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.

Abstract: Let \(A\) be an abelian variety defined over a number field. A theorem of Rémond states that for any two finite subgroup schemes \(G, H\), the Faltings height of the four isogenous abelian varieties \(A/G, A/H, A/(G+H), A/(G\cap H)\) are linked by an elegant inequality. The goal of the talk is to present an analogous inequality for abelian varieties defined over function fields, and discuss some applications in diophantine geometry. This is joint work with Richard Griffon and Samuel Le Fourn.

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

Abstract: Vafa-Witten invariants are virtual counts of Higgs pairs on a complex smooth projective surface. For positive geometric genus, they are conjecturally governed by universal functions with modular properties. Together with T. Laarakker and L. Göttsche, we conjectured blow-up and symmetry equations for these universal functions. Expressing them in terms of Nekrasov partition functions reduces these to blow-up and symmetry equations for Nekrasov partition functions. The former were proved by Kuhn-Leigh-Tanaka, and we prove the latter using mixed Hodge modules. Joint work with N. Arbesfeld and T. Laarakker.

Abstract:Let \(S\) be a smooth projective surface with \(p_g>0\) and \(H^1(S,\mathbb{Z})=0\). We consider the moduli spaces \(M=M_S^H(r,c_1,c_2)\) of \(H\)-semistable sheaves on \(S\) of rank \(r\) and with Chern classes \(c_1,c_2\). Associated a suitable class \(v\) the Grothendieck group of vector bundles on \(S\) there is a deteminant line bundle \(\lambda(v)\in Pic(M)\), and also a tautological sheaf \(\tau(v)\) on M. We give conjectural formulas for the virtual Verlinde numbers, i.e. the virtual holomorphic Euler characteristics of all determinant bundles \(\lambda(v)\) on M, and for Segre invariants associated to \(\tau(v)\). The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality.

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Abstract: A space of matrices of constant rank is a vector subspace V, say of dimension n+1, of the set of matrices of size axb over a field k, such that any nonzero element of V has fixed rank r. It is a classical problem to look for examples of such spaces of matrices, and to give relations among the possible values of the parameters a,b,r,n. In this talk I will report on my latest joint projects (and work in progress) with D. Faenzi, D. Fratila and P. Lella, where we use techniques involving vector bundles on projective spaces, finitely generated graded modules, and representation theory.

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Abstract: Consider the classical problem in enumerative geometry of counting rational plane curves through a fixed configuration of points. The problem may be considered over any base field and the point conditions might be scheme theoretic points. Recently, Kass--Levine--Solomon--Wickelgren have used techniques from \(\mathbb{A}^1\)-homotopy theory to define an enumerative invariant for this problem which is defined over a large class of possible base fields. This new theory generalizes Gromov-Witten invariants (base field = complex numbers) and Welschinger invariants (base field = real numbers) simultaneously. In this talk I will report on work in progress which explores the tropical approach to computing these new invariants. More specifically, for point conditions which are defined over (at most) quadratic extensions of the base field, we are developing a tropical correspondence theorem which expresses the KLSW-invariant as a tropical count. This result generalizes earlier correspondence theorems by Mikhalkin and Shustin and allows us to effectively compute some quadratically enriched invariants which were not known before. This is joint work in progress with Andrès Jaramillo-Puentes, Hannah Markwig, and Sabrina Pauli.

Abstract: We outline the proof that the self-intersection of the Arakelov canonical sheaf of the classical modular curves \(X_0(N)\) is asymptotic to \(3g\log N\), for \(N\) that tends to infinity and coprime with 6, where \(g\) is the genus of \(X_0(N)\).

Abstract: In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I showed how his general strategy can be adapted to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite diameter of hedgehogs. Motivated by Mahler's work on root separation, I gave an elementary proof of Dubinin's Theorem, albeit with a worse numerical constant. In this talk, I will report on joint work in progress with Harry Schmidt. We find new upper bounds for the transfinite diameter of some finite topological trees. We construct these trees using the Hubbard tree of a post-critically finite map. They are more attuned to the dynamical setting than hedgehogs. As a consequence, we can cover new cases of the dynamical Schinzel--Zassenhaus Conjecture.

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Abstract: We consider the cones of divisors that are ample in codimension k on a variety of dimension n, for all possible values of k. Taking closures, for k=0 and k=n-1 we obtain the pseudoeffective and the nef cone respectively. For intermediate values of k, little is known about these cones. In this talk I will show that for Mori dream spaces (MDSs) all such cones are rational polyhedral. I will then discuss duality between such cones and cones of k-moving curves. For MDSs, a Weak Duality property holds, already proved by Payne and Choi. I will show that a Strong Duality property is satisfied by an interesting family of varieties. This is a joint work with O. Dumitrescu, C. Brambilla and L. Santana-Sánchez.

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