Algebra & Geometry Seminar - Previous years: 2021-22
Algebra & Geometry Seminar - Previous years: 2021-22
Date: 13/07/2022 Time: 14:00 Room: 713
Title: The fiber-full scheme and applications
Abstract: We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We also present some applications that can be derived from the existence of the fiber-full scheme. This talk is based on joint works with Ritvik Ramkumar.
Date: 28/07/2022 Time: 15:15 Room: 704
Title: Formulas for symbolic powers of ideals
Abstract: In this talk we provide a couple of formulas to compute symbolic powers of unmixed, generically complete intersection ideals in Cohen-Macaulay rings. For instance, we give a multiplicity-based characterization for the m-th symbolic power I^(m) of an ideal I, a formula to obtain I^(m) as the saturation of I^m with respect to an explicit ideal only depending on I (and not m), and effective bounds for the exponent achieving the saturation. We plan to discuss the connection with a conjecture by Eisenbud and Mazur on ann(I^(m)/I^m), which we prove in a generalized form, and a couple of applications of the formulas.
Date: 25/05/2022 Time: 16:15 Room: 509
Title: Some divisors in the moduli space of Debarre-Voisin varieties
Abstract: Projective cubic fourfolds provide a remarkable connection between different areas in algebraic geometry. The rationality question of such varieties has been conjectured to be controlled by properties coming from their Hodge structure and derived category. Debarre-Voisin (DV) hypersurfaces share some fundamental properties with cubic fourfolds, among which the fact that one can naturally associate to these Fano varieties some hyper-Kahler fourfolds. As in the case of cubics, the hyper-Kahler fourfold turns out to encode the rational Hodge structure of DV hypersurfaces. In this talk I will focus the attention on some divisors of the moduli space of DV varieties. By studying the geometry of such divisors and the associated hyper-Kahler, I will show how to deduce the integral Hodge conjecture for DV hypersurfaces. This is a joint work with Jieao Song.
Date: 25/05/2022 Time: 17:30 Room: 509
Title: On Rees algebras of determinantal ideals of general matrices
Abstract: Let M be a matrix of linear forms and let I be its ideal of maximal minors. Assume that M is general enough, e.g. that I has the expected codimension. The study of the Rees algebra R(I) is an important topic in commutative algebra, with open questions including: What are the degrees of the defining relations of R(I)? If the powers of I have linear resolutions, is R(I) a Koszul algebra? What are the singularities of R(I)?
In this talk we will give a fairly complete picture for the case of matrices M with two rows, which includes objects of interest such as varieties of minimal degrees, 2-regular algebraic sets, and small schemes. Our analysis of their Rees algebras builds heavily on previous contributions (mostly from Genoa) and involves squarefree Groebner degenerations and a stratification of the Hilbert scheme in terms of Kronecker-Wierestrass normal forms.
This is a preliminary report on joint work with Ritvik Ramkumar.
Date: 18/05/2022 Time: 16:15 Room: 509
Title: K-theoretic sheaf counting
Abstract: Donaldson-Thomas theory is an enumerative theory counting sheaves on smooth 3-folds. The “classical” setup involves sheaves of rank 1 (ideal sheaves of subschemes of codimension at least 2). We will give an overview on the subject with special focus on the known results in the higher rank case. In particular, we will see how to use Quot schemes to compute higher rank DT invariants in several flavours: enumerative, motivic, cohomological, K-theoretic.
Date: 11/05/2022 Time: 16:15 Room: 509
Title: Supersolvable posets and aspherical Abelian arrangements
Abstract: A long-standing open problem in the theory of arrangements of hypersurfaces is to classify which arrangement complements are aspherical. This is unsolved even for arrangements of linear hyperplanes, where asphericity is known mainly for reflection arrangements (Deligne 1972; Bessis 2011; Paolini-Salvetti 2021) and for "fiber-type" arrangements (Falk-Randell 1985).
In this talk I will review the history and the basics of this problem. I will then present some recent advances in the case of arrangements in connected abelian Lie groups (including complex tori and products of elliptic curves). I will state a combinatorial condition on the partially ordered set of intersections of such an arrangement that implies an inductive fibration of the arrangement's complement. This gives a combinatorially determined class of aspherical, toric and elliptic arrangements.
Time permitting I will detail some of the intriguing properties of this class, including factorizations of the Poincaré polynomial and its relationship with the LCS of the fundamental group. (This is joint work with Christin Bibby.)
Date: 27/04/2022 Time: 16:15 Room: 509
Title: Finitisic dimensions over commutative DG-rings
Abstract: Over a ring A, the big projective finitistic dimension FPD(A) is defined to be the supremum of projective dimensions of all A-modules of finite projective dimension. Classical results of Bass and Raynaud-Gruson showed it is always equal to the Krull dimension of A. In this talk I will discuss the corresponding problem for commutative DG-algebras and its solution. This is based on a joint work with Isaac Bird, Prashanth Sridhar and Jordan Williamson.
Date: 13/04/2022 Time: 16:15 Room: 509
Title: Ideals in a local ring under small perturbations
Abstract: Let I be an ideal of a Noetherian local ring R. We study how properties of I change for small perturbations, that is, for ideals J that are the same as I modulo a large power of the maximal ideal. In particular, assuming that J has the same Hilbert function as I, we show that the Betti numbers of J coincide with those of I. We also compare the local cohomology modules of R/J with those of R/I.
Date: 30/03/2022 Time: 16:15 Room: 509
Title: A crash course on q-matroids
Abstract: Matroid theory originates from a seminal paper by Whitney (1935) as a connection among linear algebra, combinatorics and finite geometry, which generalizes the basic linear algebra notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, through different objects, each of which with its personal axiom system. These equivalent axiom systems are called cryptomorphisms. The idea of "q-analogue" in combinatorics, essentially means generalizing combinatorial objects by taking, instead of finite sets, finitely generated vector spaces over GF(q). In particular, we will deal with the q-analogue of a matroid, a concept that has been anticipated by Crapo and then rediscovered by Jurrius-Pellikaan. In this talk, we will take a tour of the world of q-matroids, keeping an eye to what's different with respect to the classical case of matroids.
This is based on joint works with E. Byrne, R. Jurrius, S. Ionica and E. Saçikara.
Date: 23/03/2022 Time: 16:15 Room: 509
Title: Cohen-Macaulay binomial edge ideals
Abstract: In the last decades the connections between Commutative Algebra and Combinatorics have been extensively explored. In this perspective, many authors have considered classes of ideals in a polynomial ring that can be naturally associated with combinatorial objects, and have studied their algebraic invariants exploiting this combinatorial connection. In this talk we are interested in the so-called binomial edge ideals, which are ideals generated by binomials corresponding to the edges of a finite simple graph. They can be viewed as a generalization of the ideal of the maximal minors of a generic matrix with two rows, where only some minors are considered. After reviewing some results about these ideals, we will present a conjecture for a combinatorial characterization of Cohen-Macaulay binomial edge ideals. We will identify sufficient and necessary conditions for Cohen-Macaulayness, both of which can be read off from the underlying graph. Moreover, we will show that these conditions are indeed equivalent for large classes of graphs settling the conjecture in these cases.
This is joint work with Davide Bolognini and Antonio Macchia.
Date: 16/03/2022 Time: 16:15 Room: 509
Title: K-stability of Fano weighted hypersurfaces
Abstract: K-stability is a fundamental notion in algebraic geometry, needed to construct moduli spaces of Fano varieties. In this talk I will present recent results on the K-stability of Fano weighted hypersurfaces. This is based on a joint work with Taro Sano.
Date: 09/03/2022 Time: 16:15 Room: 509
Title: Fano 4-folds with a small contraction
Abstract: Let X be a smooth, complex Fano 4-fold. We will talk about the following result: if X has a small elementary contraction, then the Picard number of X is at most 12. This result is based on a careful study of the geometry of X, from the point of view of birational geometry and of families of rational curves. I will give an idea of the context and of the main tools: the properties of the faces of the effective cone; the classification of fixed prime divisors; the detailed study of rational contractions of fiber type; the study of families of lines and the construction of divisors covered by lines.
Date: 23/02/2022 Time: 16:15 Room: 509
Title: Topological Data Analysis: From Scratch To Algebra
Abstract: Topological Data Analysis (in short, TDA) is a recent discipline between mathematics and computer science that faces the never-ending challenge of extracting descriptive and compact information from huge datasets often characterized by noise and redundancy.
Specifically, TDA tackles this problem by adopting geometric (or, more properly, topological) tools capable of revealing features of a dataset such as its connectivities or the presence of loops and holes.
In this talk, I will introduce through examples the basic tools of TDA and we will discuss together their connections with algebra trying to understand if and how these two worlds can benefit each other.
Date: 09/02/2022 Time: 16:15 Room: 704
Title: Binomial edge ideals of weakly-closed graphs
Abstract: In a recent work Herzog et al. characterize closed graphs as the graphs whose binomial edge ideals have quadratic Groebner bases (with respect to a diagonal term order). In this talk, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs).
Building on known results about Knutson ideals of generic matrices, we characterize weakly-closed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly-closed graphs in positive characteristic.
Date: 26/01/2022 Time: 16:15 Room: 704
Title: Singularities of irreducible theta divisors
Abstract: The question of studying the singularities of theta divisors has attracted a lot of interest both old and new. Its importance is amplified when connecting it to the problem of describing interesting loci on the moduli space of abelian varieties. The main goal of this talk is to introduce a folklore conjecture about the behaviour of the multiplicity at a point for irreducible theta divisors and explain some new results in this direction.
Date: 19/01/2022 Time: 16:15 Room: 704
Title: Secant non-defectivity via collisions of fat points
Abstract: A classical problem in algebraic geometry is the classification of algebraic varieties whose secant varieties are defective, i.e., they have dimension strictly smaller than the expected. For polarized varieties (X,L), the dimension of the r-secant variety corresponds to the dimension of the linear system of divisors in L passing through r 2-fat points (i.e., having r singularities) in general position. By semi-continuity, non-defectivity can be proved via degeneration techniques in which such fat points are chosen to be in special position on X. In the late ‘90s, Evain studied the case in which fat points are degenerated to collide together: this gives rise to a 0-dimensional scheme with a special non-reduced structure. After a general introduction, I will present a recent work with Francesco Galuppi where we use collisions to provide a criterion for secant non-defectivity. We applied this criterion to complete a work started by H. Abo and M.C. Brambilla to prove that all Segre-Veronese embeddings of Pm x Pn of bi-degree (c,d) are non-defective whenever c and d are both at least 3.
Date: 15/12/2021 Time: 16:15 Room: 509
Title: Famiglie di rivestimenti di Galois della retta
Abstract: La famiglia di tutti i rivestimenti di Galois della retta proiettiva viene normalmente costruita sfruttando lo spazio di Teichmüller. Oltre ad essere poco elementare, questo metodo rende meno chiaro che la famiglia è algebrica e come essa dipende dai dati combinatorici. In questo seminario spiegherò una costruzione topologica molto elementare di queste famiglie, che rende molto facile dimostrare l'algebricità delle famiglie e controllare la loro dipendenza dai dati combinatorici.
È un lavoro in collaborazione con Carolina Tamborini.
Date: 24/11/2021 Time: 16:15 Room: 509
Title: Varietà di Fano di tipo K3 e loro proprietà
Abstract: Le varietà di Fano di tipo K3 sono una speciale classe di Fano studiate per il collegamento con la geometria hyperkaehler, le loro interessanti proprietà di razionalità, e molto altro ancora. In questo talk ripercorreremo alcuni risultati recenti, ottenuti in una serie di lavori in collaborazione con Bernardara, Manivel, Mongardi e Tanturri, focalizzati alla costruzione esplicita di esempi e allo studio delle loro proprietà Hodge-teoretiche.
Date: 17/11/2021 Time: 18:00 Room: Online
Title: Bounds on the number of generators of prime ideals
Abstract: Let S be a polynomial ring over any field k, and let P inside S be a non-degenerate homogeneous prime ideal of height h. When k is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in P which only depends on h. We significantly extend this result by proving that the number of minimal generators of P in any degree j can be bounded above by an explicit function that only depends on j and h. In addition to providing a bound for generators in any degree j, not just for quadrics, our techniques allow us to drop the assumption that k is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal. [This is a joint work with Alessandro De Stefani]
Date: 17/11/2021 Time: 16:15 Room: 714
Warning: only the first 13 participants to confirm their presence on this doodle will be guaranteed a seat!
Title: Mori Dream Regions and C*-actions
Abstract: In this talk we establish a correspondence between Mori Dream regions, that is some cones of divisors for which the multisection ring is finitely generated, and C*-actions on polarized pairs (X,L), where X is a normal projective variety and L is an ample line bundle on X. We will begin by giving an introduction to the theory of C*-actions; we will then sketch the construction behind this correspondence, discussing the ideas, some questions and the results obtained so far. This is a joint work in progress with E. Romano (University of Genova), L. Solà Conde (University of Trento) and S. Urbinati (University of Udine).
Date: 10/11/2021 Time: 16:15 Room: 509
Title: N-fiber-full modules
Abstract: "N-fiber-full modules" is a generalization of the "fiber-full modules" introduced by M. Varbaro. In this talk we present some properties of N-fiber-full modules. In particular, we have that being N-fiber-full is related to the flatness of Ext_A^i(M, N). In the end we show some applications of this result.
Date: 27/10/2021 Time: 16:00 Room: 509
Title: Toric ideals arising from probability trees
Abstract: A staged tree model is a statistical model which can be represented by a tree. The model can be described algebraically as the variety of a homogeneous ideal. In this talk we will discuss the connections between the combinatorics of the model and its associated ideal. In particular we want to understand if the ideal is toric, possibly after a linear change of coordinates. The talk is based on a joint work with Christiane Görgen and Aida Maraj.
Date: 20/10/2021 Time: 16:00 Room: 714
Warning: only 13 participants are allowed. Fill in this doodle to reserve a seat!
Title: On the existence of Ulrich bundles on smooth surfaces.
Abstract: In my talk I will discuss the existence of rank r indecomposable Ulrich bundles on a general surface S in P^3 of degree d. More concretely, I will address the problem of characterizing the set {(r,d) | exists a rank r Ulrich bundle on a general surface S of degree d in P^3}. This problem is related to the problem of determining whether an homogeneous form F in n+1 variables of degree d (or one of its powers) can be written as the determinant of a matrix with linear entries.
In my talk, I will summarize what it is known so far about this problem.
Date: 13/10/2021 Time: 16:00 Room: 509
Title: Sumsets and projective curves.
Abstract: We present a relationship between additive combinatorics and the geometry of projective curves. We associated to a finite set of non-negative integers A a monomial projective curve C_A such that the Hilbert function of C_A and the cardinalities of sA agree. The singularities of C_A determines the asymptotic behavior of the cardinalities of sA.