Specialization for the pro-etale fundamental group
We provide a natural extension of Grothendieck's theory of specialization for the etale fundamental group in the context of rigid-analytic geometry. More precisely, for a formal scheme of finite type over a complete dvr, we define a specialization map from the de Jong fundamental group of the rigid-analytic generic fiber to the pro-etale fundamental group of the special fiber. We apply similar ideas to study the notion of tameness for etale coverings of non-archimedean spaces, which requires the use of logarithmic geometry in the form of the logarithmic Abhyankar's lemma.
Hilbert scheme of points and global singularity theory
Abstract: Global singularity theory is a classical subject in geometry which classifies singularities of maps between manifolds, and describes topological reasons for their appearance. I explain how tautological integrals over Hilbert schemes of points provide a natural framework to tackle two of its central problems and I report on recent major developments (joint with A. Szenes).
The Chow rings of moduli spaces of elliptic surfaces
For each nonnegative integer N, Miranda constructed a coarse moduli space of elliptic surfaces with section over the projective line with fundamental invariant N. I will explain how to compute the Chow rings with rational coefficients of these moduli spaces when N is at least 2. The Chow rings exhibit many properties analogous to those expected for the tautological ring of the moduli space of curves: they satisfy analogues of Faber's conjectures, and they exhibit a stability property as N goes to infinity. When N=2, these elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice. I will explain how the computation of the Chow ring confirms a special case of a conjecture of Oprea and Pandharipande on the structure of the tautological rings of moduli spaces of lattice polarized K3 surfaces. This is joint work with Bochao Kong.
Abstract: We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes and show that many of the properties of motives of smooth schemes also continue to hold for motives of smooth algebraic stacks. As an application we show that for quotient stacks motivic cohomology is isomorphic to the Edidin-Graham-Totaro Chow groups with integer coefficient. This is joint work with U. Choudhury and A. Hogadi.
I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program.
Moduli spaces of stable objects in the Kuznetsov component of cubic threefolds
We will first discuss a general criterion that ensures a fractional Calabi-Yau category of dimension less than or equal to 2 admits a unique Serre-invariant stability condition up to the action of the universal cover of GL+(2, R). This result can be applied to a certain triangulated subcategory (called the Kuznetsov component) of the bounded derived category of coherent sheaves on a cubic threefold. As an application, we will prove (i) a categorical version of the Torelli theorem holds for cubic threefolds, and (ii) the moduli space of Ulrich bundles of fixed rank r greater than or equal to 2 on a cubic threefold is irreducible. The talk is based on joint work with Laura Pertusi and a group project with A. Bayer, S.V. Beentjes, G. Hein, D. Martinelli, F. Rezaee and B. Schmidt.
Non-isomorphic smooth compactifications of moduli of cubic
The moduli space of complex cubic surfaces admits a natural GIT compactification, which is singular, and a natural resolution is provided by the Kirwan desingularization procedure. Alternatively, the moduli space of complex cubic surfaces admits a ball quotient model, which has a natural Baily-Borel compactification, whose singularities are naturally resolved by the toroidal compactification of the Ball quotient. It turns out that the Kirwan compactification and the toroidal compactification have the same Betti numbers, but we show that they are not isomorphic nor even K-equivalent.
This is based on joint work in progress with S. Casalaina-Martin, K. Hulek, and R. Laza
Conic bundles via algebraic stacks
Conic bundles arise in a natural construction involving algebraic stacks. After a review of some notions from stack geometry, I'll describe this construction and present applications from joint work with Hassett and Tschinkel to rationality questions for conic bundles.
A unipotent local to global principle
One says a scheme, or an algebraic stack, has the resolution property if every coherent sheaf is the quotient of a locally free sheaf. We explain the two most important sources of non-examples: (1) affine group schemes G/S which cannot be embedded into GL_n but which are forms of embeddable group schemes, and (2) cohomological Brauer classes which are not represented by Azumaya algebras.
After describing a new way to construct non-trivial vector bundles on schemes and stacks, we introduce the notion of an R-unipotent morphism and characterize it geometrically. We then present a surprising local to global principle: a locally R-unipotent morphism over a base with enough line bundles is globally R-unipotent. To conclude, we explain why the unipotent analogues of (1) and (2) above cannot occur. This is joint work with Daniel Bragg and Jack Hall.
Artin fans, piecewise polynomials and the double ramification cycle.
Abstract: The double ramification cycle -- roughly, the locus of curves admitting a rational function with given ramification over 0 and infinity -- is a cycle class on the moduli space of curves, with connections to both Gromov-Witten theory and Abel-Jacobi theory. Some years ago, a remarkable formula for the DR cycle in terms of tautological classes was conjectured by Pixton, and subsequently proven in work of Janda, Pandharipande, Pixton and Zvonkine. However, in order to make further progress in this line of study -- for instance, in order to study the Gromov-Witten theory of more complicated targets -- it is necessary to study a refinement of the DR cycle coming from logarithmic geometry, for which no formulas exist. In the talk, I want to explain the key tools that, in joint work with Holmes,Pandharipande, Pixton and Schmitt, allow us to obtain such formulas. The moduli space of tropical curves, the cohomology rings of certain algebraic stacks and the compactified Jacobians of Kass-Pagani will play a crucial role.
Wall-crossing on fine compactified universal Jacobians
If C is a smooth curve of genus g, there is a natural Brill-Noether cycle W_d of codimension g-d in the moduli space of degree-d line bundles J^d, which consists of those line bundles that admit a nonzero global section. This definition extends to families, producing a codimension g-d universal Brill-Noether cycle W_{g,n}^d in the degree-d universal Jacobian J^d_{g,n} over the moduli space of curves M_{g,n}.
The moduli space of curves M_{g,n} can be extended to a complete moduli space parametrising stable curves. Several different compactifications of the universal Jacobian exist, and the cycle W_{g,n}^d can be extended to each of them by means of the Thom-Porteous formula. In this talk we will discuss how to compare two extensions on two different compactifications in terms of some natural classes. This is a joint work with Alex Abreu.
Brauer groups of moduli of hyperelliptic curves and their compactifications
While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields.
I will report on recent works with A. Di Lorenzo. Using the theory of cohomological invariants we completely describe the Brauer group of the moduli stacks of hyperelliptic curves over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It is generated by elements coming from the base field, cyclic algebras, an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. There are two natural compactifications, the first obtained by taking stable hyperelliptic curves and the second by taking admissible covers. It turns out that the Brauer group of the first is trivial, while for the second it is almost as large as in the non-compact case.
Compactifying moduli of pointed projective spaces up to automorphisms
In this talk, we consider the compactification problem for point configurations modulo automorphisms, and we construct a geometric and functorial compactification for points in the projective plane. The boundary parametrizes the n-pointed central fibers of Mustafin joins associated to one-parameter degenerations of n points in P^2 (joint work with J. Tevelev). Finally, we discuss the first steps towards extending this to projective spaces of arbitrary dimension, which led to the study of a possible generalization of the Losev-Manin moduli space (joint work with S. Di Rocco).
Level structures on logarithmic (and tropical) curves
Level structures are extra data that can be added to some moduli problems in order to rigidify the situation. For example, in the case of curves, they yield smooth Galois covers of the moduli space M_g, and the problem of extending this picture to the boundary was studied by several authors, using in particular admissible covers and twisted curves. I will report on some work in progress with M. Ulirsch and D. Zakharov, in which we consider a tropical notion of level structure on a tropical curve. The moduli space of these is expected to be closely related to the boundary complex of the stack of G-admissible covers. As usual, logarithmic geometry stands in the middle and provides a convenient language to bridge the two worlds.
On logarithmic compactification of stratas of Hodge bundle
The projectivized Hodge bundle on M_g can be naturally stratified by the pattern of zeros and poles, and then a natural question is to find a nice, e.g. smooth and modular, compactification of these strata over the nodal boundary of M_g. Bainbridge, Chen, Gendron, Grushevsky and Möller studied this question in a series of papers by using complex-analytic techniques: first they described an incidence compactification, obtained by taking the schematic closure in the projectivized Hodge bundle of \barM_{g,n}, then they extended these results to k-differentials, and in the last work they refined the incidence compactification to a smooth compactification with a modular interpretation.
A different proof of the characterization of incidence compactification was found in my work with I. Tyomkin and it was very recently extended by U. Brezner to k-differentials. These proofs are based on Berkovich geometry, but they have many common features with [BCGGM1] and [BCGGM2], first of all due to patching via certain good coordinates, though they provide a new interpretation of the global residue condition. In this talk I will briefly outline this story and then proceed to a work in progress with Tyomkin, where we construct a modular compactification by tools of log geometry. Nothing is written down yet, but as it seems now one will obtain a simple construction of a log modular compactification, which applies in any characteristic. It is (combinatorially) coarser than the construction of [BCGGM3] and is hopefully log smooth, but certainly not smooth. Also, it makes no use of good coordinates and residue conditions.
Geometry of fine compactified Jacobians in genus 1
The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian. We focus on the case of genus 1 and obtain a combinatorial classification for fine universal compactified Jacobians, which enables us to construct new examples of them. The description we obtain for universal fine compactified Jacobians of genus 1 also yields a formula for their rational cohomology.
This is joint work with Nicola Pagani (Liverpool).
Framed logarithmic curves and the formality conjecture
The famous "formality theorem" proven by Tamarkin states that the algebraic E2 operad (also known as the operad of chains on the topological operad of little disks) is formal. Explicit formality equivalences are in bijection with universal deformation quantizations, associators and other useful algebraic objects. An old folklore belief is that any construction of a canonical formality equivalence must involve "transcendental" methods and that no "purely topological" or "purely geometric" proof exists. I will explain that this folklore belief is incorrect, so long as one is allowed to use logarithmic geometry. The proof involves a new kind of moduli space of logarithmic curves with "framed" boundary components.
On the Universal Jacobian: algebraic, tropical and logarithmic aspects
I will start by reviewing how to compactify the universal Jacobian stack, parametrizing pointed curves endowed with a line bundle, over the moduli stack of stable pointed curves. Then I will discuss logarithmic universal Jacobians and their tropicalization morphism towards the corresponding tropical universal Jacobians. Finally, I will discuss the connection with the Molcho-Wise's logarithmic Picard and tropical Picard stacks.
This is a joint work with M. Melo, S. Molcho, M. Ulirsch, J. Wise.
Compact Kähler manifolds with no projective specialization
We show the existence of a compact Kähler manifold which does not fit in a proper flat family over an irreducible base with one projective (possibly singular) fiber. This strengthens our earlier counterexamples to the Kodaira algebraic approximation problem, where we considered only smooth families. Our main tool is the limit Hodge structure on cohomology as constructed by Steenbrink, which allows to study the singular fibers.