Algebraic Geometry
in Roma Tre
June 14-17, 2022
On the occasion of
Sandro Verra's 70th birthday



The Ceresa cycle
Arnaud Beauville
(Université de Nice)

Let $C$ be a curve of genus $\geq 3$, embedded in its Jacobian $JC$. The cycle $[C]-[(-1)^*C]$ is cohomologous to zero in $JC$; is it algebraically equivalent to zero? The answer is negative for $C$ general (Ceresa, 1983) and for some very particular curves, and positive (trivially) for hyperelliptic curves. I will explain an example, obtained with C. Schoen, of a non-hyperelliptic curve $C$ for which $[C]-[(-1)^*C]$ is torsion modulo algebraic equivalence.

Fano manifolds with Lefschetz defect 3
Cinzia Casagrande
(Università di Torino)

We will talk about a structure result for some (smooth, complex) Fano varieties $X$, which depends on the Lefschetz defect $\delta_X$, an invariant of $X$ defined as follows. Consider a prime divisor $D$ in $X$ and the restriction $r\colon H^2(X,\mathbb{R})\to H^2(D,\mathbb{R})$. Then $\delta_X$ is the maximal dimension of $\ker(r)$, where $D$ varies among all prime divisors in $X$. If $\delta_X\geq 4$, then $X\cong S\times T$, where $S$ is a surface. When $\delta_X=3$, $X$ does not need to be a product, but we will see that it still has a very rigid and explicit structure. More precisely, there exists a smooth Fano variety $T$ with $\text{dim} T=\text{dim} X-2$ such that $X$ is obtained from $T$ with two possible explicit constructions; in both cases there is a $\mathbb{P}^2$-bundle $Z$ over $T$ such that $X$ is the blow-up of $Z$ along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. This structure theorem allows to complete the classification of Fano 4-folds with Lefschetz defect at least 3. This is a joint work with Eleonora Romano and Saverio Secci.

The connected components of the space of nodal K3 surfaces, and of nodal quintic surfaces.
Fabrizio Catanese
(Universität Bayreuth)

To a nodal complex projective surface one associates a binary code $K^\prime$ .

Theorem 1: The connected components of the space of nodal K3 surfaces are in bijection with the admissible codes $K^\prime$ , which we classify.

I will explain how $K^\prime$ determines an even lattice $L$, and, using Nikulin's theory of quadratic forms, one can determine when does $L$ embed in the K3 lattice. The rest follows from the Torelli theorem and a reality condition.

An interesting corollary is the irreducibility of the space of Togliatti cubics and quintics.

In the second part I shall describe some work (partly joint with Sandro Verra) on the codes and the varieties of nodal quintic surfaces in 3-space.

Irrational nef rays at the boundary of the Mori cone for very general blowups of the plane
Ciro Ciliberto
(Università di Roma Tor Vergata)

In this talk I will first recall what is the conjectural structure of the Mori cone of a general blowup of the plane. Then I will explain a technique for discovering (non-effective) irrational rays at the boundary of the Mori cone, and I will give examples of such irrational rays. This is joint work with R. Miranda and J. Roé.

Construction and classification of QQ-Fano 3-folds via mirror symmetry: a status report.
Alessio Corti
(Imperial College London)

I will explain where we are with this and give some demonstration of computational tools.

Intermediate Jacobians of Verra and Gushel-Mukai threefolds
Olivier Debarre
(Université Paris Diderot)

The Prym map, which takes a double étale cover of smooth projective connected curves to its Prym variety, is not injective in any dimension: Donagi's tetragonal construction (published in 1981) proves that it is 3-to-1 on the locus of double covers of tetragonal curves. Verra then noted in 1992 (and belatedly published in 2004) that the Prym map is also 2-to-1 on the locus of double covers of plane sextic curves. The reason for that is that a general Verra threefold $T$ (a divisor of bidegree (2,2) in ${\bf P}^2\times {\bf P}^2$) has two conic bundle structures, with discriminants plane sextic curves, and the intermediate Jacobian $J(T)$ of $T$ is isomorphic to the two associated Prym varieties. We will explain the links between the 9-dimensional intermediate Jacobian $J(T)$, whose theta divisor was studied in great details by Verra, and the 10-dimensional intermediate Jacobians of Gushel-Mukai threefolds. The ultimate aim is to understand the theta divisors of the latter and prove a Torelli-type theorem for Gushel-Mukai threefolds.

The birational geometry of $M_g$ via non-abelian Brill-Noether theory and tropical geometry.
Gavril Farkas
(Humboldt Universität zu Berlin)

I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program. This is joint work with Verra respectively with Jensen-Payne.

Contractions of hyper-Kähler fourfolds and the Brauer group
Bert van Geemen
(Università di Milano)

The exceptional locus of a birational contraction on a hyper-Kähler fourfold of $K3^{[2]}$-type is a conic bundle over a K3 surface. These conic bundles are projectivized (twisted) rank two vector bundles. We discuss the associated Mukai vectors, Brauer classes (B-fields) and Heegner divisors. We also give various examples of such conic bundles.

Moduli of cubic surfaces revisited
Klaus Hulek
(Leibniz Universität Hannover)

Moduli of cubic surfaces have been studied by very many authors. There are two approaches to construct these moduli spaces, namely either by GIT methods or by Hodge theory. The latter leads to ball quotient models. It is known that the (compact) GIT model and the Baily-Borel compactification of the ball quotient coincide (Allcock, Carlson, Toledo). Due to its construction, this space has two natural (partial) desingularizations, namely the Kirwan blow-up and the toroidal compactification. The main point of this talk is the observation that these compactifications have the same topology, but otherwise behave quite differently. In particular, they are not (naturally) isomorphic. We will also briefly touch on the analogue question for cubic threefolds. This is joint work with Sebastiano Casalain-Martin, Sam Grushevsky and Radu Laza.

Segre cubic 4-fold
Shigeru Mukai
(RIMS, Kyoto University)

The cyclic triple covering of the projective 4-space with branch the Segre cubic is characterized by 10 cusps as the Segre 3-fold is so by 10 nodes. The Fano variety of 2-planes is birationally equivalent to the Hilbert square of a K3 surface studied by Vinberg(1983). I will discuss the birational automorphism of this holomorphic symplectic 4-fold.

Theta groups and projective models of hyperkähler varieties
Kieran O'Grady
(Sapienza Università di Roma)

We show how to associate a non trivial theta group to a simple coherent sheaf $\mathcal{F}$ on a hyperkähler manifold $X$ of Kummer type or OG6 type, provided $g^{*}(\mathcal{F})$ is isomorphic to $\mathcal{F}$ for every automorphism $g$ of $X$ acting trivially on $H^2(X)$. This condition is satisfied if $\mathcal{F}$ is invertible, if $\mathcal{F}$ is one of the rank $4$ stable vector bundles on general polarized HK fourfolds with certain discrete invariants recently constructed by myself, or if $\mathcal{F}$ is the tangent bundle. The guiding idea is that if the theta group is a Heisenberg group, then there are stringent constraints on the equations of the image of $X$ under the map associated to global sections of $\mathcal{F}$.

Global injectivity of the Prym for ramified double coverings.
Angela Ortega
(Humboldt Universität zu Berlin)

Given a finite morphism between smooth projective curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. It is a classical result that the Prym map for étale double coverings over curves of genus at least 7 is generically injective but never injective (Donagi's tetragonal construction).

I will show that, unlike the unramified case, the Prym map is injective for ramified double coverings over curves of genus $g \geq 1$ and ramified in at least 6 points. I also explain why it is never injective for double coverings ramified in at most 4 points. This is a joint work with J.C. Naranjo.

Cohomological Abel-Jacobi theory in three parts
Rahul Pandharipande
(ETH Zürich)

There are three levels in the cohomological study of the Abel-Jacobi map: the standard double ramification cycle, the universal double ramification cycle, and the logarithmic double ramification cycle. I will discuss the log DR cycle and its recent calculation (joint work with Holmes, Molcho, Pixton, Schmitt). The method involves the geometry of the universal Jacobian over the moduli space of curves, which I will also discuss.

Measures of irregularity of Abelian varieties
Gian Pietro Pirola
(Università di Pavia)

Let $X$ be a complex abelian variety of dimension $n.$ We will consider some irrational measures for $A$, and in particular:

  1. the degree of irrationality $d_i(A)$: that is the minimal degree of a rational dominant map $f:A$ - - - > $\mathbb P^n;$
  2. (covering) gonality $gon(A)$: The minimal gonality of a normalization of an irreducible curve lying on $A;$
  3. minimal genus $m_g(A)$}: the minimal geometric genus of a curve lying on $A.$
We will discuss some old and recent estimates of these invariants. We collect some of them in the following:

If $A$ is a very general complex abelian variety of dimension $n>2$ then:
  1. $m_g(A)> \frac {(n-1)(n-2)}{2}.$     [ P. (1993)]

  2. $gon(A)\geq \frac{n+2}{2}.$     [O.Martin] .

  3. $d_i(A)\geq \frac{3n+1}{2} $     [E.Colombo-O.Martin- J.C. Naranjo-P.].
Rationality of admissible cubic fourfolds versus unirationality of the moduli spaces of associated K3 surfaces
Francesco Russo
(Università di Catania)

In the first part I will report on recent results with Giovanni Staglianà showing direct connections between the rationality of admissible cubic fourfolds in $\mathcal C_{d}$, $d=26, 38, 42$, and the explicit unirationality of $\mathcal C_d$ and of the moduli space $\mathcal F_{\frac{d+2}{2}}$ of associated K3 surfaces. In the second part I will present a geometric description of some codimension 2 loci of rational cubic fourfolds containing a plane studied by Hassett and of the corresponding loci of K3 surfaces, ending with some remarks on very general cubic fourfolds containing a plane.

Stable cohomology of complements of discriminants and moduli spaces
Orsola Tommasi
(Università di Padova)

The discriminant of a space of functions is the closed subset consisting of the functions which are singular in some sense. Specifically, we will consider the the space of non-singular sections of a very ample line bundle L on a fixed non-singular variety. In this set-up, Vakil and Wood proved a stabilization behaviour for the class of complements of discriminants in the Grothendieck group of varieties. In this talk, I will discuss a topological approach for obtaining the cohomological counterpart of Vakil and Wood's result, which implies that the k-th cohomology group of the space of non-singular sections remains the same if one takes a sufficiently high power of the line bundle L. As an application, I will discuss a result by my former PhD student Angelina Zheng on the stabilization of the cohomology of the moduli space of trigonal curves.

New invariants in equivariant birational geometry (joint with A. Kresch).
Yuri Tschinkel
(New York University)

Abstract: A well-known problem in geometry is to classify different embeddings of a finite group into the Cremona group, up to conjugation. I will discuss new invariants that allow to distinguish previously indistinguishable actions of finite groups on rational varieties.

Yuri Tschinkel (New York University)