharov),
and how this comparison may possibly relate to other results in
Hurwitz theory. This is joint work with Renzo Cavalieri and
Jonathan Wise and will be, for the mostpart, an easygoing
continuation of my talk from last year.

September 20 
William D. Gillam Brown University 
Quotient schemes and stable pairs
Let E be a rank two vector bundle over a Riemann surface C. The
moduli space of stable pairs on E, in the sense of
PandharipandeThomas, provides an alternative to the space of
stable maps to E for the purpose of "counting" curves in the
threefold E. The scaling action on E induces a torus action on
the stable pairs moduli space. The moduli spaces of torus
fixed stable pairs can be described as closed subschemes of
products of quotient schemes of symmetric powers of E. The
description is somewhat compatible with the obstruction
theories. In favourable situations this can be used to express
stable pairs invariants in terms of quotient scheme
invariantsthe latter being wellunderstood. In the
"favourable situations" one thus obtains "explicit formulas"
for the full descendant stable pairs theory of E.

September 27 
Davide Fusi University of Utah 
Geometry of varieties of small rational degree 
October 4 
María Pe Pereira Ins. de Math. de Jussieu 
Nash problem for surfaces
Nash formulated this problem in an attempt to understand
resolution of singularities of a variety X in relation with the
space of arcs in X centered at the singular locus. The space of
arcs is an infinite dimensional algebraic variety given by the
inverse limit of the spaces of njets, which are finite
dimensional algebraic varieties. Consider a resolution of
singularities of X and take the decomposition of the
exceptional divisor \(E = \cup_i E_i\). Given any arc
\(\gamma \colon (\mathbb{C}, 0) → (X, SingX)\)
one can consider the lifting
\(\gamma \colon (\mathbb{C}, 0) → (X, E)\).
Nash considered the set of arcs whose lifting \(\gamma\) meets
a fix divisor \(E_i\) , that is \(\gamma(0) \in E_i\) and
proved that these are irreducible sets of the space of arcs.
Nash's question is whether for the essential divisors \(E_i\)
they are in fact irreducible components of the space of arcs or
not (an essential divisor appears by definition in any
resolution of X up to birational mapping). He conjectured that
the answer was yes for the case of surfaces (for which there
exists a minimal resolution that has only essential divisors)
and suggested the study in higher dimensions. In 2003, Ishii
and Kollár gave an example of a variety of dimension 4 for
which some of these sets are not. Hence the case of dimension 2
and 3 remained opened.
Recently we solved the conjecture for the surface case in a joint work with J. Fernández de Bobadilla. I will give an introduction to the problem and details a of the proof for the normal surface case. After works of M. Lejenune Jalabert, A. Reguera and J. Fernández de Bobadilla the problem deals with holomorphic 1parameter families of convergent arcs. The key of our approach is to work with representatives of appropriate arc families and find a topological obstruction to their existence. The obstruction is expressed as a bound for the euler characteristique of the normalization of the representative of the generic member of the family, which we know is a disc. 
October 18 
Yusuf Mustopa University of Michigan 
Ulrich Bundles on del Pezzo Surfaces
Ulrich bundles occur naturally in a variety of algebraic and
algebrogeometric topics, including determinantal and Pfaﬃan
descriptions of hypersurfaces, the computation of resultants,
and the representation theory of generalized Cliﬀord algebras.
In this talk I will discuss the connection between the
existence of rank r Ulrich bundles on a degreed del Pezzo
surface X, the geometry of curves of degree dr on X, and points
on these curvesand how del Pezzo surfaces are the only
arithmetically Gorenstein surfaces for which this connection
can hold. This is joint work with Emre Coskun and Rajesh
Kulkarni.

October 28 Friday, 2pm JWB 333 
Andrei Căldăraru University of Wisconsin, Madison 
The Hodge theorem as a derived selfintersection
The Hodge theorem is one of the most important results in
complex geometry. It asserts that for a complex projective
variety X the topological invariants \(H^*(X, \mathbb{C})\) can
be refined to new ones that reflect the complex structure. The
traditional statement and proof of the Hodge theorem are
analytic. Given the multiple applications of the Hodge theorem
in algebraic geometry, for many years it has been a major
challenge to eliminate this analytic aspect and to obtain a
purely algebraic proof of the Hodge theorem. An algebraic
formulation of the Hodge theorem has been known since
Grothendieck's work in the early 1970's. However, the first
purely algebraic (and very surprising) proof was obtained only
in 1991 by Deligne and Illusie, using methods involving
reduction to characteristic p. In my talk I shall try to
explain their ideas, and how recent developments in the field
of derived algebraic geometry make their proof more geometric.

November 8 
Brian Lehmann Rice University 
Algebraic bounds on analytic multiplier ideals
A classical theorem of Kodaira states that ample line bundles
are characterized by the positivity of their curvature form.
More generally, one expects that the geometric "positivity" of a
line bundle L can be detected on the metrics carried by L. The
key tool relating these two concepts is the multiplier ideal. I
will introduce multiplier ideals and explain how to obtain
bounds on the behavior of analytic multiplier ideals using
algebraic constructions.

November 15 
Dave Anderson University of Washington 
Okounkov bodies, toric degenerations, and polytopes
Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and in the last few years, this construction has been developed further in work of KavehKhovanskii and LazarsfeldMustata. In general, the Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. As an application, I'll describe some toric degenerations of flag varieties and Schubert varieties, and explain how the Okounkov bodies arising generalize the GelfandTsetlin polytopes.

November 22 
Nathan Ilten University of California, Berkeley 
On the Hilbert Scheme of Degree Twelve Fano Threefolds
Hilbert schemes provide a useful tool for moduli problems, but
are difficult to explicitly describe in most situations. In my
talk, I will discuss a specific example, namely the Hilbert
scheme of degree 12 Fano threefolds. Among its many irreducible
components, there are four special components which correspond
to different families of smooth Fano threefolds. I will
describe the geometry of these special components and their
intersection behavior. Motivation coming from mirror symmetry
for studying this particular Hilbert scheme will also be
discussed. This project is joint work with J. Christophersen.

November 29 
Wenbo Niu Purdue University 
A regularity bound for normal surfaces
CastelnuovoMumford regulairy of varieties has drawn
considerable attention in recent twenty years. There are two
main general results about smooth curves and surfaces.
GrusonPeskineLazarsfeld showed that for a smooth curve X, one
has \(reg X \leq deg X  codim X+1\). And Lazarsfeld showed
that for a smooth surface the above result is still true. This
regularity bound was formulated and conjectured by
EisenbudGoto for any variety. In this talk we use duality
theory to first give a quick proof for smooth curves and
surfaces and then prove this bound for normal surfaces which
have rational, Gorenstein elliptic or log canonical
singularities. This is joint work with Lawrence Ein.

December 6 
Jonathan Wise Stanford University 
Infinitesimal deformation theory and Grothendieck topologies
To probe the infinitesimal structure of a moduli space of
geometric objects, one seeks to understand families of those
objects over "fat points". Remarkably, these deformation
problems tend to admit cohomological solutions of a common
form: obstructions in H^2, deformations in H^1, and
automorphisms in H^0. I will offer an explanation for this
common form, coming from some exotic Grothendieck topologies.
We will see how this point of view works in several examples.
No prior knowledge about Grothendieck topologies or deformation
theory will be assumed.

Archive of previous seminars.
This web page is maintained by Roi Docampo Álvarez and Steffen Marcus.