Koszul modules, resonance schemes and applications

Koszul modules, introduced by Ş. Papadima and A. Suciu in a topological setup represent infinitesimal versions of Alexander invariants. They are finitely generated graded modules over a polynomial algebra and are defined with the help of Koszul complex, starting from a vector subspace in a second exterior power. Their geometric counterparts, specifically, annihilator supports are called resonance schemes. I will report on some recent joint works with G. Farkas, Ş. Papadima, C. Raicu, A. Suciu and J. Weyman on several applications of these objects to algebraic geometry and geometric group theory.

$P=W$ phenomena on abelian varieties

On syzygy schemes

Prym-Brill-Noether Theory

Brill-Noether Theory is a valuable tool for obtaining significant results concerning the geometry of curves and Abelian varieties. I will present the analogue situation for the case of Prym curves (i.e. etale double covers). In particular, I will compute the class of the Prym-Brill-Noether divisor and I will show the irreducibility of the universal Prym-Brill-Noether locus.

Relative Polar Geomerty of Algebraic Data

Foliations on homogeneous spaces

A codimension-1 foliation on a projective manifold X can be though of as a corank-1 (saturated) subsheaf F of the tangent bundle TX which is stable under the Lie bracket. After fixing the determinant of F, the set of such foliations is a locally closed subset of the space of 1-forms on X with values in some fixed line bundle L. We investigate the set of these foliations when X is a projective homogeneous space of Picard rank 1, for the simplest possible choice of L. A special focus will be on Grassmannians, most notably Grassmannians of lines, where our foliations are particularly easy to describe. In collaboration with V. Benedetti and A. Muniz.

The Minimal Resolution Conjecture on points on generic curves

The Minimal Resolution Conjecture predicts the shape of the resolution of general sets of points on a projective variety in terms of the geometry of the variety. We present an essentially complete solution to this problem for general curves. Our methods also provide a proof (valid in arbitrary characteristic) of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1. Joint work with E. Larson.

Ulrich bundles on some 3-fold scrolls over Hirzebruch surfaces

In this talk I will report on a recent joint work with M.L. Fania, where we investigate on Ulrich bundles on some 3-fold scrolls whose Hilbert schemes have been extensively studied in previous papers by the authors together with G. M. Besana. Inspired by some results by M.L. Fania, M. Lelli Chiesa and J. Pons Llopis, concerning existence of Ulrich bundles of rank one and two on scrolls over Hirzebruch surfaces F_0 and F_1, arising as 3-folds of low degree, we explicitely describe components M(r) of moduli spaces of Ulich bundles of any rank r>0 on 3-fold scrolls over Hirzebruch surfaces F_e, for any non-negative integer e, showing that such components M(r) are generically smooth, of computed dimension and whose general point corresponds to a slope-stable bundle. This in particular allows us to compute the Ulrich complexity of such 3-folds and to give an effective proof that these 3-folds are of Ulrich wild type.

Degenerations of singularities and Severi varieties of surfaces with nodes and triple points

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.

Unirationality of certain moduli spaces parametrizing ramified double covers of curves

Vector bundles on Fano threefolds

A celebrated part in the classification of Fano threefold is Mukai's vector bundle method. One of the main result is an existence (and rigidity) result for vector bundles with rank dividing the genus, for prime Fano threefolds of index 1. Unfortunately, in the literature, the proof has a gap. I will present joint work with Arend Bayer and Alexander Kuznetsov, where we fill this gap.

Taylor varieties

Stable cohomology of line bundles on flag varieties

A fundamental problem at the confluence of algebraic geometry and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. By embedding smaller flag varieties as Schubert subvarieties in larger ones, one can compare cohomology groups on different spaces and study their eventual asymptotic behavior. I will describe a sharp stabilization result, as well as explicit stable cohomology calculations in a number of cases of interest. Joint work with Keller VandeBogert.

Coinvariants of vertex algebras and abelian varieties

Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on my recent preprint arXiv:2301.13227.

Correspondences acting on constant cycle curves

Footnotes on Pfaffian cubic fourfolds with generalizations

• Davide BRICALLI: On the Hessian of cubic hypersurfaces, Università di Pavia
• Giusi CAPOBIANCO: Tropical and algebraic Prym varieties: the Abel-Prym map, Università di Roma Tor Vergata
• Dario FARO: Gauss-Prym maps on Enriques surfaces, Università di Pavia
• Filippo FAVALE: Jacobian rings of cubics and Lefschetz properties, Università degli studi di Pavia
• Sara Angela FILIPPINI: Residual intersections and Schubert varieties, Università del Salento
• Miguel GUERRERO: Stratification of the moduli space of curves by the corank of the Wahl map, National Autonomous University of Mexico
• Martina MISERI: The Prym-canonical Clifford index, Università di Roma Tre
• Federico MORETTI: Brill-Noether theory and degree of irrationality of surfaces, Humboldt university (Berlin)
• Giovanni PASSERI: Stable Measures of Associations, Stony Brok University
• Simone PESATORI: The moduli space of rational elliptic surfaces, Università di Roma Tre
• Federico PIERONI: Coble surfaces, Università di Roma Tre
• Samal PRAJWAL: Artinian Rings and Calabi-Yau Manifolds, Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)
• Elena SAMMARCO: Construction of a divisor in the moduli space of cubic fourfolds, Università di Roma Tre
• Angelina ZHENG: Rational cohomology of moduli spaces of curves, Università degli studi di Roma Tre