Cycle of seminars organized jointly by E. Angelini, A. Bernardi, L. Chiantini, M. Mella, G. Ottaviani, E. Turatti.

To receive information on this cycle of seminars, contact the organizers by email or subscribe to the mailing list geometria.

Seminars typically take place every three weeks, between Bologna and Firenze .

Due to Covid-19 restrictions, seminars are currently taking place both in presence and online. Write to ettore.teixeiraturatti@unifi.it to get the online access codes.

or in presence, Math department, Università di Bologna.

Title: Decompositions of powers of quadrics

Abstract: The study of the decompositions of the powers of a quadratic form, also called representations, is a very classical problem. Many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, that in the case of the powers of quadrics results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

or in presence, aula Tricerri, DIMAI, Università di Firenze.

Title: Complete intersections on Veronese surfaces

Abstract: In "Commentationes Geometricae" Euler asked when a set of points in the plane is the intersection of two curves, that is, using the modern terminology, when a set of points in the plane is a complete intersection. In the same period, Cramer asked similar questions so that this type of questions is presently known as the Cramer-Euler problem. In this paper, we consider a generalization of the Cramer-Euler problem: characterize the possible complete intersections lying on a Veronese surface, and more generally on a Veronese variety. The main result describes all possible reduced complete intersections on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces.

or in presence, room Bombelli, Math department, Università di Bologna.

Title: Defectivity of Segre-Veronese varieties

Abstract: Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the base points to collapse together. We deduce a general result which we apply to prove a conjecture by Abo and Brambilla: for c ≥3 and d ≥3, the Segre-Veronese embedding of (P^m)x(P^n) in bidegree (c,d) is non-defective.

or in presence, room 201, DIMAI, Università di Firenze.

Title: Lower bounds for algebraic branching programs via intersection theory

Abstract: Algebraic branching programs (ABP) define one of the algebraic versions of the class of problems which can be solved in polynomial time. In this seminar, I will explain how to characterize ABPs via restrictions of a special class of polynomials and how geometric methods allow us to determine lower bounds in this class. In particular, I will focus on recent work with Ghosal, Ikenmeyer and Lysikov, where we use intersection theoretic properties of an associated hypersurface to give lower bounds on the ABP complexity.

or in presence, room Tricerri, DIMAI, Università di Firenze.

Title: Linear systems on blow-ups of projective spaces and their birational geometry

Abstract: I will give an introduction to polynomial interpolation problems in several variables with assigned multiple points and to their geometric formulation. I will give an overview of long standing conjectures as well as of partial results obtained via a thorough study of the base loci of linear systems on blow-ups of projective spaces in points in general position. When the latter are Mori dream spaces, we will see how an action of the Weyl group governs their birational behaviour and compare two types of chamber decompositions of the effective and movable cones of divisors. Based on joint work with C. Brambilla, O. Dumitrescu and L. Santana Sánchez.

or in presence, room Seminario II, Math department, Università di Bologna.

Title: Singular vector tuples of tensors and Kalman varieties

Abstract: Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. I will talk about some families of Kalman varieties, extending previous work of Ottaviani and Shahidi to the partially symmetric context, highlighting the special role of isotropic vectors in the spectral theory of tensors. I will indicate how to describe the totally isotropic Kalman variety as a dual variety and how to obtain a generating function whose coefficients are the degrees of these varieties. This is based on a joint work with Shahidi and Sodomaco.

or, in presence, room Tricerri, first floor, DIMAI, Università di Firenze.

Title: Uniform determinantal representations and spaces of singular matrices

Abstract: The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and many more areas. In this talk I will introduce the notion of "uniform determinantal representation", and derive a lower bound on the size of the matrix, showing a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. I will also relate uniform determinantal representations to vector spaces of singular matrices, in particular compression spaces. The paper I will report on is a joint work with van Doornmalen, Draisma, Hochstenbach, and Plestenjak.

or, in presence, room Pincherle, second floor, Math department, Università di Bologna.

Title: On the strength of homogeneous polynomials

Abstract: The strength of a homogeneous polynomial is the smallest length of an additive decomposition as sum of reducible forms. It is called slice rank if we additionally require that the reducible forms have a linear factor. Geometrically, the slice rank corresponds to the smallest codimension of a linear space contained in the hypersurface defined by the form. Due to this relation, it is well-known and easy to compute the value of the general slice rank and also to show that the set of forms with bounded slice rank is Zariski-closed. In this talk, I will present the following results from recent joint works with A. Bik, E. Ballico and E. Ventura: (1) the set of forms with bounded strength is not always Zariski-closed: this is an asymptotic result in the number of variables proved by using the theory of polynomial functors; (2) for general forms, strength and slice rank are equal: this is proved by showing that the largest component of the secant variety of the variety of reducible forms is the secant variety of the variety of forms with a linear factor.

or, in presence, room Seminario, Math department, Università di Bologna.

Title: Decomposition algorithms for tensors and polynomials (joint work with Antonio Laface and Rick Rischter)

Abstract: We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present new methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic.