Algebraic Geometry Seminar
Spring 2012
Regular Day: Tuesday
Regular Time: 3:30PM - 4:30PM
Regular Location: LCB 222
| Date | Speaker | Talk Information | |
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January 10 |
Qile Chen
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The theory of stable log maps was developed recently by
Abramovich-Chen-Gross-Siebert 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.
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| View details |
January 17 |
Noah Giansiracusa
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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 Mori-theoretic
information is completely determined by variational GIT in a
natural way. In this talk I will discuss joint work with Dave
Jensen and Han-Bom 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.
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| View details |
January 24 |
Yi Zhu
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A basic question in arithmetic geometry is whether a given
variety defined over a non-closed 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
Graber-Harris-Starr, 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.
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| View details |
January 27 Friday, 3:30pm LCB 225 |
Morgan Brown
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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.
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| View details |
February 14 |
Chenyang Xu
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The famous Yau-Tian-Donaldson conjecture says that a polarized
variety (X,L) admits a constant scalar curvature metric if and
only if it is K-polystable. The last notion is a completely
algebraic notion which I will concentrate on in this talk. More
precisely, we will study the K-stability 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 DF-invariants 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.
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| View details |
February 21 |
Ching-Jui Lai
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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 Borisov-Alexeev-Borisov conjecture which asserts
boundedness of the set of n-dimensional \(\epsilon\)-klt log
\(\mathbb Q\)-Fano varieties.
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| View details |
February 28 |
Dung Nguyen
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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.
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| View details |
March 6 |
Xiaodong Jiang
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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.
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| View details |
March 27 |
Yuchen Zhang
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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.
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| View details |
April 3 |
Yi Hu
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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 so-called
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.
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| View details |
April 10 |
Melissa Liu
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Moduli spaces of semi-stable 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
semi-stable 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.
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| View details |
April 17 |
Alberto Chiecchio
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| View details |
April 24 |
David Steinberg
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Donaldson-Thomas theory provides a virtual count of curves on a
smooth Calabi-Yau threefold X. When X is singular,
Donaldson-Thomas 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 Donaldson-Thomas 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.
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