abstracts for 2019-20 seminars

(The seminar webpage is here.)

Isabel Vogt (Stanford)

September 27, 2019

Stability of normal bundles of space curves

The normal bundle controls the deformation theory of a curve embedded in projective space. In this talk we study the stability of normal bundles of curves in P^3 using degeneration. This is joint work with Izzet Coskun and Eric Larson.

Jordan Ellenberg (Wisconsin)

October 4, 2019

Heights of rational points on (some) stacks

The study of rational points on algebraic stacks over global fields is in many respects very similar to the familiar world of Diophantine geometry of schemes. But one key element of that world is missing; a theory of heights. I will propose such a theory and explain how it recovers many already-in-use notions of complexity for points on stacks, while also generating new ones. I talked about this project in the Stanford number theory seminar in spring 2018; I will explain the basic ideas again and then talk about some examples we understand better now than we did then, such as points on the moduli stack of abelian varieties, and discuss some questions that remain open for future work.

Stefan Schreieder (Munich)

October 11, 2019

Stably irrational hypersurfaces of small slopes

We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension n>2 and degree at least log_2(n)+2 is not stably rational. This improves earlier results of Koll\’ar and Totaro, who proved the same result under a linear bound on the degree.

Jim Bryan (University of British Columbia)

October 25, 2019, 2:30 pm

K3 surfaces with symplectic group actions, enumerative geometry, and modular forms

The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the “G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as chi_y genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.

Nathan Pflueger (Amherst)

October 25, 2019, 4 pm

Relative Richardson Varieties

A Richardson variety is an intersection of two Schubert varieties defined by transverse flags in a vector space. Richardson varieties have many nice geometric properties; for example, a theorem of Knutson, Woo, and Yong shows that their singularities are completely determined by those of Schubert varieties. I will discuss a generalization of this theorem to a relative context, where the two transverse flags are replaced by a moving pair of flags in a vector bundle that become non-transverse at some points. I will also discuss a theorem describing the cohomology of the resulting relative Richardson variety. I will describe an application to Brill-Noether theory, and some related conjectures. This is joint work with Melody Chan.

Rachel Pries (Colorado State)

November 8, 2019

The intersection of the Torelli locus with the non-ordinary locus in PEL-type Shimura varieties

Around 1980, mathematicians developed several techniques to study the Newton polygon stratification of the moduli space of principally polarized abelian varieties in positive characteristic p.  In 2004, Faber and Van der Geer used these techniques to prove that the Torelli locus of Jacobians of smooth curves intersects every p-rank stratum.  In 2013, Viehmann and Wedhorn proved that every Newton polygon satisfying the Kottwitz conditions occurs on Shimura varieties of PEL-type.  In most cases, it is still not known whether the Torelli locus intersects these Newton polygon strata.  We provide a positive answer for the mu-ordinary and non-mu ordinary strata in infinitely many cases.  As an application, we produce infinitely many new examples of unusual Newton polygons which occur for Jacobians of smooth curves.  This is joint work with Li, Mantovan, and Tang.

Eric Riedl (Notre Dame)

November 15, 2019

Linear subvarieties of hypersurfaces and unirationality

The de Jong-Debarre Conjecture predicts that the space of lines on any smooth hypersurface of degree d <= n in P^n has dimension 2n-d-3. We prove this conjecture for n > 2d, improving on the previously-known exponential bounds. We prove an analogous result for k-planes, and use this generalization to prove that an arbitrary smooth hypersurface is unirational if n > 2^{d!}. This is joint work with Roya Beheshti.

Maddie Weinstein (Berkeley)

November 22, 2019

Metric Algebraic Geometry

Metric algebraic geometry is a term proposed for the study of properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. In this talk, we introduce metric algebraic geometry through discussion of Voronoi cells, bottlenecks, offset hypersurfaces, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

Carl Lian (Columbia)

January 10, 2020

Enumerating pencils with moving ramification on curves

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

Geoffrey Smith (Harvard)

January 24, 2020, 2:30pm

Covering gonalities in positive characteristic

The covering gonality of an irreducible projective variety over the complex numbers is the minimum gonality of a curve through a general point on the variety. This definition has two reasonable generalizations to positive characteristic, the covering gonality and the separable covering gonality. Of the two, separable covering gonalities are much easier to bound, and I’ll give an easy lower bound for smooth hypersurfaces essentially due to Bastianelli-de Poi-Ein-Lazarsfeld-Ullery. I’ll then give an analogous bound for the covering gonality of very general hypersurfaces, using a Chow-theoretic argument that extends work of Riedl-Woolf.

Jackson Morrow (Emory)

January 24, 2020, 4pm

Non-Archimedean entire curves in varieties

The classical conjectures of Green—Griffiths—Lang—Vojta predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, and groupless. In his thesis, Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the “correct” version, as it does not mirror complex hyperbolicity. In recent work, Javanpeykar and Vezzani introduced a new non-Archimedean notion of hyperbolicity, which fixed this issue and also stated a non-Archimedean version of the Green-Griffiths-Lang-Vojta conjecture.
In this talk, I will discuss algebraic, complex, and non-Archimedean notions of hyperbolicity and a proof of the non-Archimedean Green—Griffiths—Lang–Vojta conjecture for closed subvarieties of semi-abelian varieties and projective surfaces admitting a dominant morphism to an elliptic curve.

Izzet Coskun (UIC)

January 31, 2020, 4pm

The stabilization of the cohomology of moduli spaces of sheaves on surfaces

The Betti numbers of the Hilbert scheme of points on a smooth, irreducible projective surface have been computed by Gottsche. These numbers stabilize as the number of points tends to infinity. In contrast, the Betti numbers of moduli spaces of semistable sheaves on a surface are not known in general. In joint work with Matthew Woolf, we conjecture these also stabilize and that the stable numbers do not depend on the rank. We verify the conjecture for large classes of surfaces. I will discuss our conjecture and provide the evidence for it.

Jack Huizenga (Penn State)

February 7, 2020, 4pm

Moduli of sheaves on Hirzebruch surfaces

Let X be a Hirzebruch surface.  Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter.  On the other hand, many other basic properties of sheaves on Hirzebruch surfaces are unknown.  I will discuss two different problems on this topic.  First, what is the cohomology of a general sheaf on X with fixed numerical invariants?  Second, when is the moduli space
actually nonempty? The latter question should have an answer reminiscient of the Drezet-Le Potier classification of semistable sheaves on the projective plane; in particular, there is a fractal-like hypersurface in the space of numerical invariants which bounds the invariants of semistable sheaves.  This is joint work with Izzet Coskun.

Sarah Frei (Rice)

February 14, 2020, 4pm

Derived equivalence and rational points

It is natural to ask which properties of a smooth projective variety are recovered by its derived category. In this talk, I will consider the question: is the existence of a rational point preserved under derived equivalence? In recent joint work with Nicolas Addington, Ben Antieau, and Katrina Honigs, we show that over Q, the answer is no. We give two counterexamples: an abelian variety and a torsor over it, and a pair of hyperkaehler fourfolds.

Alex Perry (IAS)

February 21, 2020, 2:30pm

The integral Hodge conjecture for 2-dimensional Calabi-Yau categories

I will formulate a version of the integral Hodge conjecture for categories, discuss its proof for categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and explain how this implies new cases of the usual integral Hodge conjecture for varieties.

Daniel Litt (UGA)

February 21, 2020, 4pm

The section conjecture at the boundary of moduli space

Grothendieck’s section conjecture predicts that over arithmetically interesting fields (e.g. number fields), rational points on a smooth projective curve X of genus at least two can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter.

Alex Smith (Harvard)

February 28, 2020, 2:30pm

8-class ranks of imaginary quadratic fields and 4-Selmer groups of elliptic curves

We prove that the two-primary subgroups of the class groups of imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra-Gerth heuristic. In this talk, we will detail our method for proving the 8-class rank portion of this theorem and will compare our approach to one that uses the governing fields predicted by Cohn and Lagraias. We will also connect this work to related questions on the 4-Selmer groups of elliptic curves in quadratic twist families.

Anand Patel (Oklahoma State)

February 28, 2020, 4pm

Projection and Ramification

When a projective variety is linearly projected onto a projective space of the same dimension, a ramification divisor appears. In joint work with Anand Deopurkar and Eduard Duryev, we study basic questions about the map which sends a projection to its ramification divisor. I will present proven results, open problems, and if time permits, some curious numerology.

Dinesh Thakur (University of Rochester)

March 13, 2020, 4pm

Multizeta and motives in function fields

The study of multizeta values in intimately connected with current developments in non-abelian and homotopical directions in number theory.  We explain some background with multizeta values and their interpretation as periods in terms of higher-dimensional generalizations of Drinfeld modules (called mixed t-motives), as well as conjectures on relations and non-relations among various multizeta values.  We also discuss the implications for huge Galois representations and  lifting of multizeta values to functions on (pro)algebraic groups.  Finally, we describe new results for higher-genus curves as well as several open questions.

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