# Algebraic Geometry – Satellite Conference of 7ECM

programme

## Location

The talks on **Wednesday** will take place in the **Kali-Chemie-Saal (# 202) **in the chemistry building (# 2501** **on the map here)**.**

**Directions**: With public transport, take tram line 4 or 5 to the stop „Schneiderberg / Wilhelm-Busch-Museum“. Follow the street „Am Schneiderberg“ and turn right onto „Callinstraße“, entering into # 9.

On foot, just go to the main building of the university (Welfenschloss, see below) and walk through the park behind it until you hit either of the above streets.

The talks on **Thursday through Saturday** will take place in the main building of the university where also the mathematical institutes are located (# 1101 on the map here).

**Thursday, Friday**: room **F428.**

**Saturday**: room **E001.**

**Directions**: take tram line 4 or 5 to the stop "Leibniz Universität"

## Schedule

Wed | Thu | Fri | Sat | ||

10:00 – 11:00 | Kovács | Voisin | Cantat | 09:30 – 10:30 | Huybrechts |

11:30 – 12:30 | Bayer | Saccà | Viehmann | 11:00 – 12:00 | Petersen |

12:15 – 13:15 | Esnault | ||||

14:00 – 15:00 | Verra | van Geemen | Ruddat | ||

15:30 – 16:30 | Nicaise | Liedtke | Ottem | ||

17:00 – 18:00 | Maulik | Charles | Tschinkel |

The conference dinner will take place on Thursday evening at 19:00 at Gustino.

## Titles and abstracts

**Brill-Noether questions on K-trivial surfaces via wall-crossing** **(Arend Bayer)**

I will explain how wall-crossing can be used to answer Brill-Noether type questions for curves on K3 or abelian surfaces. This both gives a natural setting to reprove well-known results, but also leads to new results. In particular, it leads to a complete description of the Brill-Noether behaviour of the primitive linear system on generic abelian surfaces. This result is joint work with Chunyi Li.

**Dynamical degrees of birational transformations (Serge Cantat)**

Let f be a birational transformation of a projective variety X.

Iterating f, one gets a sequence of birational transformations f^n. Typically,

the degrees of the formulae defining f^n grow exponentially fast with the number

n of iterations. The exponential growth rate of this sequence is the ‘dynamical degree’

of f. Dynamical degrees turn out to be interesting numbers, with nice arithmetic

properties. I shall describe several results and questions concerning them.**Rational curves on hyperkähler manifolds (François Charles)**

We will survey various construction techniques for rational curves on compact hyperkähler manifolds and draw some conclusions regarding rational equivalence for zero-cycles. Some results are joint with Pacienza.

**Lefschetz theorem for F-overconvergent isocrystals and l-companions (Hélène Esnault)**

We show a Lefschetz type theorem for F-overconvergent isocrystals. This enables one to prove the crystalline analog of Deligne’s theorem: the local polynomials of irreducible l-adic sheaves with torsion determinant at closed points over a finite field is a number field, and the analog of Drinfeld’s theorem: there are l-adic companions. (Joint with Tomoyuki Abe.)

**A very special EPW sextic (Bert van Geemen)**

Recently Donten-Bury and Wiśniewski constructed an IHS (irreducible

holomorphic symplectic manifold) as a desingularization of the quotient of an

abelian fourfold by a finite group. In collaboration with them and with

Grzegorz and Michał Kapustka, we show first of all that the abelian fourfold

has a natural principal polarization and as such is isomorphic to the Debarre-

Varley abelian fourfold. The IHS is a double EPW-sextic, and this sextic has a singular locus consisting of 60 planes, 20 of which form a complete family of incident planes. Moreover, we show that the DBW IHS is birationally isomorphic to the Hilbert square of Vinberg's K3 surface, which is a double cover of the projective plane

branched over a special configuration of six lines.**Derived categories of cubic fourfolds and K3 surfaces (Daniel Huybrechts)****On the boundedness of slc surfaces of general type (Sándor Kovács)**

This is a report on joint work with Christopher Hacon. We give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field. We also highlight some important consequences of this result.**On morphisms to Brauer-Severi varieties (Christian Liedtke)**We classify morphisms over non-algebraically closed fields and morphisms to Brauer-Severi varieties, which generalises the classical correspondence between globally generated invertible sheaves and morphisms to projective spaces. As an application, we study the arithmetic and the geometry of Brauer-Severi varieties and del Pezzo surfaces over non-closed fields.

**Refined stable pair invariants and Higgs bundles (Davesh Maulik)**

In a series of papers, Hausel and Rodriguez-Villegas proposed an explicit conjectural formula for the Betti numbers (among other things) of the moduli space of stable Higgs bundles on a smooth projective curve. More recently, Diaconescu and collaborators have given an also-conjectural reinterpretation of the HRV formula using so-called refined invariants of the moduli space of stable pairs on a certain non-compact Calabi-Yau threefold. In this talk, I plan to explain this circle of ideas and discuss the proof of some of these statements as well as some generalizations (work in progress with A. Pixton).**Poles of maximal order of Igusa zeta functions (Johannes Nicaise)**

Igusa’s p-adic zeta function Z(s) attached to an integer polynomial f in N variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation f=0 modulo powers of a prime p. It is expressed as a p-adic integral, and Igusa proved that it is rational in p^{-s} using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of Z(s) is at most N, the number of variables in f. In 1999, Wim Veys conjectured that the real part of every pole of order N equals minus the log canonical threshold of f. I will explain a proof of this conjecture, obtained in collaboration with Chenyang Xu. The proof is inspired by non-archimedean geometry and Mirror symmetry, but the main technique that is used is the Minimal Model program in birational geometry. **Effective cones of cycles on blow-ups of projective space (John Christian Ottem)**

While the cones of curves and effective divisors are important tools in algebraic geometry, the cones of subvarieties of intermediate codimension remain much more mysterious, mainly due to the lack of examples. In this talk, I will outline two explicit computations of such cones; for blow-ups of projective space and for certain hyperkahler fourfolds. In the first case, we determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that in some cases, the higher codimension cones behave better than the cones of divisors. In the case of hyperkahler fourfolds, we show the cone of nef 2-cycles can be strictly larger than the cone of pseudoeffective 2-cycles, showing that the usual intuition from divisors do not generalize to higher codimension.**A spectral sequence associated to a stratification, and a conjecture of Vakil-Wood (Dan Petersen)**

Given a space with a stratification, there is a well known spectral sequence computing the Borel-Moore homology of the total space in terms of the BM homology of the open strata. I will describe a "dual" spectral sequence, which seems not to have been considered before, that takes as input the BM homology of closed strata and calculates the BM homology of an open stratum. Many familiar spectral sequences arise as special cases. I will apply it to the cohomology of configuration spaces of points, significantly generalizing representation stability results of Church, and proving a strengthening of a conjecture of Vakil-Wood (which had previously been proven by Kupers-Miller-Tran).

**Two Applications of log Gromov-Witten invariants** **(Helge Ruddat)**

Counting rational curves in P^2 totally tangent to a smooth elliptic curve is a so-called relative Gromov-Witten invariant. It was proved by Graber-Hasset that this count relates to the "local Gromov-Witten invariant": counting rational stable maps to the canonical bundle of P^2. In a joint on-going work with van Garrel and Graber, we show this generalizes to X smooth projective with a smooth ample divisor D. A key technology to prove this is log Gromov-Witten theory that I will explain in this talk. I will then give a second application which is a correspondence theorem of tropical curve counts with descendent log Gromov-Witten invariants for toric and conjecturally cluster varieties.

**Intermediate Jacobians and hyperKahler manifolds (Giulia Saccà)**

In recent years, there have been more and more connections between cubic 4folds and hyperkahler manifolds (aka holomoprhic symplectic manifolds). The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic 4folds X is holomorphic symplectic. The aim of the talk is to describe another instance of this phenomenon, which is carried out in joint work with R. Laza and C. Voisin.

Given a general cubic 4fold X, we can consider the universal family Y_U \to U of smooth hyperplanes sections of X and the relative Intermediate Jacobian fibration f: J_U \to U. In 1995 Donagi and Markman constructed a holomorphic symplectic form on J_U, with respect to which the fibration f is Lagrangian. Since then, there have been many attempts to find a smooth hyperkahler compactification of J_U. This was conjectured to exist and to be deformation equivalent to O'Grady's 10--dimensional exceptional example. With R. Laza and C. Voisin, we solve this conjecture by using relative compactified Prym varieties.**Rationality problems (Yuri Tschinkel)**

I will discuss recent advances in the study of rationality properties

of higher-dimensional algebraic varieties (joint work with B. Hassett and A. Pirutka).**On the birational geometry of some families of K3 surfaces (Sandro Verra)**

The talk considers families of K3 surfaces, polarized in genus g, which are quotients of K3 surfaces by a cyclic group of automorphisms of order n. The moduli of these families, more precisely their irreducible components for each n and g, are presently studied in order to understand more of their birational geometry and more on the projective models of these surfaces. The talk focuses on moduli of quotients of K3 surfaces by a symplectic involution in low genus g. In particular the (uni)rationality problem is considered for the moduli space N_g of the so called Nikulin surfaces, in genus g < 9. After a geometric description of N_g, for g < 8, the case g = 8 is studied and the rationality of such a moduli space is proved. The proof relies on classical geometric properties of singular cubic threefolds, including six nodal ones. Some possible developments in higher genus or higher order can be discussed.

**Geometry of Newton strata (Eva Viehmann)**

The Newton stratification is one of the central tools to study the reduction of Shimura varieties.

I will introduce this stratification and discuss recent results on the geometry of strata,

including very foundational ones such as non-emptiness, dimensions, irreducible components and closure relations.**On the torsion points of sections of Lagrangian fibrations (Claire Voisin)**

We prove that for a Lagrangian fibration of a hyper-Kähler fourfold X, and for any line bundle on X which is topologically trivial along the fibers, the set of fibers on which the restricted line bundle is torsion is dense in the base. This result holds as well for elliptic surfaces which are not isotrivial fibrations. We will explain an application to the Chow ring of such varieties.