Titles and Abstracts
COURSES
Università di Firenze
giorgio.ottaviani [at] unifi.it
The Hessian Map
The Hessian map is the rational map associating to each homogeneous polynomial in r+1 variables its Hessian, which is the determinant of its Hessian matrix. Gordan and Noether proved in 1876 that the Hessian map is not defined exactly on cones for r<= 3 and provided examples of polynomials in more variables with vanishing Hessian which are not cones. It has been understood, thanks to classical results by Franchetta, and modern updates by Russo and others, that the indeterminacy locus of the Hessian map consists of exactly two irreducible components for r=4, one of them is given by cones. After this striking result, the situation becomes immediately more intricate for r>= 5. We discuss a couple of different approaches to these results, including its connections with the strong and weak Lefschetz properties for Artinian Gorenstein rings.
The Hessian map is equivariant for the action of SL(r+1), indeed the Hessian is the main classical example of a covariant, we briefly discuss the invariant-theoretical point of view. An interesting open problem, where we expose some results jointly with Ciliberto, is if the Hessian map is birational on its image. If time allows, we also touch a surprising connection with persistent tensors, introduced recently by Gharahi and Lysikov in the setting of quantum information.
Università degli Studi di Napoli Federico II
francesco.polizzi [at] unina.it
Surface braid groups, Galois covers, and double Kodaira fibrations
We show how to use braid group techniques in order to construct new examples of double Kodaira fibrations as branched Galois covers of the product Σ_b × Σ_b, where Σ_b is a closed Riemann surface of genus b. Some of these results are joint work with A. Causin and P. Sabatino.
CONTRIBUTED TALKS
Università degli Studi di Trieste
beorchia [at] units.it
Jacobian schemes and hessians of highly singular plane curves
We shall illustrate the classification of planar conic-line arrangements with maximal and almost maximal global Tjurina number. Such curves turn out to admit a linear Jacobian syzygy, which allows to give a geometric description of the polar map and of the Hessian curve as a ramification divisor. Such an approach can be performed also in other cases. The results are in collaboration with R. M. Mirò Roig.
Politecnico di Torino
stefano.canino [at] polito.it
How many points can stand on the tip of a needle? Superfat points and associated tensors.
In this talk we introduce a new class of 0-dimensional schemes: the m-symmetric schemes. We start by discussing their first properties and their strict relationship with fat points, of which they are a generalization. After that, we restrict to 2-symmetric schemes in P^2, which we call 2-squares, and we show how to define some new varieties via Veronese and Segre-Veronese embeddings of 2-squares, and we also show how these new varieties allow to obtain some information about secant and osculating varieties of Veronese and Segre-Veronese varieties. Finally, we study the defectiveness of these varieties by using the apolarity theory. Time permitting, we will also see something about the good postulation of a generic union of 2-squares via the Horace method.
Università degli Studi di Trento
federico.fallucca [at] unitn.it
Explicit description of coverings of the projective line from spherical systems of generators
A G-covering of P^1 is a Riemann surface C together with an action of a finite group G such that C/G is P^1.
A classic approach to studying coverings of P^1 is through the so-called spherical systems of generators. In this talk, we briefly revisit the connection between these objects. In particular, we show that, according to the Riemann Existence Theorem (RET), a spherical system of generators defines a covering of P^1, although providing a clear description of it proves to be very challenging due to the non-constructiveness of RET.
Therefore, we will explore alternative techniques to explicitly describe coverings of P^1 from spherical systems of generators, using equations in suitable weighted projective spaces.
If time permits, I will also give an example of the utility of describing such coverings through equations, which has led to new results into an open question regarding the degree of the canonical map of surfaces of general type.
Universidade de Évora
pmm [at] uevora.pt
Reducible families of Artinian Gorenstein algebras
We study local Artinian Gorenstein (AG) algebras and consider the set of Jordan types of elements of the maximal ideal, i.e. the partition giving the Jordan blocks of the respective multiplication map.
In a joint work with Tony Iarrobino, we construct examples of families Gor(T) of local AG algebras with given Hilbert function T, and use obstructions that the symmetric decomposition of the associated graded algebra of an AG algebra A imposes on the Jordan type of A to study their irreducible components.
Partially supported by CIMA – Centro de Investigação em Matemática e Aplicações, Universidade de Évora, project UIDB/04674/2020 (Fundação para a Ciência e Tecnologia), https://doi.org/10.54499/UIDB/04674/2020.
CAMGSD, Instituto Superior Técnico, Universidade de Lisboa
mmendeslopes [at] tecnico.ulisboa.pt
Fibrations with few singular fibres
Let S be a smooth complex projective surface and B a smooth projective curve. A fibration of S over B is a surjective morphism of S onto B such that the general fibre is a smooth connected curve. In this talk some properties of surface fibrations with general fibre of genus greater than 2 will be discussed, focusing in particular on the question of the minimal number of singular fibres for semi-stable fibrations, i.e, fibrations such that no (-1)-curve is contained in a fibre and such that the only possible singularities of a fibre are nodes. In particular some recent joint work with Alexis Zamora will be discussed.
Università degli Studi di Trento
roberto.pignatelli [at] unitn.it
Fibrations in (1,2)-surfaces
The (1,2)-surfaces, minimal surfaces of general type with volume 1 and geometric genus 2, played a special role in the theory of the surfaces of general type. They have recently shown to have a special role also in the theory of threefolds of general type, namely in the proof of the Noether inequality for threefolds due to J. Chen, M. Chen and C. Jiang. In fact, that proof shows that fibrations onto a curve with general fibre a (1,2)-surface play, in the geography of threefolds, a role similar to the one played by fibrations in genus 2 curves in the theory of surfaces.
In this talk I will introduce the notion of "simple" fibration in (1,2)-surfaces and show how through this notion we obtained a complete classification of canonical threefolds on the Noether line (i.e. realizing the equality in the Noether inequality) and we described their moduli space, remarking analogies and differences with the analogous case in dimension 2. If time permits I will also discuss some applications and some conjectures on the behaviour of the moduli space of canonical threefolds "near" the Noether line.
This is a joint work with S. Coughlan, Y. Hu and T. Zhang.