The conference focuses on recent advances in algebraic geometry in a broad sense.
The participation of young researchers is strongly encouraged.
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
Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product. Finally, work in progress on similar phenomena in other setups will be included. This is joint work with Alex Küronya and Martin Ulirsch.On syzygy schemes
If $X$ is a projective variety cut out by quadrics, the p^th Syzygy scheme $Syz_p(X)$ is the scheme cut out by quadrics involved in a p^th syzygy of $X$, and it turns out to capture refined geometrical properties of $X$ in its embedding. We report on joint work in progress with M. Aprodu and E. Sernesi, concerning the second Syzygy scheme $Syz_2(C)$ of a smooth curve in case $C$ is embedded either by the canonical line bundle or by a nonspecial line bundle $L$, aiming at a classification of all $(C,L)$ such that $Syz_2(C)$ strictly contains $C$PrymBrillNoether Theory
BrillNoether 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 PrymBrillNoether divisor and I will show the irreducibility of the universal PrymBrillNoether locus.
Relative Polar Geomerty of Algebraic Data
Tangential properties of embedded varieties have proven to be effective measures of their shape and topology in projective geometry. These properties are captured by the socalled polar classes. In this context, degrees and intersections of polar classes provide valuable numerical bounds, offering insights into critical points of rational functions and estimating the appropriate density of a pointsample of the variety. During our discussion, we will revisit these results and introduce similar bounds specifically for the case when the locus of interest is a subvariety rather than the entire variety. This extension allows us to examine the shape and topology of specific regions within the variety. The presentation is based on (on going) joint work with L. Gustafsson and L. Sodomaco and previous work with D. Eklund and M. Weinstein.Foliations on homogeneous spaces
A codimension1 foliation on a projective manifold X can be though of as a corank1 (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 1forms 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>2r1. Joint work with E. Larson.
Ulrich bundles on some 3fold 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 3fold 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 3folds of low degree, we explicitely describe components M(r) of moduli spaces of Ulich bundles of any rank r>0 on 3fold scrolls over Hirzebruch surfaces F_e, for any nonnegative integer e, showing that such components M(r) are generically smooth, of computed dimension and whose general point corresponds to a slopestable bundle. This in particular allows us to compute the Ulrich complexity of such 3folds and to give an effective proof that these 3folds 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
I will discuss some work in progress with M. LelliChiesa and Alessandro Verra concerning unirationality of the moduli spaces R_{g,2n}, parametrizing double covers of smooth genusg curves ramified at 2n points, for low genera and n=1,2,3. The proofs exploit curves on socalled nonstandard Nikulin surfaces.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
Consider the set of rational functions where the number of variables and the degrees of numerator and denominator are fixed. Such functions can be approximated, as usual, by their Taylor polynomials. The coefficients of these polynomials satisfy some natural algebraic constraint and define what we call a Taylor variety. We were surprised to find that some Taylor varieties have dimension smaller than the expected one (defective cases). The classification of defective cases is known only in particular cases and it is open in general. Another interesting phenomenon is that many Taylor varieties have vanishing Hessian. This is joint work with A. Conca, S. Naldi and B. SturmfelsStable 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 BorelWeilBott theorem and is wellunderstood, 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 infinitedimensional Lie algebra. I will show how these spaces globalize to twisted Dmodules 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
Constant cycle curves on K3 surfaces X have been introduced by Huybrechts as curves whose points all define the same class in the Chow group. In this talk we introduce correspondences $Z subseteq Xtimes X$ over $mathbb{C}$ that act on the group $mbox{ccc}(X)$ of cycles generated by constant cycle curves. We construct for any $ngeq 2$ and any very ample line bundle $L$ a locus $Z_n(L)subseteq Xtimes X$ of expected dimension $, which yields a correspondence that acts on the group $mbox{ccc}(X)$, when it has the expected dimension. We provide examples for low $n$.Footnotes on Pfaffian cubic fourfolds with generalizations
The first part of the talk revisits some line geometry of Pfaffian cubic fourfolds in a complex 5dimensional projective space and the related family of K3 surfaces of genus 8. Then, moving from a seminal paper of BeauvilleDonagi, the case is considered of hypersurfaces defined by a (2r+2) x (2r+2) Pfaffian of linear forms, in a projective space of dimension 2r+1. Some natural generalizations are presented, involving the Grassmannian of lines G(1, 2r+1) and its CalabiYau linear sections. Some other possible generalizations higher dimensional cubic hipersurfaces are also considered. Joint work with Michele Bolognesi.
During the conference there will also be short presentations by young participants.
All talks will be hosted by DISIM in the “Alan Turing” Building, Room A1.7 (Via Vetoio SNC, 67100 L'Aquila AQ)
18/07 Tuseday 
19/07 Wednesday 
20/07 Thursday 
21/07 Friday 


9:30  10:30  Flamini  Galati  Tarasca  Faenzi 
10:30  11:00  Coffee break  Coffee break  Coffee break  Coffee break 
11:00  12:00  Verra  Aprodu  Macrì  Bruno 
12:15  13:15  Torelli  Knutsen  Raicu  Ottaviani 
13:15  15:00  Lunch break  Lunch break  Lunch break  Lunch break 
15:00  16:00  Bud  Di Rocco  Farkas  
16:00  16:30  Coffee break  Coffee break  Coffee break  
16:30  17:30  Short presentations  Short presentations  Bolognese  
20:30  Social dinner 
For late registrations please email directly the organizers and fill the form available here.
This conference is funded by