Algebraic Geometry Seminar
Fall 2014 — Tuesdays 3:304:30, location LCB 225
Date  Speaker  Title — click for abstract (if available) 
Semptember 2 
YuChao Tu University of Utah 
Higher ramification loci over homogeneous spaces
Consider a branched covering map from an irreducible projective variety X
to a smooth variety Y. In 1980, Gaffney and Lazarsfeld extended the notion
of branched locus to “higher ramification loci” which stratifies the
branched locus, and they proved that when Y is a projective space, there
are anticipated codimension bounds for the loci. I will explain their
method with some background, and show that this result is also true when Y
is a projective homogeneous space with Picard number 1.

September 9 
Karl Schwede University of Utah 
Fsingularities and weak ordinarity
I will discuss recent work of Bhargav Bhatt, myself and Shunsuke
Takagi relating several open problems and generalizing work of Mustata
and Srinivas. First: whether a smooth complex
variety is ordinary after reduction to characteristic $p > 0$ for
infinitely many $p$. Second: that multiplier ideals reduce to test ideals
for infinitely many $p$ (regardless of coefficients). Finally, whether
complex varieties with Du Bois singularities have $F$injective
singularities after reduction to infinitely many $p > 0$.

September 16  Tiankai Liu
University of Utah 
Cuspidal curves and Cremona transformations
We抣l discuss various constraints on rational curves homeomorphic to their normalizations in the complex projective plane, and consider the question of whether every such curve is necessarily equivalent to a line, under a birational automorphism of the plane.

September 23  Herbert Clemens
University of Utah 
A global model for heterotic  Ftheory duality with Wilson line symmetry breaking
We construct a global geometric model for heterotic  Ftheory duality,
that is, a CalabiYau threefold with two flat E_8 bundles with SU(5)symmetry
(heterotic side) and a CalabiYau fourfold with SU(5) root symmetry (Ftheory side),
both equipped with compatible involutions that, in the respective quotients, break
the SU(5)symmetry to the SU(3)xSU(2)xU(1)symmetry of the physics 'standard model.'
The duality is realized by an equivariant semistable degeneration of the
CalabiYau fourfold to the union of two components, each of which is a socalled
DP9bundle containing a common CalabiYau threefold (along which the two components
are joined). A result of FriedmanMorganWitten, gives a dictionary between flat
E_8 bundles and DP9bundles.
(NB: The terminology DP9 stands for 'Del Pezzo 9surface' which the string
theorists use for something that is, in fact, not a Del Pezzo surface but what you
get when you blow up the onepoint base locus of K in a Del Pezzo surface with
(K)^2 = 1.) This is joint with S. Raby and T. Pantev.

September 30 
Katrina Honigs UC Berkeley 
Zeta Functions of Derived Equivalent Varieties
In this talk I will present results demonstrating that derived equivalence between varieties over finite fields that are either abelian or surfaces implies equality of zeta functions.

October 7 
Alexander Polishchuk University of Oregon 
Moduli of curves and Ainfinity structures
I will describe an elementary construction of invariants of higher genus curves
which are analogous to
the jinvariant of an elliptic curve. Then I will discuss how the jinvariant and
its higher genus analogs appear in the study
of certain Ainfinity algebras associated with algebraic curves. I will also explain
how a similar construction
in the case of genus 1 curves with n marked points leads to a nonstandard modular
compactification of M_{1,n} studied earlier
by Smyth (this is a joint work with Yanki Lekili).

October 14  FALL BREAK  FALL BREAK 
October 21 
Rachel Webb Brigham Young University 
LandauGinzburg mirror symmetry
I will present the LandauGinzburg mirror symmetry conjecture and its proof for all invertible singularities. The proof includes a reconstruction theorem for the Amodel (FJRW theory) and the Bmodel (SaitoGivental theory) which states that the allgenus potential is determined by certain terms of the the genuszero fourpoint function. This work is joint with Weiqiang He, Si Li, and Yefeng Shen.

October 28 4:00 pm 
Dmitry Zakharov
Courant Institute of Mathematical Sciences, New York University 
The AbelJacobi map and relations in the tautological ring
Given an algebraic curve with marked points and integer weights on the markings, we can consider the corresponding divisor. This defines an AbelJacobi map from the moduli space of marked curves of compact type to the universal Jacobian variety. Pulling back relations from the universal Jacobian for various values of the weights gives a plethora of tautological relations on the space of curves of compact type. These relations include those discovered by Faber, Getzler, Belorousski and Pandharipande, and Tavakol, and this method was also used by Hain to compute the double ramification cycle. I will talk about the extension of the AbelJacobi map to a larger moduli space of marked curves, the corresponding extended formula for the double ramification cycle, and Pixton's conjectures about the extensions of these relations to the entire DeligneMumford compactification.

November 4 
Jesse Kass University of South Carolina 
What is the limit of a line bundle on a nonnormal variety?
On a nonnormal variety, the limit of a family of line bundles is not
always a line bundle. What is the limit? I will present an answer to this question
and give some applications. Time permitting, I will discuss connections with Néron
models, autoduality, and recent work of R. Hartshorne and C. Polini.

November 11  Brian Lehmann
Boston College 
Positivity for cycles and asymptotic invariants
Divisors exhibit a close relationship between numerical "positivity" and asymptotic invariants. I will discuss the analogous principle for cycle classes of arbitrary codimension.

November 18 
Dawei Chen Boston College 
Effective divisors on moduli space of curves
The cone of effective divisors contains important information about the birational geometry of a variety. In this talk I will give an introduction to this subject, with a focus on the case when the ambient variety is the moduli space of curves. If time permits, I will also talk about effective cycles of higher codimension.

November 25  Drew Johnson
University of Utah 
Strange Duality for Del Pezzo Surfaces
Strange duality is a conjectural perfect pairing between sections of determinant (or theta) line bundles on moduli spaces of sheaves. Marian and Oprea resolved this conjecture in the affirmative for curves by constructing a finite Quot scheme whose points gave a basis for the image of the strange duality map. We attempt to use this strategy on surfaces, the most accessible case being del Pezzo surfaces. Some of the Quot schemes we construct will be Quot schemes in a tilted category. I will discuss some interesting computations and partial results that support our conjecture.

December 2 
Sebastian CasalainaMartin University of Colorado at Boulder 
Geometry of compactified Jacobians
There is a close connection between the geometry of a
smooth curve, and the geometry of its Jacobian. In this talk I will
discuss joint work with J. Kass and F. Viviani where we investigate
the connection between the geometry of a stable curve and its
compactified Jacobians, constructed by Caporaso, OdaSeshadri,
Pandharipande, and Simpson. Specifically, we describe the
singularities of the spaces, as well as singularities of their theta
divisors in the integral case. Applications to the birational geometry
of universal compactified Jacobians will also be discussed.

December 9  Mark McLean
Stony Brook University 
Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits.
Abstract: Let A be an affine variety inside a complex N dimensional
vector space which has an isolated singularity at the origin. The
intersection of A with a very small sphere turns out to be a manifold
called the link of A. The link has a natural hyperplane distribution
called a contact structure. If the singularity is numerically \Q
Gorenstein then we can assign an invariant of our singularity called
the minimal discrepancy. We relate the minimal discrepancy with the
contact geometry of our link. As a result we show that if the link of
A is contactomorphic to the link of \C^3 and A is normal then A is
smooth at 0. This generalizes a Theorem by Mumford in dimension 2.

This web page is maintained by Mark Shoemaker, Nicola Tarasca, Sofia Tirabassi, and Yi Zhu.