Algebraic Geometry Seminar

An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, we will restrict to the case of log Calabi-Yau pairs (X,\Delta). We prove that if (X,\Delta) admits a polarized endomorphism that preserves the boundary structure, then (X,\Delta) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.

The F-signature of a local singularity is an invariant in positive characteristics that measures the asymptotic properties of the Frobenius map. It can be used to detect regularity, strong F-regularity and other finiteness properties in positive characteristics. Thus, the F-signature seems to play a role analogous to the local volume of KLT singularities over the complex numbers. This talk concerns the behavior of the F-signature under the process of reduction to characteristic p >> 0 of a fixed complex singularity. Motivated by applications to the sizes of local fundamental groups, Carvajal-Rojas, Schwede and Tucker conjectured that the F-signatures remain uniformly bounded away from zero when we reduce a complex KLT singularity to large characteristics . We will present joint works with Yuchen Liu, and with Anna Brosowsky, Izzet Coskun and Kevin Tucker, in which we prove this conjecture in many new cases including for cones over low degree smooth hypersurfaces and most three dimensional KLT singularities. We will present some of the key ideas in the proof, which come from the K-stability theory of Fano varieties.

A lot of moduli spaces have the property that they are locally cut out by zeros of a section of a vector bundle over smooth spaces. These moduli spaces could be highly singular, but possess certain properties called quasi-smoothness. Using this, one can define on the moduli "virtual structure sheaf", which is deformation-invariant. The moduli of stable maps and quasimaps (as well as many other curve counting moduli) satisfy this property, and can be used to define quantum K-theory (or Gromov-Witten theory, Donaldson-Thomas theory etc.). In this talk I will talk about results on moduli of stable maps to a point and to Calabi-Yau threefolds.

Contact manifolds are odd-dimensional analogues of symplectic manifolds. The main aim of this talk is to present some structural results on contact structures on smooth complex projective log varieties. I will generalize a few standard results on rational curves on smooth projective varieties to the logarithmic case. Then, I will use these results to study Mori-type log contractions of contact projective log varieties.

I plan to survey some results on the analogues of Simpson's correspondence for varieties defined over an algebraically closed field of positive characteristic. Special attention will be given to quasi-projective varieties, where our study leads to some interesting problems concerning standard notions of positivity of vector bundles.

A smooth hypersurface of dimension n and degree d in complex projective space always has finite automorphism group when the degree d is at least 3, unless (n,d) = (1,3) or (2,4). Moreover, for each fixed pair (n,d), there is a finite upper bound on the order |Aut(X)|. Previously, this bound was only known explicitly for certain small values of n and d. In this talk, I'll describe results (joint with Jennifer Li) that exactly determine the upper bound on |Aut(X)| for every pair (n,d) and show that the Fermat hypersurface always achieves the upper bound apart from a finite number of exceptional cases.

The minimal exponent of a hypersurface is an invariant of singularities introduced by Morihiko Saito via D-module theory, which refines the log canonical threshold. I will give an introduction to this invariant and then I will present some work in progress with Qianyu Chen in the direction of a birational description of this invariant.

Geometric realizations (as etale projective bundles) of Brauer classes on surfaces have many applications, such as the arithmetic of surfaces over non-closed fields and rationality of fourfolds (even over the complex numbers). In this talk, I will discuss joint work with Jack Petok and Anthony Varilly-Alvarado, in which we consider constructions of these etale projective bundles for Brauer classes on K3 surfaces. This builds on recent results of van Geemen and Kaputska, who show that some 2-torsion Brauer classes on K3 surfaces have realizations as the exceptional locus of a divisorial contraction on a hyperkahler fourfold.

Mori dream spaces, named for their ideal behavior in the Minimal Model Program, are (for the sake of this talk) projective, normal varieties with finitely-generated Cox rings. While there has been much progress in the identification of Mori dream spaces, no complete classification yet exists. Due to their relationship with toric varieties, we will narrow our search to a class of spaces called (projectivized) toric vector bundles. Here, we will describe both positive and negative results of toric vector bundles as Mori dream spaces, including some combinatorial and computational approaches.

Abstract: The KSBA moduli space of stable pairs (X,B), introduced by Kollár--Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves for higher dimensional varieties. This moduli space is described concretely only in a handful of situations. For instance, if X is a toric variety and B=D+\epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. More generally, for stable pairs of the form (X,D+\epsilon C) with (X,D) a log Calabi-Yau variety and C an ample divisor, it was conjectured by Hacking--Keel--Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Arguz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.

In this talk, we will address the question for which pairs of integers (n,n') the variety Hilb^n_X is stably birational to Hilb^n'_X, when X is a surface with H^1(X,O_X)=0. In order to do so we will relate the existence of degree n' effective cycles on X with the existence of degree n ones using curves on X. We will then focus on geometrically rational surfaces, proving that there are only finitely many stable birational classes among the Hilb^n_X 's. If time permits, we will prove the rationality of a generalization of the Hasse-Weil zeta function Z(X, t) in K_0(Var/k)/([A^1_k])[[t]] when char(k) = 0 and X is geometrically rational.