Algebraic Geometry Seminar
Fall 2011
Regular Day: Tuesday
Regular Time: 3:30PM - 4:30PM
Regular Location: LCB 222
| Date | Speaker | Talk Information | |
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September 6 |
Jie Wang
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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).
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September 13 |
Steffen Marcus
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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 Grushevsky-Zakharov),
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 most-part, an easy-going
continuation of my talk from last year.
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September 20 |
William D. Gillam
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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
Pandharipande-Thomas, 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
invariants---the latter being well-understood. In the
"favourable situations" one thus obtains "explicit formulas"
for the full descendant stable pairs theory of E.
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September 27 |
Davide Fusi
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October 4 |
María Pe Pereira
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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 n-jets, 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 1-parameter 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.
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October 18 |
Yusuf Mustopa
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Ulrich bundles occur naturally in a variety of algebraic and
algebro-geometric 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 degree-d del Pezzo
surface X, the geometry of curves of degree dr on X, and points
on these curves---and 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.
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October 28 Friday, 2pm JWB 333 |
Andrei Căldăraru
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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.
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November 8 |
Brian Lehmann
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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.
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November 15 |
Dave Anderson
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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 Kaveh-Khovanskii and Lazarsfeld-Mustata. 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 Gelfand-Tsetlin polytopes.
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November 22 |
Nathan Ilten
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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.
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November 29 |
Wenbo Niu
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Castelnuovo-Mumford regulairy of varieties has drawn
considerable attention in recent twenty years. There are two
main general results about smooth curves and surfaces.
Gruson-Peskine-Lazarsfeld 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
Eisenbud-Goto 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.
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December 6 |
Jonathan Wise
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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.
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