# Talks

## Preliminary schedule

Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|

9:00-9:50 | Hoffmann | Salberger | Casalaina-Martin | Colliot-Thélène | |

Registration | Coffee | Coffee | Coffee | Coffee | |

10:30-11:20 | Brüdern | Pieropan | Fortuna | Tian | Debarre |

11:30-12:20 | Ottem | Tschinkel | Laza | Mboro | |

Lunch | Lunch | Lunch | Lunch | ||

15:00-15:50 | Loughran | Frei | Jahnel | ||

Coffee | Coffee | Coffee | |||

16:30-17:20 | Vial | von Bothmer | Bayer | ||

18:30- | Dinner |

## Titles and abstracts

**Arend Bayer (University of Edinburgh):***Moduli spaces for Kuznetsov categories of cubic threefolds and fourfolds*

Moduli spaces of sheaves on (or stable objects in the derived category of) higher-dimensional varieties are badly behaved in many ways. In the case of cubic threefolds and fourfolds (and other Fano varieties), one can get rid of these pathologies by restricting ourselves to stable objects that lie in a certain semiorthogonal component of the derived category, called Kuznetsov category. I will also explain in examples how one can still study concrete geometry within this seemingly abstract setting.

The underlying foundational results are the construction of stability conditions for the Kuznetsov category of cubic fourfolds (joint with Lahoz, Macri and Stellari), and the notion and construction of stability conditions in families of varieties (joint with Lahoz, Macri, Nuer, Stellari and Perry).**Hans-Christian Graf v. Bothmer (Universität Hamburg):***Stable nonrationality of*Q*-Fano threefolds with anticanonical models of small codimension*

We report on joint work with Christian Böhning (still very much in progress) on rationality properties of some classes of Q-Fano threefolds given in a Gorenstein format in the sense of Reid et al. Besides proving stable non-rationality for very general members of families of such Fanos in new cases, this provides a unified view on and recovers results of Ahmadinezhad-Okada-Totaro et al. Time permitting, we will in addition say a few words about singular models of cubics in the context of the degeneration method.**Jörg Brüdern (Universität Göttingen):***Rational points on cubic threefolds - the role of analytic number theory*

This survey talk is based on joint work with Blomer and Salberger concerned with the Manin-Peyre predictions for the distribution of rational points on a certain cubic threefold. In particular we will discuss what analytic number theory is able to contribute to this area of research, and beyond.**Sebastiano Casalaina-Martin (University of Colorado):***Geometry and topology of moduli spaces of cubic threefolds*

The moduli space of smooth cubic threefolds admits a natural GIT compactification. In this talk I will discuss several other birational models of the moduli space, the geometric relationship among the spaces, and the topology of the spaces. This is joint work with Samuel Grushevsky, Klaus Hulek, and Radu Laza.**Jean-Louis Colliot-Thélène (Université Paris-Sud / CNRS):****Brauer-Manin obstruction for Markoff type surfaces**

Giving sufficient conditions for existence and density of integral points (Zariski density, strong approximation) on affine cubic surfaces is a very much open problem. Ghosh and Sarnak have recently studied integral points on Markoff type surfaces, which are affine cubic surfaces defined by an equation*x*^{2}+*y*^{2}+*z*^{2}-*xyz*=*m*, for*m*an integer. These are special, inasmuch as a discrete group of automorphisms acts on them, and there is in general a bounded fundamental domain for the set of integral solutions. In their work, Ghosh and Sarnak have shown that strong approximation and integral Hasse principle often fail. Some of their arguments rely on the quadratic reciprocity law. In joint work with Fei Xu and Dasheng Wei, we investigate the exact extent to which the integral Brauer-Manin obstruction accounts for such failures, and "how many" such failures occur. In simultaneous and independent work, Daniel Loughran and Vladimir Mitankin have also addressed the same question.**Olivier Debarre (Université Paris 7 / ENS):***Periods of cubic fourfolds and of DV varieties*

Beauville and Donagi showed that the primitive Hodge structure of a smooth complex cubic hypersurface of dimension four is isomorphic to that of its variety of lines, a smooth hyperkähler variety of dimension also four. This gives a way to study the period map for cubic fourfolds, a rational map from the projective GIT moduli space of semistable cubic fourfolds to the period domain. As shown by Hassett, this map is not defined at the very special (singular but semistable) point corresponding to the determinantal (or chordal) cubic, but it is well defined (generically) on the blow up of this point and the exceptional divisor maps onto a well understood divisor (called a Hassett—Looijenga—Shah, or HLS divisor) in the period domain.

In this talk, I would like to describe an analogous situation (although it has no direct relationship with cubic hypersurfaces): to a general 3-form on a complex vector space of dimension 10, one can associate a smooth hyperkähler variety of dimension four (called a Debarre—Voisin variety). We exhibit several HLS divisors and identify the corresponding very special 3-forms.

This is work in progress in collaboration with Frédéric Han, Kieran O'Grady, and Claire Voisin.**Mauro Fortuna (Leibniz Universität Hannover):***Cohomology of the moduli space of non-hyperelliptic genus four curves*

In this talk, I will present the intersection Betti numbers of the moduli space of non-hyperelliptic Petri-general genus four curves. This space has a canonical compactification as GIT quotient, which was proved to be the final step in the Hassett-Keel log MMP for stable genus four curves. The strategy of the cohomological computation relies on a general method developed by F. Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on stratifications, a partial desingularisation and the decomposition theorem.**Christopher Frei (University of Manchester):***Counting rational points of bounded non-anticanonical height*

A conjecture of Batyrev and Manin predicts the asymptotic behavior of rational points of bounded height on smooth projective varieties over number fields. We discuss some new cases of this conjecture for conic bundle surfaces equipped with certain non-anticanonical height functions. As a special case, the conjecture is verified for some smooth cubic surfaces with height functions associated to some ample line bundles. This is joint work with Dan Loughran.**Norbert Hoffmann (Mary Immaculate College):****Universal torsors over degenerating del Pezzo surfaces**

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor under the Neron-Severi torus. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. It is joint work with Ulrich Derenthal.**Jörg Jahnel (Universität Siegen):***On cubic surfaces violating the Hasse principle*

While the geometry of cubic surfaces was studied intensively in the 19th century, their arithmetic was investigated mainly in the second half of the 20th century. The first example of a cubic surface violating the Hasse principle is due to Swinnerton-Dyer (1962), generalisations have been given by Mordell and others.

In this talk, I will report on further generalisations of Mordell's examples. This is joint work with Andreas-Stephan Elsenhans (Paderborn). We show, in particular, that counterexamples to the Hasse principle, due to the Brauer-Manin obstruction, form a Zariski-dense subset of the moduli scheme of all cubic surfaces.

An analogous result is true for del Pezzo surfaces of degree four, and was obtained jointly with Damaris Schindler (Utrecht).**Radu Laza (Stony Brook University):***Cubics with an Eckardt point*

I will discuss a surprisingly rich picture associated to a cubic hypersurface with an Eckardt point (i.e. a cubic with a hyperplane section isomorphic to the cone over a lower dimensional cubic). I will discuss potential connections to hyper-Kaehler manifolds and rationality questions.

This is joint work with G. Pearlstein and Z. Zhang.**Daniel Loughran (University of Manchester):***Arithmetic of Cubic Threefolds*

We prove a generalisation of Clemens-Griffiths' famous Torelli theorem to non-algebraically closed fields. This is achieved by studying the intermediate Jacobian as a morphism of algebraic stacks. We give applications to integral points on stacks and versions of the Shafarevich conjecture for cubic threefolds. This is joint work with Ariyan Javanpeykar.**René Mboro (Universität Wien):***Decomposition of the diagonal for cubic threefolds over a field of positive characteristic*

Adapting arguments of Voisin, we show that for a cubic threefold over an algebraically closed field of characteristic > 2, to admit a cohomological decomposition of the diagonal is equivalent to admit a Chow-theoretic decomposition of the diagonal. We deduce that such a cubic threefold admits a Chow-theoretic decomposition of the diagonal if and only if the minimal class of its intermediate Jacobian is algebraic. Then, on the closure of a finite field of characteristic p > 2, the Tate conjecture for divisors on surfaces defined over finite fields of characteristic p predicts that any cubic threefold admits a Chow-theoretic decomposition of the diagonal.**John Christian Ottem (University of Oslo):***Curve classes on irreducible holomorphic symplectic varieties*

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds. This is joint work with G. Mongardi.**Marta Pieropan (EPFL):***Cox rings and Galois descent of complete intersections*

Given a variety Y defined over a nonclosed field k and a subvariety X that is geometrically a complete intersection, one can ask if X is a complete intersection of subvarieties of Y over k. We generalize the notion of strict complete intersection subvariety from projective spaces to varieties with Cox rings, and we prove Galois descent for strict complete intersections.**Per Salberger (Chalmers University of Technology):***Counting rational points of cubic hypersurfaces*

Let N(X;B) be the number of rational points of height at most B on an integral cubic hypersurface X over Q. It is then a central problem in Diophantine geometry to study the asymptotic behavior of N(X;B) when B growths. We present some recent results on this for various classes of cubic hypersurfaces.**Zhiyu Tian (Université de Grenoble):***Stability of spaces of rational curves*

I will explain a Floer type heuristic from symplectic geometry about the homotopy types of moduli spaces of rational curves on a rationally connected variety (due to Cohen, Jones, and Segal), and its connection with counting rational curves over finite fields.**Yuri Tschinkel (New York University):***Rationality problems*

I will discuss new results and constructions in the study of rationality of higher-dimensional algebraic varieties over algebraically closed and nonclosed field.**Charles Vial (Universität Bielefeld):***The generalized Franchetta conjecture for hyperKaehler varieties*

The original Franchetta conjecture, established by Harer, predicts that the restriction of a line-bundle on the universal family of smooth projective curves of given genus g>1 to a fiber is a multiple of the canonical line-bundle. Recently, O'Grady proposed an analogue of that conjecture for codimension-2 cycles on the universal family of polarized K3 surfaces of given degree. In this talk I will propose a version of the Franchetta conjecture for hyperKaehler varieties (and their powers) and provide some evidence, most notably by focusing on the Fano variety of lines on a smooth cubic fourfold. This is joint work with Lie Fu, Robert Laterveer and Mingmin Shen.

## Talks outside the workshop

*Thursday, 23 August 2018, 14:15-15:15 in G117***DongSeon Hwang (Ajou Universität):***On cascades of log del Pezzo surfaces*

A normal projective surface with quotient singularities is called a log del Pezzo surface if the anticanonical divisor is ample. There have been extensive work devoted to classify log del Pezzo surfaces.To achieve this goal, we proposed an approach by using the concept of a 'cascade', which can be viewed as a generalization of the classification of the smooth del Pezzo surfaces. So far this approach has been reasonably successful in the case when the log del Pezzo surface is toric or is of Picard number one.

In this talk, I will briefly review the known classification of log del Pezzo surfaces and explain how our approach leads to the classification. Some application of this approach relies on computing power. If time permit, I will mention a web-based computer program realizing this approach.-
*Tuesday, 4 September 2018, 11:00-12:00 in G117*

Toshiyuki Katsura (Tokio, Hosei Universität):*Construction of numerically trivial automorphisms of Enriques surfaces in characteristic 2*

Let S be an Enriques surface defined over an algebraically closed field in characteristic p. An automorphism f is called numerically trivial if it acts identically on the second l-adic cohomology group of S. In case that p = 0 of p>2, S. Mukai, Y. Namikawa, I. Dolgachev and G. Martin examined numerically trivial automorphisms of Enriques surfaces and showed that there exists no non-trivial numerically trivial automorphism of odd order for Enriques surfaces. In this talk, we will construct some examples of Enriques surfaces S with numerically trivial automorphisms of odd order >1 in characteristic 2.