Research

Valery Alexeev (University of Georgia)

Degenerations of $K3$ surfaces and 24 points on the sphere

I will discuss Kulikov and stable degenerations of $K3$ surfaces and describe explicit, geometric compactifications of their moduli spaces in several interesting cases. Based on joint work with Philip Engel and Alan Thompson.

Kenny Ascher (Princeton University)

Wall crossings for $K$-moduli and compactifying moduli spaces of plane curves

here are various methods to compactify the moduli space of smooth plane curves of degree at least 4 (e.g. GIT and MMP) and so it is natural to ask how they are related. We consider pairs $(P^2, aC)$ where $C$ is a plane curve of degree at least 4, and $a$ is a weight. We construct $K$-moduli spaces of log Fano pairs and establish a wall-crossing framework to show that as the weight $a$ increases, these spaces give a natural way to interpolate between the GIT and moduli space of stable pairs considered first by Hacking. We show that when a is small, the $K$-moduli compactification is isomorphic to the GIT moduli space, and study the subsequent wall crossings as a series of weighted blowups and flips. This is joint with K. DeVleming and Y. Liu.

Igor Dolgachev (University of Michigan)

15-nodal quartic surfaces : geometry, automorphisms and moduli

We use a realization of a 15-nodal quartic surface as the focal surface of a line congruences of bi-degree (2,3) to show that the moduli space of such surfaces is unirational and helps to explore the rich geometry and automorphism groups of such surfaces. We will compare these results with the known results about jacobian Kummer surfaces which are specializations of 15-nodal quartic surfaces.

Gavril Farkas (Humboldt-Universität zu Berlin)

Algebraic curves, syzygies and topological invariants of groups

I will discuss a surprisingly deep connection between nilpotence properties of Alexander invariants of groups and Green's Conjecture on syzygies of canonical curves. This has led to a universal upper bound on the nilpotence index of metabelian groups with vanishing resonance and to a simpler proof of Green's Conjecture for generic curves in characteristic zero, as well as to a proof in positive characteristic $p>(g+1)/2$. Joint work with Aprodu, Papadima, Raicu and Weyman.

Gerard van der Geer (University of Amsterdam)

Modular forms on a ball quotient and curves of genus three

We discuss a ball quotient related to moduli of curves of genus three and the associated modular forms. This is joint work with Jonas Bergstroem and Fabien Clery.

Brendan Hassett (Brown University)

Moduli of rational threefolds

We consider moduli problems of geometrically rational threefolds, with a view toward analyzing rationality behavior over non-closed ground fields. One central example is complete intersections of two quadrics.

Klaus Hulek (Leibniz Universität Hannover)

The Mori fan of the Dolgachev–Nikulin–Voisin family in genus 2

Gross, Hacking, Keel und Siebert have started a program to construct compactifications of moduli spaces of $K3$ surfaces. Their staring point is the mirror family of polarized $K3$ surfaces of given genus (degree). This mirror family is known as the Dolgachev-Nikulin-Voisin (DNV) family. It is a 1-dimensional family of lattice-polarized $K3$ surfaces. One of the main ingredients of the GHKS program is the Mori fan of the DNV family. In this talk we discuss the Mori fan of the DNV family in genus 2. This is joint work with Carsten Liese.

Radu Laza (Stony Brook University)

Symmetric cubic fourfolds

I will discuss about the classification of automorphism groups of cubic fourfolds (joint work with Z. Zheng), and possible connections to the rationality question, and the construction of hyper-Kaehler manifolds (joint work with G. Pearlstein and Z. Zhang).

Shohei Ma (Tokyo Tech)

Universal $K3$ surfaces and orthogonal modular forms

I prove that the graded ring of pluricanonical forms on the universal family of polarized $K3$ surfaces, including those with $A-D-E$ points, is naturally isomorphic to the ring of orthogonal modular forms of weight divisible by 20, twisted by character det and with vanishing condition at the $(-2)$-Heegner divisor, for the associated stable orthogonal groups. For a certain class of compactification, if exists, this extends to an isomorphism with the log canonical ring. I also explain the local model of the universal family (around the $A-D-E$ locus) which enables the analysis of the singularity.

Shigeru Mukai (MCM Chinese Academy of Sciences / Kyoto University)

Extremal Enriques surfaces and their mod 3 reductions

As an analogue of extremal elliptic surfaces, I will define and classify extremal Enriques surfaces in characteristic zero. After finding their defining equations, I consider their mod 3 reductions, which may give us a better view.

Yuji Odaka (Kyoto University)

Compactified moduli of $K3$ surfaces via Tropical $K3$ surfaces and their relatives

We explain explicit and canonical compactification of moduli of polarized $K3$ surfaces (for any degree), and of Kähler $K3$ surfaces, whose boundary should parametrize all differential geometric (compact) limits of the associated hyperKähler metrics on $K3$ surfaces which we wish to classify. For instance, a part of the boundary parametrizes tropical $K3$ surfaces, i.e., metrized spheres. Joint with Yoshiki Oshima. (arXiv:1805.01724, 1810.07685).

Kieran O’Grady (Università degli studi di Roma “La Sapienza”)

Natural sheaves on HK manifolds

Holomorphic vector bundles on K3's, and more generally torsion free sheaves, play a key role in many non trivial results. We do not expect that torsion free sheaves on higher dimensional hyperkaehler (HK) manifolds behave as well as they do on K3 surfaces. I will describe a special class of sheaves on HK varieties, and I will motivate the expectation that they behave (almost) as well as sheaves on K3's.

Kristian Ranestad (University of Oslo)

Divisors in the moduli space of cubic fourfolds

Cubic fourfolds that contain a surface that is not a complete intersection of two hypersurfaces form the Noether-Lefschetz locus in the moduli space. This locus is a countable union of divisors. A general cubic fourfold $V(F)$ gives rise to two different hyperkähler fourfolds, the variety of lines in the fourfold and the powersumvariety of $F$ that parametrizes ten-tuples of linear forms that appear in presentations of $F$ as a sum of cubic powers of linear forms. The incidence correspondence induces an isomorphism between the Hodge structure on the cohomology of degree 4 of $V(F)$ and the Hodge structure on the cohomology of degree 2 of the variety of lines. There are no such relation between $V(F)$ and the powersumvariety of $F$. On the other hand the construction of a powersumvariety allows natural definitions of divisors in the moduli of cubic fourfolds. I shall give a survey of old and recent results on Noether Lefschetz divisors and divisors of cubic fourfolds with special powersum presentations.

Colleen Robles (Duke University)

What representation theory can tell us about the cohomology of a hyperkähler manifold

The cohomology (with complex coefficients) of a compact kähler manifold M admits an action of the algebra $sl(2,C)$, and this action plays an essential role in the analysis of the cohomology. In the case that M is a hyperkähler manifold Verbitsky and Looijenga?Lunts showed there is a family of such $sl(2,C)$'s generating an algebra isomorphic to $so(4,b_2-2)$, and this algebra similarly can tell us quite a bit about the cohomology of the hyperkähler. I will describe some results of this nature for both the Hodge numbers and Nagai's conjecture on the nilpotent logarithm of monodromy arising from a degeneration. This is joint work with Mark Green, Radu Laza and Yoonjoo Kim.

Alessandro Verra (Università Roma Tre)

Coble cubics, genus 10 Fano threefolds and the theta map

The talk deals with the relations between two different moduli. From one side the moduli space $F$ of Fano threefolds $X$ of genus 10 is considered. Since the intermediate Jacobian of $X$ is a genus two jacobian $JC$, the assignement $X \rightarrow C$ defines a rational map $f: F \longrightarrow M$, $M$ being the moduli space of genus 2 curves. From the other side the moduli of semistable rank 3 vector bundles on $C$ with trivial determinant is considered and the branch divisor $B$ of its theta map. $B$ is the sextic dual to the Coble cubic, the unique cubic hypersurface singular along the embedding of $JC$ by its 3-theta linear system. Relying on properties of theta map and of $B$, a description of the map $f$ and of its fibres is given. The main result is that the fibre of $f$ is naturally birational $B$. This is a joint work with Daniele Faenzi.

Qizheng Yin (Peking University)

A compact analogue of the $P=W$ conjecture

The $P=W$ conjecture of de Cataldo, Hausel, and Migliorini relates the topology of Hitchin systems to the Hodge theory of character varieties via Simpson's nonabelian Hodge theory. Here $P$ and $W$ stand for the perverse and weight filtrations on the cohomology of the corresponding moduli spaces. While the original conjecture remains open, the $P=W$ phenomenon can be observed in a much broader context. In recent joint work with Junliang Shen, we proved a compact version of the $P=W$ conjecture, which relates the topology of holomorphic Lagrangian fibrations to the Hodge theory of compact hyper-Kaehler manifolds. I will present the circle of ideas as well as some applications of our result.

Susanna Zimmermann (Université d'Angers)

Cremona groups in higher dimensions

A Cremona group is the group of birational selfmaps of a projective space. It is an old object of study, and we already find Enriques wondering about its normal subgroups in 1895. Finally, in 2013 it was shown by Cantat–Lamy that the plane Cremona group is not simple. I present a collaboration with Blanc and Lamy which shows that also all Cremona groups in higher dimension are not simple.