This is joint work with Michael Tëmkin (Jerusalem) and Jarosław Włodarczyk (Purdue), a side product of our work on functorial semistable reduction. A similar result was discovered by G. Marzo and M. McQuillan. Given a variety $X$, one wants to blow up the worst singular locus, show that it gets better, and iterate until the singularities are resolved. Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci – it goes into a loop. We show that there is a functorial way to resolve varieties using weighted blowings up, in the stack-theoretic sense. To an embedded variety $X \subset Y$ one functorially assigns an invariant $(a_1,\ldots,a_k)$, and a center locally of the form $(x_1^{a_1} , \ldots , x_k^{a_k})$, whose stack-theoretic weighted blowing up has strictly smaller invariant under the lexicographic order.

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of polarized varieties, in particular on the base of families of Q-Fano varieties (that is, Fano varieties with klt singularities). It is conjectured that the CM line bundle yields a polarization on the conjectured moduli space of K-polystable Q-Fano varieties. This boils down to showing semi-positivity and positivity statements about the CM-line bundle for families with K-semi-stable and K-polystable Q-Fano fibers, respectively. I present a joint work with Giulio Codogni where we prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for very general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I also present applications to fibered Fano varieties.

A question of Orlov is whether the derived category of the Grothendieck--Knudsen moduli space of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group S_n. I will present an approach towards answering this question. This is joint work with Jenia Tevelev.

There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known. In the last part of the talk I will discuss the Chow ring of the stack of stable curves of genus 2, which has been recently calculated by Eric Larson. I will present a different approach to the calculation, which offers an interesting point of view on stack of stable curves of genus 2. This part is joint work with Andrea Di Lorenzo.

We consider rationality questions for threefolds over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics. (joint with Tschinkel)

Brody hyperbolic projective varieties over the complex numbers are extremely rich in properties. For instance, such varieties have only finitely many automorphisms, and every surjective endomorphism is actually an automorphism of finite order. Consequently, if one believes the conjectures of Green-Griffiths, Lang or Vojta, these properties should be shared by varieties which are “arithmetically” hyperbolic or “algebraically” hyperbolic. In this talk, we will see how to establish some of these properties, and thereby verify many of the predictions made by the conjectures of Green-Griffiths and Lang. I will report on joint works with Raymond van Bommel, Ljudmila Kamenova, Alberto Vezzani, and Junyi Xie.

In recent years, the combinatorial systematic treatment of degenerations of classical linear series within the theory of tropical linear series has seen spectacular developments and has led to many important results on algebraic curves. On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical (compactified) counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and in particular of their compactifications. In this talk, which is based on joint work with Lucia Caporaso and Marco Pacini, I will explore this principle for Cornalba's moduli space of spin spin curves. In particular, I will describe a stratification of this moduli space and introduce a tropical interpretation for the skeleton of its analytification. Time permitting, I will mention a number of interesting connections and applications of our work.

This is joint work with Massimo Bagnarol and Fabio Perroni. We compute the motive in question, which depends on the curve, the integers n and rank E, but not the specific bundle. If time allows, I will outline applications by Bagnarol and Perroni (which motivated this work).