38406501359372282063949 & all that.

A Fano problem is an enumerative problem concerning linear subspaces of complete intersections in Pⁿ​​. For suitably chosen parameters (r,n,d)​​ a general complete intersection X_d​​ of type d=(d₁,...,d_k)​​ in Pⁿ​​ contains finitely many r-planes, in other words the Fano scheme F_r(X_d)​​ is finite.

I will talk about the monodromy groups of Fano problems, that is about the permutations of the set F_r(X_d)​​ induced by moving the variety X_d​​ in loops in a suitable moduli space. This calculation is unusual in that it avoids studying local monodromy and uses only global arguments. This is joint work with Sachi Hashimoto.

Stable pairs with a twist

The notion of a stable log variety or stable pair is the higher dimensional analogue of a stable pointed curve. The existence of a proper moduli space of stable pairs in any dimension has been established thanks to the last several decades of advancements in the minimal model program. However, the notion of a family of stable pairs remains quite subtle, and in particular a deformation-obstruction theory for this moduli problem is not known. Building on the work of Abramovich-Hassett, I will describe an approach to this question using a certain Deligne-Mumford stack canonically associated to the stable pair (X,D) and mention some applications of this approach. This is joint work with G. Inchiostro.

Intersection theory of the stable pair compactification of the moduli space of six lines in the plane

A result due to Keel gives an explicit presentation of the Chow ring of the moduli space of stable n-pointed rational curves. I will discuss recent work generalizing Keel's presentation to describe the intersection theory of the first non-trivial higher-dimensional case, the stable pair compactification of the moduli space of six lines in the plane.

Based on https://arxiv.org/abs/2009.06056

Calabi-Yau threefolds and torsion in cohomology

I’ll discuss some explicit examples of Calabi-Yau threefolds (CY3s) with torsion in various cohomology groups and consequences of such torsion; in particular, I’ll present computations using p-adic Hodge theory which answer some outstanding questions about CY3s over finite fields. I will not assume familiarity with Calabi-Yau manifolds or p-adic Hodge theory.

Compactifications of moduli of points and lines in the projective plane

Projective duality identifies the moduli space B_n​​ parametrizing configurations of n​​ general points in the projective plane with X(3,n)​​, parametrizing configurations of n​​ general lines in the dual projective plane. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces. In this work, we consider Gerritzen-Piwek's compactification of B_n​​ and Kapranov's Chow quotient compactification of X(3,n)​​, and we show they have isomorphic normalizations. We also construct an alternative compactification parametrizing all possible n​​-pointed central fibers of Mustafin joins associated to one-parameter degenerations of n​​ points in the projective plane, which was proposed by Gerritzen and Piwek. This is joint work with Jenia Tevelev.​

Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space.

Based on https://arxiv.org/abs/2002.07127

The top weight cohomology of A_g​

I will discuss recent work calculating the top weight cohomology of the moduli space A_g​​ of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g​​ and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

Preceded by a Happy Zoom Hour on Oct 13 @ 6:30 pm.

Masur-Veech volumes and higher genus meanders

This is a joint work with V. Delecroix, P. Zograf and A. Zorich. A meander is a topological configuration of a pair of simple closed curves on the sphere intersecting transversally. They appear in various areas of mathematics, theoretical physics and computational biology, and their enumeration remains an open problem. Similarly higher genus meanders are topological configurations of pairs of simple closed curves on higher genus surfaces. This talk will focus on the enumeration of meanders of fixed combinatorics (fixed number of bigons), and its large genus asymptotics.

This counting problem is strongly related to the counting of square-tiled surfaces and the evaluation of Masur-Veech volumes; the results follow from different recent advances in the evaluation of these volumes for large genus surfaces. As an ingredient of the proof, I will present a formula for the Masur-Veech volumes of moduli spaces of meromorphic quadratic differentials as a sum over stable graphs of some intersection numbers.

The locus of post-critically finite maps in the moduli space of self-maps of Pⁿ

A degree d>1 self-map f of Pⁿ is called post critically finite (PCF) if its critical hypersurface C_f is pre-periodic for f, that is, if there exist integers r ≥ 0 and k>0 such that f^r+k(C_f) is contained in f^r(C_f).

I will discuss the question: what does the locus of PCF maps look like as a subset of the moduli space of degree d self-maps on Pⁿ? I’ll give a survey of many known results and some conjectures in dimension 1 (i.e. for n=1). I’ll then present a result, joint with Joseph H. Silverman and Patrick Ingram, that suggests that in dimensions two or greater, PCF maps are comparatively scarce in the moduli space of all self-maps.

Fundamental group of a ball quotient

We shall talk about an arithmetic lattice M in PU(13,1) acting on the the unit ball B in 13-dimensional complex vector space. Let X be the space obtained by removing the hypersurfaces in B that have nontrivial stabilizer in M and then quotienting the rest by M. The fundamental group G of the ball quotient X is a complex hyperbolic analog of the braid group. We shall state a conjecture that relates this fundamental group G and the monster simple group and describe our results (joint with D. Allcock) towards this conjecture.

The discrete group M is related to the Leech lattice and has generators and relations analogous to Weyl groups. Time permitting, we shall give a second example in PU(9,1) related to the Barnes-Wall lattice for which there is a similar story.

Covering gonalities of complete intersections 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.

Configuration spaces of smooth complex curves

The configuration space Conf_n(X) of a space X (where n is a positive integer) is simply the open subset of Xⁿ which parametrizes n-tuples of points that are pairwise distinct. For X a smooth complex curve, this space has an interesting Hodge theory of which the middle cohomology, or of its rather the FⁿHⁿ part appears in the description of the space of conformal blocks. We review this and discuss some of its topological repercussions.​

Geometrically irreducible p-adic local systems are de Rham up to a twist

Inside the category of \overline{Q}_p-local systems on a smooth algebraic variety X over a p-adic field K there is a subcategory of de Rham local systems. All local systems arising from relative etale cohomology of families of algebraic varieties over X are de Rham and to any de Rham local system one can canonically associate a filtered vector bundle of the same rank on X equipped with a flat connection satisfying Griffiths transversality. One might think of de Rham local systems as being analogous to complex variations of Hodge structures.

It turns out that any \overline{Q}_p-local system L such that the base change of L to X_{\overline{K}} is irreducible, becomes de Rham after twisting by a character of the Galois group of K. The proof uses p-adic Riemann-Hilbert and Simpson correspondences studied by Liu and Zhu, their decompleted versions and the action of Sen operator.

Brill--Noether theory over the Hurwitz space

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.