Tsutomu Nakamura (Università degli Studi di Verona): Pure derived categories and weak balanced big Cohen-Macaulay modules. In this talk, we establish a foundation of an infinite version of Cohen-Macaulay representation theory over a Cohen-Macaulay local ring. One of our motivations is to find an important role of pure-injective modules over this ring, like the case of representation theory of artinian algebras. Such a research direction was originally considered by Gena Puninski (2018), and our study is mainly devoted to develop his work.
Laura Pertusi (Università degli Studi di Milano): Stability conditions on Gushel-Mukai fourfolds. An ordinary Gushel-Mukai fourfold X is a smooth quadric section of a linear section of the Grassmannian G(2,5). Kuznetsov and Perry proved that the bounded derived category of X admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Xiaolei Zhao, where we construct Bridgeland stability conditions on the K3 subcategory of X. Then I will explain some applications concerning the existence of a homological associated K3 surface and hyperkaehler geometry.
Juan Francisco Pons Llopis (Politecnico di Torino): Instanton bundles on the complete flag variety F(0,1,2). Instanton bundles on P^3 were defined as the algebraic counterpart to SU(2)-connections with self-dual curvature on the real sphere S^4. Their study prompted the development of many techniques that have become central in algebraic geometry (monads, loci of jumping lines,study of the geometric properties of moduli spaces,...). Later on, Instanton bundles have been defined and studied in other Fano threefolds of Picard number one by Faenzi, Kuznetsov and Sanna . Recently, Casnati, Coskun, Genc and Malaspina have proposed a general definition of instanton bundle on Fano threefolds of arbitrary Picard number. Motivated by the fact that the complete flag variety F(0,1,2) is the only case, besides P^3, of projective twistor space associated to a real 4-manifold, we pursue early work by Donaldson and Buchdahl to study instanton bundles on this Fano threefold (which has Picard number two), underlining the similarities and differences with the classical case. Joint work with F. Malaspina and S. Marchesi.
Enrico Schlesinger (Politecnico di Milano): The maximum genus problem for locally Cohen-Macaulay curves in projective 3-space. The classical maximum genus problem asks what is the maximum genus G(d,s) of a smooth curve of degree d in P^3 that is not contained in a surface of degree < s. In this talk I will review what is known about the analogous problem when the class of smooth curves is replaced by the class of locally Cohen-Macaulay curves. This is joint work with V. Beorchia and P. Lella.
Lisa Seccia (Università degli Studi di Genova): Knutson ideals in characteristic zero and determinantal ideals of Hankel matrices. Motivated by a work of Knutson, in a recent paper Conca and Varbaro defined a new class of ideals (called “Knutson ideals”) starting from a polynomial f with squarefree leading term. In this talk I will describe the properties of these ideals and I will explain how the main property of this class is preserved when one extends the definition of Knutson ideal to polynomial ring over fields of characteristic zero. Then I will show that a certain class of determinantal ideals are Knutson ideals for a suitable choice of the polynomial f.
Fabio Tanturri (Università degli Studi di Torino): On Abelian surfaces, Coble cubics, and orbital degeneracy loci. Starting from a general 3-form omega in 9 variables, it is possible to construct an Abelian surface A in P^8 which is the singular locus of a cubic hypersurface, known as the Coble cubic. In this talk I will show how, from the same omega, we can realize several other varieties which can be interpreted in terms of A and the Coble cubic; among them, we can find the generalised Kummer fourfold Kum^2(A) and the Hilbert scheme Hilb^2(A). This perspective allows us to describe the group law of A, in a strikingly similar way to how the group structure of a plane cubic can be defined in terms of its intersection with lines. Joint work with V. Benedetti and L. Manivel.
Francesco Veneziano (Università degli Studi di Genova): An effective criterion for periodicity of l-adic continued fractions. The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. For example, in the l-adic case, rational numbers may have a periodic non-terminating expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction expansion is periodic was still open. In our paper (joint work with Laura Capuano and Umberto Zannier) we investigate the l-adic continued fraction expansions of rationals and quadratic irrationals using the definition introduced by Ruban. We give general explicit criteria to assess the periodicity of the expansion in both the rational and the quadratic case.
