Seminars

**Abstract**: Demailly introduced the notion of algebraic hyperbolicity as an algebraic version of the classical complex-analytic hyperbolcity notions due to Brody and Kobayashi. In this talk we will introduce this notion and its links to arithmetic as well as geometric questions. Then, we will focus on some new cases where we can prove algebraic hyperbolicity in the context of Campana's program. This is joint work with C. Gasbarri, E. Rousseau and J. Wang.

**Abstract**: Consider X a smooth algebraic projective hypersurface of dimension 2n and Y a subvariety of X of codimension n. Movasati and SertÃ¶z pose interesting questions on whether the low degree equations vanishing on Y can be recovered from the generators of the Artinian Gorenstein ideal associated to its Hodge class. The question is still open in general: for instance, a complete intersection always has this property, but there are also counterexamples (e.g. a smooth rational quartic on a surface of degree 4 in the projective space cannot be reconstructed by the cubics in the ideal associated to the Hodge cycle). I will present the general construction and partially answer the problem in any dimension.

**Abstract**: A. Treibich and J.L. Verdier introduced a special class of projective curves of arbitrary genus and used them to study KP differential equations. I will describe the main properties of these Treibich-Verdier curves, the most remarkable being that they are Brill-Noether general, and their connection with current research topics.

**Abstract**: The moduli spaces of complex curves together with a meromorphic differential with prescribed multiplicities of zeros and poles are the phase spaces of the action of GL(2,R) in Teichmuller dynamics, and are natural interesting geometric subvarieties of the moduli of curves with marked points. The geometry and topology of the strata remain mysterious, and we will present perhaps the first topological result beyond the number of connected components: that every connected component of every stratum in genus at least 2 has only one end; equivalently, this says that the boundary of a smooth compactification is connected. Based on joint work with Ben Dozier.

**Abstract**: Prima parte: Richiamerò risultati noti su spazi di moduli di fasci (semi)stabili su superfici proiettive liscie, in particolare sulle superfici K3. Seconda parte: Presenterò alcuni miei risultati su spazi di moduli fibrati vettoriali stabili su varietà hyperkähler di tipo \(K3^{[n]}\). Questi risultati possono essere considerati l'analogo di classici risultati di Mukai su fibrati vettoriali sferici su superfici K3.

**Abstract**: In 1938 U. Morin, improving on earlier results by G. Fano (1918), stated a projective classification theorem for varieties of dimension $n\geq 3$ whose general surface sections are rational. Although Morin's result is correct, his proof is wrong. In the first part of this talk I will explain how to fix Morin's argument by using ideas from Mori's theory already exploited by F. Campana and H. Flenner to attack a quite similar problem. This part is joint work with C. Fontanari. In the second part of the talk I will make some application to rationality of Fano threefolds.

**Abstract**: Given a projective variety \(X \subset \mathbb{P}^N\), it is said to be extendable if there exists \(Y \subset \mathbb{P}^{N+1}\) which is not a cone and such that \(X\) is a linear section of \(Y\). This talk focuses on the list of the weighted projective 3-spaces which are Gorenstein and their extendability in their anticanonical model. More precisely we focus on those for which a construction of their extensions was not known. We will see how to construct some of their maximal extensions from the study of the K3 surfaces inside them and their linear curve sections and how this provides maximal extensions for some examples of canonical curves.

**Abstract**: The goal of the talk consists of providing an overview of the theory of 2-dimensional cohomological Hall algebras (COHAs) and its relations to the study of the topology of moduli spaces (Nakajima quiver varieties, moduli spaces of framed sheaves on the complex projective plane, moduli spaces of Gieseker-semistable sheaves on smooth projective complex surfaces) and to the study of quantum groups and vertex algebras. Two examples will be addressed in detail: the COHA of the complex affine plane and the COHA of a minimal resolution of an ADE type singularity.

**Abstract**: This talk will be devoted to a joint work in progress with Ciro Ciliberto where we study Severi varieties of surfaces in threefolds with nodes and triple points. In particular, we will report about certain results of degenerations of singularities of surfaces, obtained by limit linear systems techniques.

**Abstract**: Classical Brill--Noether theory studies linear systems on a general curve in the moduli space \(\mathcal{M}_g\) of genus \(g\) curves. A refined Brill-Noether theory studies the linear systems on curves with a given Brill-Noether special linear system. As a first step, one would like to understand the stratification of \(\mathcal{M}_g\) by Brill-Noether loci, which parameterize curves with a particular projective embedding. In this talk, we'll introduce the Maximal Brill-Noether loci conjecture, and discuss recent progress via the established refined Brill-Noether theory for curves of fixed gonality and via studying unstable Lazarsfeld-Mukai bundles on K3 surfaces. This is joint work with Asher Auel and Hannah Larson.

The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2022" is a Winter meeting in Algebra and Geometry in Rome that takes place in the rione Monti. For more info here is the Link to the Official Webpage

Info

The seminar is usually held every **Thursday at 14:15-15:45** in **Aula M1**. This year seminars will be held in person but you can email one of the organizers if you are interested in attending remotely (through the platform Microsoft Teams).

The seminar is organized by Luca Schaffler and Amos Turchet and maintained by the Geometry Group of the Department of Mathematics and Physics at the Roma Tre University.

We acknowledge the support of the grants PRIN2020: Curves, Ricci flat varieties and their interactions, PRIN2017: Advances in Moduli Theory and Birational Classification, and the support of the Department of Mathematics and Physics at Roma Tre University.