Schedule for: 17w5146 - New Trends in Arithmetic and Geometry of Algebraic Surfaces

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree~6 and at the same time as \'etale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus~1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface~$S$ defined over~$\mathbb{Q}$. This is a joint work with Michael Stoll.

In this talk we complete the classification of surfaces of general type with $K^2=2c_2=8$ and $p_g=q=2$ whose Albanese map is a degree $2$ cover and we prove that there are only two families of such surfaces. The first one was constructed by Penegini, the universal cover of these surfaces is the bi-disk. The second family, which we construct, consist of two surfaces $X_1, X_2$, which are rigid and complex-conjugated. Their universal cover is not the bidisk : they contain an open subset which is a quotient of the complex 2-ball by a lattice in $PU(2,1)$. Joint work with C. Rito and F. Polizzi.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Brauer groups of K3 surfaces behave in many ways like torsion points of elliptic curves. In 1996, Merel showed that torsion groups of elliptic curves are uniformly bounded across elliptic curves defined over number fields of fixed degree. I will discuss a conjecture pointing towards an analogous statement for K3 surfaces, and survey recent mounting evidence for it.

Let K be the fraction field of a dvr R. Given a symmetric bilinear space V over K, and a group G acting by isometries on V we give necessary and sufficient criteria for V to contain a unimodular lattice stabilized by G. We sketch two applications to zeta functions of varieties over finite fields. In one direction, the theorem gives restrictions on the possible characteristic polynomials of Frobenius on the middle cohomology of a smooth projective variety of even dimension over a finite field. This application generalizes (and gives a more conceptual proof of) a theorem of Elsenhans and Jahnel. In the other direction, the theorem plays a crucial role in establishing the existence of K3 surfaces over finite fields with given zeta-function.

We give an unconditional construction of K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields, under some mild restrictions on the characteristic. Previously, such results were obtained by Taelman assuming the validity of good reduction criterion for K3 surfaces. The main contribution of this talk is to make Taelman's proof unconditional.

We prove that a K3 surface with an automorphism acting on the global 2-forms by a primitive $m$-th root of unity does not degenerate if $m \neq 1,2,3,4,6$ (assuming the existence of the so-called Kulikov models). To prove this we study the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the $l$-adic cohomology groups.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will report on a GIT construction for degenerations of Hilbert schemes of points on surfaces. The motivation comes from studying degenerations of irreducible holomorphic symplectic manifolds (IHMS), where a prime examples is given by degree $n$ Hilbert schemes of $K3$ surfaces. Previously, Nagai gave a very concrete construction for degree $2$ Hilbert schemes by using ad hoc modifications of the relative Hilbert scheme. In contrast to that, Li and Wu have developed a very general theory of degenerations of Hilbert schemes (not only of $0$-cycles) using expanded degenerations. Being very general, this approach makes it hard to describe the degenerations and their geometry explicitly. Here we develop a third approach: we use GIT methods to construct degenerations of Hilbert schemes of points on surfaces (in arbitrary degree). This allows us to describe the geometry of the singular fibres very explicitly. We can further prove that this is leads to dlt-degenration and by work of Halle and Nicaise this shows that the dual complex of the degenerate fibre coincides with the Kontsevich-Soibelman skeleton. This is joint work with M. Gulbrandsen, L. Halle and (partly) Z. Zhang.

In a famous paper Allcock, Carlson and Toledo describe the moduli space of smooth cubic threefolds as a 10-dimensional ball quotient. We show how the 10-dimensional ball quotient is also the moduli space of hyperkähler fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface with non-symplectic automorphism of order three (not coming from the K3 surface). With the help of the ball quotient we completely describe the hyperkähler manifolds and we identify them with the Fano variety of lines of cubic fourfolds that are triple covers of a smooth cubic threefold. As a consequence we give a relation between the hyperkähler manifolds and the moduli space of genus five principally polarized abelian varieties.

We will describe the several unexpected examples of totally geodesic surfaces in the moduli space of curves endowed with its Teichmuller metric. Our examples arise from the study of dihedral curves and plane cubics, and they reveal surprising connections between Teichmuller theory and algebraic geometry. This work is joint with Eskin, McMullen and Wright.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The leading eigenvalue of an automorphism of a complex surface, acting on cohomology, associates an algebraic integer to each such dynamical system. In this talk we will discuss the reverse process: the synthesis of a dynamical system, starting with an algebraic integer. In particular we will explicitly describe dynamical systems of minimal entropy on K3 surfaces, rational surfaces and complex tori.

The entropy of an automorphism of a surface is the logarithm of a Salem number, that is, an algebraic integer $\lambda>1$ which is conjugate to $1/\lambda$ and all whose other conjugates lie on the unit circle. In this talk I report on work in progress on the question when some multiple $n \log \lambda$, $n \in \mathbb{N}$, arises from an automorphism of a projective K3 surface.

In this talk, I would like to give examples of projective hyperKahler manifolds of Picard number two with automorphisms of positive topological entropy. These are constructed as moduli spaces of stable sheaves on K3 surfaces. Moreover, I would like to discuss relations between automorphisms on moduli spaces and derived automorphisms of K3 surfaces and compare their topological/categorical entropy.

We prove that for q>1 a smooth surface of degree q+1 over any field has at most (q+1)(q^3+1) lines, and thus that when q is a prime power the "Hermitian" (a.k.a. diagonal) surface of that degree over a field of q^2 elements has the maximal number of lines for its degree. This was previously known only for q=3 (Rams-Schuett 2015) and of course q=2. This is one of several cases where we obtain a sharp bound; other examples are the 126 tritangents of the Fermat sextic in characteristic 5 (and likewise for other "Hermitian" curves in odd characteristic), and the 891 planes on the diagonal cubic fourfold in characteristic 2.

A famous result of Fontaine (and Abrashkin) states that there is no abelian variety over the rationals with everywhere good reduction. Fontaine's proof of this result relies on the non-existence of certain finite flat group schemes. His technique has been refined by several people (including Schoof, Brumer and Calegari) to prove non-existence of semi-stable abelian varieties over various fields. On the other hand, a result of More-Bailly states that in every genus $g$, there is a curve of genus $g$ with everywhere good reduction over $\bar{\Z}$. So, as the base field varies, we must hope to find more abelian varieties with trivial conductor. In this talk, I will present a method for finding abelian surfaces with everywhere good reduction.

We describe methods to efficiently and rigorously compute the endomorphism algebra of the Jacobian of a curve. We discuss improvements to complex analytic methods and a method that entirely avoids numerical computation using Puiseux series. This is joint work with Edgar Costa, Nicolas Mascot, and Jeroen Sijsling.

We will discuss some properties of ample line bundles in arithmetic geometry, and give analogues of well-known geometric results. We will apply these results to an arithmetic analogue of the Bertini irreducibility theorem.

I will report on joint work with Marcello Bernardara, where we show how the existence of phantom subcategories of the derived category of an algebraic surface can be viewed as a stronger measure of rationality than the existence of a decomposition of the diagonal.

We prove that if two Calabi-Yau invertible pencils in projective space have the same dual weights, then they share a common polynomial factor in their zeta functions related to a hypergeometric Picard-Fuchs differential equation. The polynomial factor is defined over the rational numbers and has degree equal to the order of the Picard-Fuchs equation. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight.

We will explore new techniques for computing zeta functions of toric K3 surfaces, and some of its applications, e.g., progress towards explicit modularity, the variation of Néron--Severi ranks. Joint work with David Harvey and Kiran Kedlaya.

Point counting is a computation approach to cohomology of varieties. In this talk I will present my experiences with this method and a few interesting examples found. Most notable are K3 surfaces with real or complex multiplication.

I will report on our results, joint with Edgar Costa and Andreas-Stephan Elsenhans, on the distribution of the geometric Picard ranks of K3 surfaces under reduction modulo various primes. In the situation that rank Pic $S_{\bar{k}}$ is even, we introduce a quadratic character, called the jump character, such that rank Pic $S_{\bar{\mathbb{F}_{\mathfrak{p}}}\to$ ~ rank Pic $S_{\bar{k}}$ for all good primes, at which the character evaluates to (-1). As an application, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank. Our investigations also provide, as a by-product, a canonical choice of sign for the discriminant of an even-dimensional complete intersection.

In characteristic 0, S. Kondo classified Enriques surfaces with finite automorphism groups into seven types. In this talk, we consider Enriques surfaces with finite automorphism groups in characteristic 2, and give the complete classification of them. We have 4 types for singular Enriques surfaces, 5 types for supersingular Enriques surfaces and 8 types for classical Enriques surfaces. We also determine the structure of automorphism groups. This is a joint work with S. Kondo and G. Martin.

Given a surface $S$ of general type, we denote by d the degree of the canonical image. d is bounded by the canonical volume $K^2$, and by the BMY inequality we have $d \leq 9 \chi$. In practice, what is the maximum $d$ for $p_g=4,5,6$? Up to now, the best lower bound is quite low (in joint work with Ingrid Bauer we achieved ball quotients with $p_g=4, K^2 = 45$, but the record $d$ for $p_g=4$ is still 28). For $p_g=6$, where one expects to have surfaces whose canonical map is an embedding, there are interesting ties with methods and questions of homological algebra (Walter's bundle Pfaffians), which led to the question whether 18 would be the upper bound (I answered by getting degree 24). I recently constructed several connected components of the moduli space, of surfaces $S$ of general type with $p_g = 5,6$ whose canonical map has image $\Sigma$ of very high degree, $d=48$ for $p_g =5, d=56$ for $p_g =6$. The surfaces we consider are SIP 's , surfaces isogenous to a product of curves $(C_1 \times C_2)/G$. Representation theory then enters the picture in several ways, the easiest one being: when does a submodule of the group algebra $\mathbb{CC}[G]$ yield a projective embedding of $G$?

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.