Arithmetic, Birational Geometry,
and Moduli Spaces
Arithmetic, Birational Geometry,
and Moduli Spaces
Conference Schedule
Conference Schedule
All talks will be held in MacMillan Hall (167 Thayer Street), Room 117. Enter on Thayer Street, between Waterman and George.
The poster session will be held in Sayles Hall Auditorium, 81 Waterman Street. Click here for walking directions between these two locations.
"...in this field, almost everything is already discovered, and all that remains is to fill a few unimportant holes." Philipp von Jolly in his recommendation to Max Planck not to go into physics.
Since 2015 I am taking part in a long project (more precisely, a series of projects) with Dan Abramovich and Jarek Wlodarczyk on resolution of singularities in characteristic zero -- a field which was (and sometimes still is) considered as accomplished up to a few unimportant holes. To our surprise it turned out that there were (and still are) quite a few fundamental things to discover in this classical and thoroughly explored field, and the new discoveries even provide a more conceptual view on what was known before we started our project. It is impossible to compress all results of this journey in one talk, but I will try to outline a unified view on most of our discoveries in these projects. If time permits in the end I will also say a couple of words about our new project in progress with Andre Belotto -- still in characteristic zero...
In favorable situations, moduli problems have particularly nice (i.e. toroidal) compactifications. These compactifications are however typically not unique nor canonical. Logarithmic intersection theory is an enrichment of traditional intersection theory that works with all these compactifications simultaneously. In the talk, I will survey some recent developments in the field, and illustrate how the ideas work through an example: Jacobians over the moduli space of stable curves. I will explain how this approach has led to the calculation of the ``higher ramification cycles" -- roughly, the cycles that control the relative or logarithmic Gromov-Witten theory of toric varieties. Time permitting, I will discuss connections to Brill-Noether theory and relations in the tautological ring.
Consider a complete toric variety X defined by a simplicial fan. If the dimension of X is even, then there is a quadratic form on the middle degree (singular) cohomology of X, defined by multiplication. Papadakis and Petrotou proved that this quadratic form is anisotropic, provided that we generalize the construction of the cohomology ring slightly. The main application of anisotropy is the Hard Lefschetz theorem for complete but not necessarily projective toric varieties.
In this talk I will explain joint work with Elizabeth Xiao, where we simplify and generalize the anisotropy theorem. We use mixed volumes and decomposition of simplicial fans to give an explicit diagonal description of the quadratic form. This description can then be used to reduce the anisotropy theorem to the case where X is the projective space. The characteristic-free formula for the quadratic form also allows specialization from the field of rational numbers to finite fields.
We explain how to define an integral structure on the sheaf of differentials $\Omega_X$ of an adic space $X$. This should be thought of as an analogue of the subsheaf $\mathcal{O}_X^+$ of the structure sheaf $\mathcal{O}_X$. This integral structure $\Omega_X^+$ can be described in terms of logarithmic differentials on a log regular model (if such a model exists). This hints at a connection of tame cohomology with log etale cohomology that is not yet totally understood.
I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the motivic monodromy conjecture for any complex polynomial f that is non-degenerate with respect to its Newton polyhedron. This conjecture, due to Igusa and Denef--Loeser, states that for every pole s of the motivic zeta function associated to f, exp(2πis) is a "monodromy eigenvalue" associated to f. On the other hand, the non-degeneracy condition on f ensures that the singularity theory of f is governed, up to a certain extent, by faces of the Newton polyhedron of f. The extent to which the former is governed by the latter is one key aspect of the conjecture, and will be the main focus of my talk.
We discuss the resolution method by the operation of cobordant blow-ups and its generalizations introduced in recent papers. The papers and the methods are based upon the ideas of the joint work with Abramovich and Temkin and a similar result by McQuillan on resolution in characteristic zero via stack-theoretic weighted blow-ups.
I will report on work in progress with Mattia Talpo and Richard Thomas on a version of the Hilbert scheme in logarithmic geometry.
In today's talk we will see how the theory of good moduli spaces developed by Alper and Alper--Halpern-Leistner--Heinloth can be applied to describe the moduli space of quiver representation. In particular, when the quiver is acyclic, we show that a determinant line bundle, naturally defined over the stack of such representations, descends to an ample line bundle L on the moduli space. Through our approach we are able to produce new effective bounds for global generation of L. This is based on a joint work with Belmans, Franzen, Hoskins, Makarova and Tajakka.
In this talk I will report on several (in many parts ongoing) projects whose overall goal is to study the basic phenomena of non-abelian Hodge theory using the explicit toolbox of combinatorial algebraic geometry. I will focus on appropriate formulations of a non-abelian Hodge correspondence as well as on $P=W$ and mirror symmetry phenomena. Central instances of this story are homogeneous bundles on abelian varieties both over Archimedean and non-Archimedean fields as well as parabolic bundles on toric varieties and wonderful compactifications of hyperplane arrangement complements. The last example gives rise to an (as of now) largely conjectural non-abelian Hodge theory of matroids.
This is based, in some parts, on joint work with B. Bolognese and A. Küronya, as well as with A. Gross, I. Kaur, and Annette Werner.
The interpolation problem is one of the oldest in mathematics. In its most broad form it asks: when can a curve of a given type be passed through a given number of general points? I'll discuss my recent joint work with Eric Larson that completely solves this problem for curves of general moduli.
A theory of heights on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For cyclotomic stacks, a particularly well-behaved class of stacks introduced by Abramovich and Hassett, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will illustrate this with the example of counting elliptic curves over a function field. This is based on joint work with Park and Satriano.
I will recall the Mordell—Lang conjecture and sketch its proof by Vojta—Faltings—Rémond. After analyzing what is needed towards a uniform result, I reduce the question to a gap principle. The rest of this talk will be a rapid introduction to the three main ingredients in the proof, with a stress on some of the key differences in higher dimensions compared to curves. This is joint work with Ziyang Gao and Lars Kühne.
Given a family of smooth curves over a punctured disk, to extend this family to a family over (a ramified cover of) the whole disk, we have to allow the central fiber to be a stable nodal curve. If we want to extend families of curves together with some “additional data” (e.g. a vector bundle over the family, a fibration in elliptic curves over the family, G-torsors, etc), then the problem becomes more complicated, and in general stable nodal curves are not enough (for example, in the case of curves with a line bundle, or admissible covers). In this talk I will present a result that says that, if we allow the central fiber to be a nodal twisted curve whose coarse space is quasi stable (i.e. it might contain rational components with two marked points), then we can always extend the original family of curves plus “additional data” (e.g. vector bundle, fibration, G-torsor,…) to a family over the whole disk, as long as the “additional data” is given by maps to quotient stacks admitting a projective good moduli space. Moreover, the algorithm for extending this family is explicit.
I will discuss two modern routes to counting curves tangent to hypersurfaces in a projective variety. The first, due to Abramovich, Cadman, and Vistoli, is to use stable maps to root stacks and produces a theory with essentially all the features of traditional Gromov-Witten theory. The second approach, due to Abramovich, Chen, Gross, and Siebert, is to work within logarithmic geometry. This latter approach has rich interactions with tropical geometry and mirror symmetry, but many aspects of the theory remain mysterious to us. As a result, while the theories do not coincide, each has moments to shine. I will give an introduction to this circle of ideas, and then share recent and ongoing work with Battistella and Nabijou that explains and controls the difference between roots and logarithms in Gromov-Witten theory.
Kontsevich introduced moduli spaces $K_{g, n}(X)$ classifying stable maps from $n$-marked genus $g$ curves to a projective variety $X$. This was generalized by Abramovich and Vistoli to allow the target $X$ to be a stack, which is natural from the point of view of classifying families of geometric objects over curves. A second generalization, due in various forms to Hassett Alexeev and Guy, and Bayer and Manin, is to allow fractional coefficients of the marked points on the curves. This is natural from the point of view of the minimal model program. In this talk I will discuss moduli spaces of weighted stable maps to stacks, which is a common generalization of all of these approaches. This is joint work with Rachel Webb.
In this talk, I will introduce the theory of punctured logarithmic R-maps, which are a special case of punctured maps in the sense of Abramovich-Chen-Gross-Siebert, further twisted by (roots) of canonical divisors of the domain curves.
Punctured logarithmic R-maps admit two different but closely related perfect obstruction theories, the canonical theory and the reduced theory. The canonical theory is a version of double ramification cycles by allowing targets and spins but without expansions. On the other hand, the reduced theory describes boundary contributions from logarithmic Gauged Linear Sigma Models, hence provides correction terms to high genus quantum Lefschetz in Gromov-Witten theory. As an example, we will show that in many interesting cases of Calabi-Yau three-fold complete intersections, the number of reduced invariants involved in these correction terms matches the number of free parameters from the BCOV B-model from physics.
This is a joint work with Felix Janda and Yongbin Ruan.
I will discuss joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia, in which we construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral cone of effective divisors, both in characteristic 0 and in prime characteristic. As a consequence, the Grothendieck-Knudsen moduli space of stable rational curves with n>=10 markings has a non-polyhedral cone of effective divisors.
The explicit description of low degree K3 surfaces leads to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these compactifications for degree two K3 surfaces was studied by Shah and Looijenga, and revisited by Laza and O’Grady, who also provided a conjectural description for the case of degree four K3 surfaces. I will discuss these results, as well as a verification of this conjectural picture using tools from K-moduli. This is joint work with Kristin DeVleming and Yuchen Liu.
With Michael Groechenig, we proved that rigid local systems on a smooth quasi-projective variety $X$ over the complex numbers yield on $X$ mod $p$ for $p$ large $F$-overconvergent isocrystals and ultimately, under certain assumptions, crystalline local systems on the $p$-adic models of $X$. We’ll review various ways to see those properties.
Various results and conjectures, such as the norm residue isomorphism theorem and the cyclicity conjecture, point to some kind of cohomological universality for projective space. I will explain the connection and describe recent work in this direction. In particular, I will discuss new techniques for simultaneously generating large subgroups of the cohomology of a variety from a single correspondence with a small projective space. As it turns out, this also gives a new way to think about surjectivity of the norm residue map. The techniques I use ultimately rely on an analysis of the Picard groups of certain stacky curves.
From Torelli's theorem, we can recover a curve from the isomorphism class of its Jacobian as a principally polarized abelian variety. But the isomorphism class of the Jacobian as just an abelian variety (let alone the isogeny class) is not always enough. We look instead at the isogeny classes of the Jacobians of unramified covers of the curve and discuss the possibility of recovering the curve from those.
Given a perfect field k with algebraic closure k' and a variety X over k', the field of moduli of X is the subfield of k' of elements fixed by elements s of the Galois group of k' over k such that the twist X_s is isomorphic to X. Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures.
In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety X with a smooth marked point p that ensure that the pair (X,p) is defined over its field of moduli.
Algebraic geometry endows the cohomology groups of moduli spaces of curves with additional structures, such as (mixed) Hodge structures and Galois representations. Standard conjectures from arithmetic, regarding analytic continuations of L-functions attached to these Galois representations, lead to striking predictions, by Chenevier and Lannes, about which such structures can appear. I will survey recent results unconditionally confirming several of these predictions and studying patterns in the appearances of motives of low weight. The latter are governed by the operadic structures induced by tautological morphisms and the cohomology of graph complexes.
Two fundamental facts about the moduli space M_g of smooth curves of genus g are what are called Harer's theorems: that the Picard group of M_g is of rank one, generated (over the rational numbers) by the Hodge class; and that the relative Picard group of the universal curve over M_g is also of rank one, generated by the relative dualizing sheaf. We can make analogous statements about the Severi variety of plane curves and the Hurwitz space parametrizing branched covers, which are still open; in fact, the former was conjectured by Enriques more than a century ago and remains open.
In this talk I'd like to describe all of these theorems/conjectures, and the implications among them, including Isabel Vogt's recent work on Severi varieties. I'll be working entirely with rational coefficients, so torsion classes, which are far more mysterious, will not enter into it.