**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 Sertöz 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.

**Abstract**: The work of Kuznetsov and Perry describes the derived category of a cyclic cover X of weighted projective space. The description is in terms of the categories of the base and of the branch locus Z and works under the assumption that both Z and X are Fano. We relax this assumption on Z and obtain an analogous result for canonically polarized branch loci. This extended framework unlocks information about two deformation families of low degree Fano threefolds which arise as double covers. We then use categorical K-theory to show that a specific component of the derived category of the threefolds determines them up to isomorphism (solving a so-called categorical Torelli problem). This is joint work with A. Jacovskis and H. Dell.

**Abstract**: The Noether-Lefschetz locus parametrizes families of surfaces in threefolds whose rank of the Neron-Severi group (the Picard rank) is higher than that of the ambient space. I will present work in progress (joint with with K. DeVleming and J. Rana) on the behavior in families of this locus. We will discuss “modularity properties” of components of both maximal and minimal codimension and present many examples.

**Abstract**:Given a dense open immersion \(U \to S\), some smooth and proper families \(X_U \to U\) do not extend to smooth proper families over \(S\). More often (but still not always), there is a "best smooth extension", the Neron model. I will talk about how to construct Neron models for families of smooth curves and their Jacobians. Neron models are not compatible with base change, so there are no "moduli spaces of Neron models", but we will see that they relate to some logarithmic moduli functors, and that their base change behaviour can be understood tropically (in terms of combinatorics of dual graphs).

**Abstract**: We give a criterion on a smooth projective variety, based on positivity of vector bundles, to characterize birationally abelian varieties. In order to look for a more elementary proof of this result, and free it from some technical hypothesis depending on the Minimal Model Program, we will ask some simpler questions related to semiampleness, and answer some of those. (part of the results presented are obtained in a joint work with Francesco Esposito).

**Abstract**: Mumford famoulsy asked to describe the ample cone of the moduli space of Deligne-Mumford stable curves \(\overline{M}_{g,n}\), or, dually, its Kleiman-Mori cone of curves. Conjecturally the answer is given by a well-known F-conjecture that states that the cone of curves is spanned by the one-dimensional boundary strata called the F-curves. A breakthrough Bridge Theorem of Gibney-Keel-Morrison reduces the positive genus case to the case of g=0. In genus 0, the F-conjecture for \(\overline{M}_{0,n}\) is implied by the strong F-conjecture which is true up to \(n \leq 7\) and fails for \( n \geq 12\) by Pixton's counterexample. I will report on joint work with Anton Mellit in which we prove the strong F-conjecture for n=8, the F-conjecture for \(\overline{M}_{g,n}\) for \(g \leq 44\) and a bit more.

**Abstract**: Virtual cycles have become one of the main tools in sheaf- and curve-counting problems. The idea is to replace the fundamental class of a moduli space by another cycle of a given "expected" dimension. This idea was established in the late 90s by Behrend-Fantechi and Li-Tian. We'll give an overview of the definition and discuss in the second part explicit applications to the moduli of Higgs sheaves.

**Abstract**: The enumerative geometry of the moduli spaces of curves, which began in the 1980s with a famous paper of Mumford, is now an extremely developed field, with perhaps thousands of papers dedicated to it. It's high time to do the higher dimensional case! The analogues of \(\overline{M}_{g,n}\) in higher dimensions are the KSBA spaces. I will introduce some characteristic classes on the KSBA spaces that are analogous to the kappa, lambda and psi classes on the moduli of curves, and present some results and speculations.

**Abstract**: The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obatined by Marché-Maurin in 2014 in the case of the square of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in \(Y(1)^2\) is tangent to finitely many modular curves (which are the one-codimensional special subvarieties) and, together with Capuano, we generalize the result of Marché-Maurin to arbitrary powers of the multiplicative group. No previous knowledge of the topic is required: we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what are the differences in the double intersection case.

**Abstract**: Let G be a connected reductive group over an algebraically closed field. A challenging task is the computation of the (intersection) cohomology of the moduli space of G-Higgs bundles over a curve. One way to tackle the problem is to study the derived direct image of the intersection complex of this moduli space along the G-Hitchin fibration. In the first part of the talk, we provide an overview of the G-Hitchin fibration for the moduli space of G-Higgs bundles. In the second part, we show that the derived direct image of the intersection complex for the meromorphic G-Hitchin fibration is determined by its restriction to the so-called elliptic locus, for any connected reductive group G. The cases GLn and SLn were already known thanks to the works of Chaudouard-Laumon, de Cataldo and Maulik-Shen. This is a work in progress jointly with Mark Andrea de Cataldo, Andres Fernandez Herrero and Mirko Mauri.

**Abstract**: The talk will report on joint work with Amos Turchet on a standard conjecture of diophantine geometry usually attributed to Vojta. This conjecture predicts that the complement in the complex projective plane of a general nodal curve, B, of degree at least 4 contains finitely many copies of \(\mathbb{C}^*\), where \(\mathbb{C}^*\) is the set of nonzero complex numbers. This is equivalent to the fact that the preimage of B under any non constant map from a smooth rational curve, X, to the plane has cardinality at most 2. Via a purely geometric approach, we prove this conjecture effectively when B has at least three irreducible components, and without the rationality assumption on X.

**Abstract**: Tratteremo l'immagine per una funzione razionale dell'insieme dei punti razionali di una varietà algebrica; un caso cruciale si realizza quando la varietà algebrica è un gruppo algebrico commutativo. Vedremo che varie questioni in questo ambito si collegano alla proprietà di Hilbert, al principio locale-globale, alla questione della densità di punti interi/razionali su varietà algebriche. I risultati originali sono stati ottenuti in collaborazione con Umberto Zannier.

**Abstract**: The exponential-algebraic closedness conjecture, due to Zilber, predicts sufficient conditions for a system of exponential-polynomial equations to have solutions in the complex numbers. It is phrased geometrically, interpreting existence of solutions to these systems as existence of points in the intersection of the graph of the complex exponential and an algebraic subvariety of the tangent bundle of the complex multiplicative group. In this talk, I will briefly recall the motivation of this question and then focus on the case of varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. These varieties correspond to systems of exponential sums equations, and the proof that the conjecture holds in this case uses tools from tropical geometry.

**Abstract**: In this talk we will consider the moduli space parametrizing roots of line bundles on algebraic curves. Our goal will be the construction of a tropical moduli space of roots of divisors on stable tropical curves. We will discuss the relation between the two moduli spaces, and we will give some applications.

**Abstract**: Generalising classical questions about regular polygons with vertices on a plane lattice, we are interested in pairs of points A,B on a lattice such that the angle \(\hat{AOB}\) is a rational multiple of \(\pi\). This problem leads to diophantine-trigonometric equations that in turn involve the study of rational points on curves of genus 0,1,2,3,5. I will present the full classification of plane lattices according to how many independent rational angles they contain and in which configurations they appear. This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier

**Abstract**: I will survey some results from the study of moduli spaces of higher dimensional varieties as well as the Hassett—Keel program for the moduli space of curves. I will then discuss applications of these techniques to the study of moduli spaces of K3 surfaces of low degree. Most of the original work presented in this talk will be based on joint work with Kristin DeVleming and Yuchen Liu.

**Abstract**: The classical Weil height machine associates heights to divisors on a projective variety. I will give a brief introduction to this machinery, how it extends to objects (closed subschemes) in higher codimension, due to Silverman, and present various ways to interpret the heights. We will then discuss several recent results in Diophantine approximation in which these ideas play a prominent and central role.

**Abstract**: The Clifford inequality is a classical upper bound on the dimension of the space of global sections for line bundles on any smooth curve in terms of the degree of the line bundle. It is easy to see that such a bound - that is, purely in terms of the total degree - is not possible if one considers reducible curves. Instead, one has to restrict to some class of multidegrees to obtain meaningful upper bounds. I will survey what is known about this question with an emphasis on recent joint work with M. Barbosa and M. Melo.

**Abstract**: The notion of a greproci set is only a few years old. It is based on a new model for classifying point sets in projective space, motivated by inverses scattering, namely: classify point sets according to the behavior of the image of the set under projection from a general point to a hyperplane. This provides many still open avenues for research. I will focus on the property that the projection is a complete intersection and discuss what we know now.

**Abstract**: The Dynamical Manin-Mumford problem is a dynamical question inspired by classical results from arithmetic geometry. Given an algebraic dynamical system (X,f), where X is a projective variety and f is a polarized endomorphism on X, we want to determine if a subvariety Y containing "unusually many" periodic points must be itself preperiodic. In a recent work in collaboration with Romain Dujardin and Charles Favre, we prove this property to hold when f is a regular endomorphism of \(\mathbb{P}^2\) coming from a polynomial endomorphism of \(\mathbb{C}^2\) of degree \(d\geq 2\), under the additional condition that the action of f at the line at infinity doesn't have periodic super-attracting points. We will introduce the problem and some of the ingredients of the proof, coming from arithmetic geometry, holomorphic and non-archimedean dynamics.

**Abstract**: Riporterò i risultati di un lavoro in collaborazione con Grushevsky, Salvati Manni e Tsimerman. In tale lavoro classifichiamo le sottovarietà olomorfe compatte massimali di \(\mathcal{A}_g\) e determiniamo la massima dimensione di una sottovarietà compatta di \(\mathcal{A}_g\) passante per un punto molto generico.

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 and the Number Theory 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, PRIN2022: Moduli spaces and birational geometry and PRIN2022: Semiabelian varieties, Galois representations and related Diophantine problems, and the support of the Department of Mathematics and Physics at Roma Tre University.