Abstract: The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. It predicts that on a curve embedded by a line bundle of sufficiently high degree, the existence of a p-th syzygy is equivalent to the existence of a certain secant to the curve. I will discuss the history of this problem, then establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial cases, that is, when the line bundles that satisfy the corresponding secant condition form a divisor in the Jacobian of the curve.

Abstract: Sasakian geometry is a vibrant field at the intersection of differential geometry, topology, complex geometry, and algebraic geometry, with applications ranging from theoretical physics to geometric analysis. In this talk, we explore closed simply connected 5-manifolds capable of hosting positive Sasakian structures.

Abstract: The conjecture of Singer on the cohomology of aspherical manifolds plays an important role in modern geometry, topology and geometric group theory. In this talk, I will give an overview of this problem in the context of algebraic geometry. The case of irregular complex surfaces will provide many concrete examples and interesting connections with classical algebraic geometry.

Abstract: The Zilber-Pink conjecture is a very general statement that implies many well-known results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang, André-Oort and Falting's Theorem. After a general introduction to the problem, I will report on joint work with Gabriel Dill in which we proved that the Zilber-Pink conjecture for a complex abelian variety \(A\) can be deduced from the same statement for its trace, i.e., the largest abelian subvariety of \(A\) that can be defined over the algebraic numbers. This gives some unconditional results, e.g., the conjecture for curves in complex abelian varieties (over the algebraic numbers this is due to Habegger and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that have transcendental \(j\)-invariant. While working on this project we realised that many definitions, statements and proofs were formal in nature and we came up with a categorical setting that contains most known examples and in which (weakly) special subvarieties can be defined and Zilber-Pink and Mordell-Lang statements can be formulated. We obtained some conditional as well as some unconditional results.

Abstract: Let us consider a complex abelian scheme endowed with a non-torsion section. On some suitable open subsets of the base it is possible to define the period map, i.e. a holomorphic map which marks a basis of the period lattice for each fiber. Since the abelian exponential map of the associated Lie algebra bundle is locally invertible, one can define a notion of abelian logarithm attached to the section. In general, the period map and the abelian logarithm cannot be globally defined on the base, in fact after analytic continuation they turn out to be multivalued functions: the obstruction to the global existence of such functions is measured by some monodromy groups. In the case when the abelian scheme is endowed with a finite surjective modular map onto some suitable universal family of abelian varieties, we show that the relative monodromy group of the abelian logarithm is non-trivial and of full rank. As a consequence we deduce a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods. (Joint work with Paolo Dolce, Westlake University.)

Abstract: In this talk we introduce a geometric refinement of Gromov-Witten invariants for P1-bundles relative to the natural fiberwise boundary structure. We call these refined invariants correlated Gromov-Witten invariants. We will introduce the correlated invariants, discuss their properties and provide some computations in the case of P1-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces. This is a joint work with Thomas Blomme.

Abstract: We consider tangencies between sections of a complex elliptic surface and use an interesting foliation to prove an "unlikely intersection" result. This leads to connections with divisibility sequences, height bounds, and the geography of algebraic surfaces. This is joint work with Giancarlo Urzua.

Abstract: Let \(X \subset \mathbb{P}^N\) be a smooth irreducible n-dimensional variety. A well-known conjecture predicts that \(X\) always carries an Ulrich vector bundle, that is a bundle \(\mathcal{E}\) such that \(H^i(\mathcal{E}(−p)) = 0\) for \(i \geq 0\) and \(1 \leq p \leq n\). In the talk we will report on three recent results in collaboration with D. Raychaudhury. The first one is that any given \(X\) carries an Ulrich bundle if and only if it contains a subvariety satisfying certain conditions. The second one is an application of this result to low rank Ulrich bundles on complete intersections of dimension \(n ≥ 5\), or on general complete intersections of dimension \(n = 4\). The third one is an application to rank 2 Ulrich bundles on general hypersurfaces of dimension \(n\) with \(2 \leq n \leq 3\).

Abstract: A theorem of Esnault states that smooth Fano varieties over finite fields always have rational points. A natural question then arises: what happens if we relax the positivity conditions on the anticanonical class? In this seminar, I will discuss the case of 3-folds with nef anticanonical class over finite fields. Specifically, we demonstrate that in the case of negative Kodaira dimension, rational points exist if the cardinality of the field is greater than 19. In the K-trivial case, we prove a similar result, provided that the Albanese morphism is non-trivial. This result draws on a combination of techniques from the Minimal Model Program, semipositivity theorems, and arithmetic considerations. This is joint work with S. Filipazzi.

Abstract: The intersection theory of the moduli space of stable curves is one of the central topics of enumerative geometry, a subject with connections to Gromov-Witten theory, integrable systems, and complex geometry among others. One of the most fruitful ways to study a curve is through its interaction with its Jacobian. The intersection theory of the universal Jacobian is however much less developed than that of curves, especially over the locus of singular curves -- which is at least partially due to the subtleties involved in compactifying spaces of line bundles of singular curves. In this talk, I will explain how ideas from logarithmic geometry allow us to study the intersection theory of compactified Jacobians systematically, leading to new results both about their geometry and the enumerative geometry of curves.

The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2024" 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