Previous years: 2019/2020
Previous years: 2019/2020
Date: 26/2/2020 Time: 10.00 Room: 713
Lecture 1. Ideal Theory of 2-dimensional normal local rings via its resolution of singularitiy, 1.
I will explain some fundamental facts about resolution of singularity and will give some examples. In particular, I will explain how to construct resolution of 2 dimensional normal graded rings.
Date: 26/2/2020 Time: 15.00 Room: 713
Title: Rational curves on O'Grady's tenfolds.
Abstract: Thanks to some recent works due to F. Charles, G. Mongardi and G. Pacienza, we know that, in order to show the existence of rational curves on irreducible holomorphic symplectic (IHS) varietis of a fixed deformation type, it is enough to do it for special points of their moduli space, thanks to the study of the monodromy group of the variety.In this talk I will start introducing the problem of finding rational curves on IHS varieties and presenting some motivation behind it; I will describe the state of the art of the problem and I will present my contribution to the OG10-case, giving an example of ample uniruled divisors on OG10-varieties.
Date: 26/2/2020 Time: 14.00 Room: 713
Title: The classification of hyperelliptic groups in dimension four
Abstract: A hyperelliptic manifold of dimension n is the quotient X=T/G of an n-dimensional complex torus T by a finite, non-trivial group G of biholomorphisms of T, which acts freely on T and contains no translations. While it follows from Hurwitz' formula that there is no hyperelliptic manifold of dimension 1, hyperelliptic surfaces have been classified Enriques-Severi and Bagnera-de Franchis. In the 3-dimensional case, Uchida-Yoshihara gave a list of the abstract groups G, which potentially occur as a group of a hyperelliptic threefold. It was later shown by Lange and Catanese-Demleitner that all of these groups indeed occur. In this talk, I will explain how one finds the list of possible groups in the 4-dimensional case and - if time allows - how to show that all of them indeed occur.
Date: 19/2/2020 Time: 14.00 Room: 713
Title: CHORDALITY AND SIMON'S CONJECTURE
Abstract: We recall a notion of chordality for simplicial complexes introduced by Bigdeli et al, extending the usual chordality of graphs. Despite the fact that this notion does not allow to extend Fröberg's theorem characterizing monomial ideals with a 2-linear resolution, they stated a conjecture ensuring the chordality of complexes associated to ideals with linear quotients. This conjecture would imply the long-standing open Simon's conjecture on extendable shellability for the skeleta of a simplex.
In this talk we disprove the stronger conjecture, providing an infinite family of ideals with linear quotients whose corresponding complexes are not chordal. This is a joint work with Bruno Benedetti.
Date: 12/2/2020 Time: 14.00 Room: 713
Title: Multiplicity VS Colength in local rings.
Abstract: For an m-primary ideal in a local ring (R, m), we will compare the multiplicity e(I) of I and the colength l(R/I). This talk is based on some of join works with N.T. Cuong, C. Huneke, L. Ma and I. Smirnov.
Date: 18/12/2019 Time: 14.00 Room: 713
Title: Realization of semigroup of modules
Abstract: Given a Noetherian commutative local ring R, the set of finitely generated R-modules equipped with direct sum forms a monoid. Similarly, given a finitely generated R-module M one can consider the submonoid +(M) of isomorphism classes of direct summands of direct sums of copies of M. This is a finitely generated Krull monoid, that is a positive normal affine semigroup. It is natural to ask which Krull monoids can be realized as +(M) for a suitable local ring R and an R-module M. In this talk, we will discuss this problem. In particular, by using an essentially geometric construction with symmetric powers of curves, we will prove that given a Krull monoid L and an integer d>2 then there exist a d-dimensional local UFD ring R and a torsion-free R-module M such that L=+(M). This generalizes a previous result by R. Wiegand who proved it for d=1,2.
Date: 18/12/2019 Time: 13.00 Room: 713
Title: Du Bois singularities, positivity and Frobenius
Abstract: In this talk we will study the Hilbert function of a standard graded algebra R with enough depth and such that X=Proj(R) has good enough singularities. Precisely, R is required to satisfy Serre's condition (S_r), in characteristic 0 X will be Du Bois, and in positive characteristic globally F-split. As a consequence, when X is smooth and the characteristic is 0, the first r coefficients of the Hilbert series of R are nonnegative and bound from below the degree of X (w.r.t. the embedding given by R). These results are part of a joint work with Dao and Ma. We will end discussing a conjecture proposed by Constantinescu, De Negri and myself regarding the possible initial ideals of a smooth projective variety, which is related to an arithmetic problem concerning the behavior of the Frobenius map.
Date: 11/12/2019 Time: 14.00 Room: 713
Title: On the regularity of binomial edge ideals.
Abstract: In this talk we compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number. Moreover we give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs, and we present a linear time algorithm to compute the Castelnuovo-Mumford regularity of the binomial edge ideal of a block graph. At the end we show that the family of block graphs contains the licci binomial edge ideals.
Bibliography:
1) J.Herzog, G.Rinaldo, On the extremal Betti numbers of binomial edge ideal of block graphs, Electron. J. Combin. Vol. 25(1), 2018,pp 1-10;
2) C. Mascia, G. Rinaldo, Krull dimension and regularity of binomial edge ideal of block graphs, To appear in J. Algebra Appl.;
3) V. Ene, G. Rinaldo, N. Terai, Licci binomial edge ideals, arXiv:1910.03612, 2019.
Date: 4/12/2019 Time: 14.00 Room: 713
Title: The range of all regularities for polynomial ideals with a given Hilbert function
Abstract: Let m and M be, respectively, the minimal and the maximal possible regularity for a (homogeneous) polynomial ideal I in the polynomial ring P in n variables over an infinite field K, such that the finitely generated graded K-algebra P/I has a given Hilbert function f. I show that, for every integer h between m and M, there is a strongly stable ideal J with regularity h and with f as Hilbert function of P/J. This result is achieved by means of arguments that prove the analogous statement for closed projective subschemes.
Date: 20/11/2019 Time: 15.00 Room: 713
Title: Saturation of Subalgebras, SAGBI bases, and U-invariants
Abstract: The purpose of my talk is to present a new paper (joint with A.M. Bigatti) concerning computations of Subalgebras of the polynomial ring. The motivation comes from classical results about invariants of the group of unipotent matrices. A recent work of Kraft and Procesi shows that this computation can be done in a way which inspired us to introduce and study saturations of Subalgebras. In the talk the merits and the limitations of our method will be shown. A particular emphasis will be given to the use of SAGBI bases.
Date: 20/11/2019 Time: 14.00 Room: 713
Title: Kobayashi-Hitchin correspondence for twisted bundles
Abstract: The Kobayashi-Hitchin correspondence for holomorphic vector bundles is an important result in complex geometry relating the stability of a vector bundle and the existence of particular metrics on it, called Hermite-Einstein metrics. In a recent paper we generalize this to twisted vector bundles.
Date: 12/11/2019 Time: 17.00 Room: 714
Title: Binomial edge ideals.
Abstract: Binomial edge ideals are a generalization of determinantal ideals and ideals generated by adjacent $2$-minors in a $2\times n$-generic matrix. In simple terms, binomial edge ideals are generated by an arbitrary collection of $2$-minors of a $2\times n$ matrix whose entries are indeterminates. It is then natural to associate a graph $G$ on the vertex set $\{1,\ldots,n\}$ whose edges are the pairs $\{i,j\}$ whenever the $(i,j)$-th minor of the matrix is a generator of the ideal. This explains the naming of this kind of ideals. In this talk we give a necessary and sufficient condition in terms of the combinatorics of the graph for the binomial edge ideal of a graph $G$ to be Gorenstein.
Date: 30/10/2019 Time: 14.00 Room: 713
Title: A new algorithm for solving multi-polynomial equations
Abstract: We present a novel method for solving any square system of multi-polynomial equations. This method generalizes the idea of solving systems of linear equations by multiplying by the inverse of the coefficient matrix. Due in part to the fact that this algorithm requires only solutions to linear systems and polynomial root finding, it is asymptotically more efficient than existing methods. It is an exact method save for the approximation of roots of a univariate polynomial.
Date: 16/10/2019 Time: 14.00 Room: 706
Title: The geometry of matroids
Abstract: Matroid theory is a combinatorial theory of independence which has its origins in linear algebra and graph theory, and turns out to have deep connections with many other fields. With time, the geometric roots of the field have grown much deeper, bearing many new fruits. The geometric approach to matroid theory has recently led to the development of fascinating mathematics at the intersection of combinatorics, algebra, and geometry, and to the solution of long-standing questions. This talk will provide an introduction to matroid theory, with an emphasis on the geometric viewpoint, and an eye towards some recent successes of the theory. It will include joint work with Graham Denham and June Huh.
The talk will be accessible to students and will assume no previous knowledge of matroids.