abstracts for 2012-13 seminars

(The seminar webpage is here.)

Zhiyuan Li (Stanford)
September 24, 2012

Modular forms and special cubic fourfolds

A special cubic fourfold is a smooth cubic hypersurface in \mathbb{P}^5 containing a rational normal scroll. Brendan Hassett has shown that the sets of special cubic fourfolds form divisors in the parameterization space \mathbb{P}^{55}. A classical geometry question is to know the degrees of these divisors in \mathbb{P}^{55}.  We solve this problem by showing that the generating series of the degree of these divisors is a modular form with respect to a level 3 congruence subgroup of SL_2(\mathbb{Z}). The modularity comes from the work of Borcherds’ generalized Gross-Kohnen-Zagier theorem. This is joint work with Letao Zhang.

Hsian-Hua Tseng (Ohio State)
October 1, 2012

Toric mirror maps revisited

For a compact semi-Fano toric manifold X, the toric mirror theorem of Givental and Lian-Liu-Yau says that a generating function of 1-point genus 0 descendant Gromov-Witten invariants, the J-function of X, coincides up to a mirror map with a function I_X which is written using the combinatorics of X. The procedure of obtaining the mirror map, which involves expanding I_X as a suitable power series, is somewhat mysterious. In this talk we’ll describe some attempts at understanding the mirror maps more geometrically.

Daniel Erman (Michigan)
October 8, 2012

Semiample Bertini Theorems over finite fields

We prove a semiample generalization of Poonen’s Bertini Theorem over a finite
field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of a corresponding morphism. This joint with Melanie Matchett Wood.

Sam Grushevsky (Stony Brook)
October 15, 2012

The stable cohomology of moduli of abelian varieties and compactifications

The stable (for g much large than the degree) cohomology of the moduli space A_g of principally polarized abelian varieties was computed by Borel, as the group cohomology of the symplectic group. Using topological methods, Charney and Lee computed the cohomology of the Satake-Baily-Borel compactification of A_g. In this talk we will discuss the question of computing the stable cohomology of toroidal compactifications, and in particular will discuss the stabilization of suitable cohomology for the perfect cone toroidal compactification, and explicit computation of some of these stable cohomology groups, by algebraic methods. Joint work in progress with Klaus Hulek and Orsola Tommasi (there is also an independent related work in progress by Jeff Giansiracusa and Gregory Sankaran, by topological methods).

Sam Payne (Yale)
Friday, October 26, 2012, 2:30 pm, 383-N

Tropicalization of the moduli space of curves

The analogy between moduli spaces of stable curves and more recently constructed moduli spaces of tropical curves is explained, canonically and functorially, through a generalized tropicalization map for toroidal embeddings of Deligne-Mumford stacks. This is joint work with D. Abramovich and L. Caporaso.

Jordan Ellenberg (Wisconsin)
October 29, 2012

Geometric analytic number theory

Analytic number theory has traditionally focused on questions about distribution of various arithmetic objects (prime numbers, ideal class groups, rational points on varieties) over Q. Many of the standard questions in that field can be expressed just as easily over a global field of characteristic p, such as \mathbb{F}_q(t). It turns out that analytic number theory in this setting often reveals unexpected connections with algebraic geometry and algebraic topology, with information flowing in both directions. I will survey some recent progress in this area, focusing on the work of myself, Akshay Venkatesh, and Craig Westerland on the Cohen-Lenstra conjectures over function fields and its relation with the stable cohomology of Hurwitz spaces. Time permitting, I will also say something about the Hardy-Littlewood method and its relationship with moduli spaces of rational curves on varieties (work of Pugin and Lee) and speculate a little about the connection between this story and “motivic analytic number theory” (a la Ravi Vakil, Melanie Matchett Wood, Dan Erman, and Daniel Litt, among others) and the theory of FI-modules (me, Tom Church, and Benson Farb, among others.) Time will not actually permit all this, so let this abstract serve as a list of things I would be interested in talking with people about during my visit, whether or not they appear during my seminar.

Bhargav Bhatt (Michigan/IAS)
Friday, November 9, 2012, 2:30 pm, 383-N

p-adic derived de Rham cohomology

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.

John Ottem (Cambridge)
November 12

Ample subschemes and partially positive line bundles

We introduce a notion of ampleness for subschemes of higher codimension and investigate geometric properties satisfied by ample subschemes, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these results, we also construct a counterexample to a conjecture of Demailly-Peternell-Schneider.

Yunfeng Jiang (Imperial College London)
November 26

On the crepant transformation conjecture

Let X and X' be two smooth Deligne-Mumford stacks. We call dash arrow X\dasharrow X' a crepant transformation if there exists a third smooth Deligne-Mumford stack Y and two morphisms \phi: Y\to X, \phi': Y\to X' such that the pullbacks of canonical divisors are equivalent, i.e. \phi^*K_{X}\cong \phi'^*K_{X'}. The crepant transformation conjecture says that the Gromov-Witten theory of X and X' is equivalent if X\dasharrow X' is a crepant transformation. This conjecture was well studied in two cases: the first one is the case when X and X' are both smooth varieties; the other is the case that there is a real morphism X\to |X'| to the coarse moduli space of X', resolving the singularities of X'. In this talk I will present some recent progress for this conjecture, especially in the case when both X and X' are smooth Deligne-Mumford stacks.

Matt Satriano (University of Michigan)
December 3 (in 380-Y)

Toric Stacks and Applications to Cycle Theory

No prior knowledge of stacks will be assumed for this talk. We will discuss the theory of toric stacks, which like toric varieties, is a class of stacks which is particularly amenable to computation. Using the theory developed, we will show that it is impossible to construct a degree map for 0-cycles on even the nicest of Artin stacks. This is based on joint work with Dan Edidin and Anton Geraschenko.

János Kollár (Princeton)
Thursday January 10, 4:15-5:15, 380-W

MRC Distinguished Lecture 1 of 2: Local topology of analytic spaces

Let M be a subset of {\mathbb C}^N defined as the common zero set of some holomorphic functions. What can one say about the local structure of M? Each point p\in M has an open neighborhood that is a cone over an (odd real dimensional) topological space called the link of p. If p is a smooth point of complex dimension n, the link is a sphere of real dimension 2n-1.

Our main interest is to understand the most complicated links. The general answer is not known but we show how to construct examples where the fundamental group of a link is nearly arbitrary. Joint work with Misha Kapovich.

János Kollár (Princeton)
Thursday January 24, 3:15-4:15, Gates B12

MRC Distinguished Lecture 2 of 2: How to recognize families of Cartier divisors?

We propose a strengthening of the Grothendieck-Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities.

Amnon Yekutieli (Ben-Gurion Univ.)
February 4

Residues and duality for schemes and stacks

Let K be a regular noetherian commutative ring. I will begin by explaining the theory of rigid residue complexes over essentially finite type K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes over finite type K-schemes. An important feature is that the rigid residue complex over a scheme X is a quasi-coherent sheaf in the etale topology of X. For any map between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map of schemes is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.

Then I will move to finite type Deligne-Mumford K-stacks. Any such stack has a rigid residue complex on it, and for any map between stacks there is a trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish by presenting two conjectures, that refer to Grothendieck duality for proper maps between DM stacks. A key condition here is that of tame map of stacks.

Xuanyu Pan (Columbia)
February 4

The Geometry of Moduli space of rational Curves on Complete intersections

For a low degree smooth complete intersection X, we consider the general fiber F of the following evaluation map ev of Kontsevich moduli space. ev: \overline{M}_{0,m}(X,m) \rightarrow X^m. I will give general structure theorems of F. It answers questions relating to
(1) Rational connectedness of moduli space
(2) Enumerative geometry
(3) The Search for a new 2-Fano variety
(4) Picard group of moduli space

Brendan Hassett (Rice)
February 22 (4 pm, 383-N)

K3 surfaces, level structure, and rational points

Moduli spaces of elliptic curves with level structure are fundamental for arithmetic and Diophantine problems over number fields in particular. For K3 surfaces, the Brauer group plays the role of the torsion points. Recently, a number of papers have systematically investigated how the Brauer group may be used to formulate criteria for the existence of rational points. However, geometric interpretations for the Brauer elements remain elusive. We present examples of several such interpretations and arithmetic applications, with a view toward putting these in a larger framework.

(joint work with V’arilly-Alvarado)

Chenyang Xu (Beijing)
February 25

Comparison of stabilities

There are different stability notions coming from different ideas of constructing the moduli space. In this talk, I will try to explore the relations among KSBA-stability, K-stabilty and asymptotic GIT stability. Smooth canonical polarized varieties are stable in all these senses, however the limiting behaviors turn out to be different. As a corollary, we answer the longstanding question that asymptotically GIT Chow semistable varieties does not form a proper family. (joint with Xiaowei Wang)

Runpu Zong (Princeton)
Thursday February 28, 3:15-4:15, Gates B12

Weak Approximation for Isotrivial Family

Following the celebrated Graber-Harris-Starr theorem that any geometrically smooth rationally connected variety over function field of curve has a rational point, there is Hassett-Tschinkel’s conjecture that whether rational points have weak approximation property of these families. While wildly open, we prove this conjecture when the general fibers of the family are isomorphic to each other, which is a large class of families that former results never covered.

Igor Dolgachev (University of Michigan)
March 11

Rational self-maps of moduli varieties

I will discuss some known examples of rational self-maps of finite degree greater than one of moduli varieties considered as stacks defining some moduli problem. A classical example of this sort is a covariant of order d on the space homogeneous forms of degree d. Other examples include moduli varieties of algebraic curves of low genus or abelian varieties with some level structure.

Nathan Ilten (Berkeley)
April 1, 2:45-3:45 pm

Equivariant Vector Bundles on T-Varieties

Klyachko has shown that there is an equivalence of categories between equivariant vector bundles on a toric variety X and collections of filtered vector spaces satisfying some compatibility conditions. I will discuss joint work with H. Suess which generalizes this equivalence to the setting of T-equivariant vector bundles on a normal variety X endowed with an effective action of an algebraic torus T. Indeed, T-equivariant vector bundles on X correspond to collections of filtered vector bundles on a suitable quotient of X.

This correspondence can be applied to show that T-equivariant bundles of low rank on projective space split, as well as to easily compute all global vector fields on rational complexity-one T-varieties.

Anand Deopurkar (Columbia)
April 1, 4-5 pm

Compactifying spaces of branched covers

Moduli spaces of geometrically interesting objects are often non-compact. They need to be compactified by adding some degenerate objects. In many cases, this can be done in several ways, leading to a menagerie of birational models, which are related to each other in interesting ways. In this talk, I will explore this idea for the spaces of branched covers of curves, known as the Hurwitz spaces. I will construct a number of compactifications of these spaces by allowing more and more branch points to coincide. I will describe the geometry of the resulting spaces for the case of triple covers.

Greg G. Smith (Queen’s University)
April 8

Nonnegative sections and sums of squares

A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when every nonnegative homogeneous polynomial is a sum of squares. In this talk, we will examine this converse for line bundles on a totally-real projective subvariety. We will uncover some unexpected connections with classical algebraic geometry and, by working with multigraded polynomial rings, provide many new examples in which every nonnegative homogeneous polynomial is a sum of squares.

Donu Arapura (Purdue University)
April 15, 3:30-4:20 pm, 383-N

Splittable objects in derived categories and Kodaira vanishing

I first want to explain some generalizations of the Kodaira vanishing theorems due to Kollár, Saito and others, which involves some fairly serious Hodge theory. Then I would outline some ideas for approaching these using characteristic p techniques, which is more elementary. Some of this is work in progress.

Vivek Shende (MIT)
April 15, 4:30-5:20 pm, 383-N

Higher Discriminants

Given a map f:X \rightarrow Y between smooth varieties, the discriminant is the locus of points y in Y such that the fibre X_y is singular. We suggest the following generalization: the i th discriminant of f is the locus of points y such that for any (i-1) dimensional disc D passing through y, the fibre X_D is singular at some point of X_y. The sense of this definition is clarified by the following two theorems in the case when f is proper and equidimensional: (1) the support of any summand of the (derived) pushforward of the constant sheaf on X is equal to a component of a higher discriminant, and (2) the pushforward of the constant function on X is equal to an integer linear combination of the Euler obstructions of components of higher discriminants.

Paolo Aluffi (Florida State University)
April 22, 4-5 pm, 381-U

Segre classes of monomial subschemes

Segre classes are fundamental intersection-theoretic invariants: many problems in enumerative geometry may be reduced to computations of Segre classes, and basic invariants such as Milnor numbers and classes may be expressed in terms of Segre classes. We propose a formula computing the Segre class of an arbitrary monomial subscheme, and prove this formula in several representative cases. The formula is reminiscent of intersection computations in terms of mixed volumes of polytopes and convex bodies as in the classical Bernstein’s theorem and more recent work of Kaveh-Khovanskii, Huh, and others.

Zhiyu Tian (Caltech)
April 29, 4-5 pm, 381-U

Weak approximation for cubic hypersurfaces

Given an algebraic variety X over a field F (e.g. number fields, function fields), a natural question is whether the set of rational points X(F) is non-empty. And if it is non-empty, how many rational points are there? In particular, are they Zariski dense? Do they satisfy weak approximation? For cubic hypersurfaces defined over the function field of a complex curve, we know the existence of rational points by Tsen’ s theorem or the Graber-Harris-Starr theorem. In this talk, I will discuss the weak approximation property of such hypersurfaces.

Andrew Morrison (ETH)
May 6, 3:30-4:30pm, 381-U

Behrend’s function is constant on \text{Hilb}^n({\mathbb{C}}^3)

We will prove that Behrend’s function is constant on \text{Hilb}^n({\mathbb{C}}^3). A calculation of motivic zeta functions shows the relevant Milnor fibers have zero Euler characteristic. As a corollary we see that \text{Hilb}^n({\mathbb{C}}^3) is generically reduced and all its components have the same dimension mod 2. Time permitting we will also extend these results to moduli schemes of curves and points in threefolds coming from resolutions of ADE singularities.

Arnav Tripathy (Stanford)
May 6, 4:45-5:45 pm, 381-U

Stabilisation of symmetric powers

Starting with a scheme X, we may consider the sequence of its symmetric powers \text{Sym}^n X. I’ll remind the audience of reasons from topology and arithmetic geometry for why we should expect various sorts of asymptotic behaviour of this sequence. I’ll then discuss a certain natural sort of stabilisation, namely that of the etale homotopy groups of this sequence.

Frank Sottile (TAMU)
May 13, 4-5 pm, 381-U

Galois groups of Schubert problems

Work of Jordan from 1870 showed how Galois theory can be applied to enumerative geometry. Hermite earlier showed that the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris used this to study Galois groups of many enumerative problems. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil’s criterion to show that all Schubert problems involving lines have at least alternating Galois group. White and I have given a new proof of this based on 2-transitivity.

My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

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