Schedule
Full schedule
Thursday 22nd
(15.00 - 16.00) . J. Herzog
(16.00 - 16.30) . coffee break
(16.30 - 17.30) . L. Brustenga
(17.30 - 18.30) . G. Favacchio
Friday 23rd
(09.00 - 10.00) . C. Bocci
(10.00 - 11.00) . Ł. Farnik & F. Galuppi
(11.00 - 11.30) . coffee break
(11.30 - 12.30) . A. Oneto
(12.30 - 15.00) . lunch break
(15.00 - 16.00) . G. Zito
(16.00 - 16.30) . coffee break
(16.30 - 17.30) . G. Rinaldo
Saturday 24th
(09.00 - 10.00) . Y. Cid Ruiz
(10.00 - 11.00) . N. Nemati
(11.00 - 11.30) . coffee break
(11.30 - 12.30) . L. Guerrieri
Titles and Abstracts
Thursday 22nd
15.00 - 16.00
Freiman’s theorem and the number of generators of powers of monomial ideals
Jürgen Herzog (U. Duisburg-Essen).
The number of generators of the square andthe higher powers of a monomial ideal can be surprisingly small, even the ideal itself may have any high number of generators. For monomial ideals in 2 variables, a sharp lower bound for this number is given in a joint paper with Eliahou and Mohammadi Saem. Assuming however that the monomial ideal is equigenerated such a lower bound is substantially bigger. This is a consequence of a celebrated theorem by Freiman from additive number theory. This and other consequences of Freiman’s theorem for the number of generators of the powers of monomial ideals will be presented in this lecture. The results are joint work with Takayuki Hibi and Guangjun Zhu.
16.00 - 16.30
coffee break
16.30 - 17.30
Waring rank for forms: binary binomials and some enumerative geometry
Laura Brustenga (U. Autònoma de Barcelona),
joint project with S. Masuti (U. di Genova).
In this talk we will present our two main results on the Waring rank of forms in S = C[x, y]. Firstly, for every homogeneous binomial xrys(axα +byα) ∈ Sr+s+α, we give an explicit formula for its Waring rank which surprisingly just depends on r, s and α and not on a and b. Secondly, for all d ≥ 4, we show that there are exactly d−1 forms in Sd with Waring rank 2 which are 2 multiples of xy(x − y).
17.30 - 18.30
In the shadows of a hypergraph: looking for associated primes of powers of squarefree monomial ideals
Giuseppe Favacchio (U. di Catania),
joint project with E. Bela (U. of Notre-Dame) and N. Tranhihieu (NUI Galway).
We study the associated primes of powers of squarefree monomial ideals. Hypergraphs and squarefree monomial ideals are strongly connected. The cover ideal J(H) is intersection of the primes corresponding to the edges of a hypergraph H. We define the shadow of H, that is a set of smaller hypergraphs related to H. We describe how the shadows of H preserve informations about the associated primes of the powers of J(H). These informations can be used to produce a class of examples of squarefree monomial ideals which fail the persistence property (the set of the associated primes loose elements from a power to the next).
Friday 23rd
9.00 - 10.00
Hadamard products of varieties and beyond
Cristiano Bocci (U. di Siena).
In this talk I will introduce the concept of Hadamard product of varieties, a natural generalization of Hadamard's product of matrices.
The results, in recent works in collaboration, will be introduced from the point of view of Algebraic Geometry, Commutative Algebra and Tropical Geometry.
10.00 - 11.00
On the unique unexpected quartic in the projective plane
Łucja Farnik (Polish Academy of Sciences) and Francesco Galuppi (U. di Ferrara),
joint project with L. Sodomaco (U. di Firenze) and B. Trok (U. Kentucky).
Unexpected curves in the projective plane arise if for a general point P the number of conditions imposed by jP on the linear system of curves of degree j+1 containing a reduced finite set of points is smaller than expected.
Recently, an example of a nine-point subscheme of P2 which admits an unexpected quartic was given by Cook II, Harbourne, Migliore and Nagel in [CHMN]. We discuss it in detail, and study which subschemes in the projective plane over the complex numbers admit an unexpected curve of degree 4. We show that the subscheme of nine points from [CHMN] gives, up to an isomorphism, the only example of an unexpected quartic.
Reference:
[CHMN] D. Cook II, B. Harbourne, J. Migliore, U. Nagel, Line arrangements and configurations of points with an unusual geometric property, preprint arXiv:1602.02300v2 [math.AG].
11.00 - 11.30
coffee break
11.30 - 12.30
Unexpected curves and line arrangements
Alessandro Oneto (U. Politècnica de Catalunya),
joint project with M. Di Marca (U. di Genova) and G. Malara (Pedagogical U. of Cracow).
In a recent paper by Cook II, Harbourne, Migliore and Nagel, the authors studied unexpected curves passing through a given set of points Z and a fat point with general support. In particular, they related the existence of unexpected curves to properties of the line arrangement dual to the set of points Z, i.e., the set of lines defined by the linear forms having as coefficients the coordinates of the points of Z. By using this characterization, we construct new examples of unexpected curves, generalizing previous examples from the paper of Cook II, Harbourne, Migliore and Nagel. In particular, we classify supersolvable line arrangements that give unexpected curves.
12.30 - 15.00
lunch break
15.00 - 16.00
On the containment problem for fat points
Giuseppe Zito (U. di Catania),
joint project with I. Bahmani (Politecnico di Torino).
Given an ideal I, the containment problem is concerned about finding the values m and n such that the m-th symbolic power of I is contained in its n-th ordinary power. In order to study this problem, it is useful, given an ideal I, to introduce an asymptotic quantity, known as resurgence and denoted by ρ(I), defined as ρ(I) = sup{m/r : I(m) is not contained in Ir }. In this talk we consider the containment problem focusing on ideals of fat points and we show how to compute the resurgence for two particular classes of schemes. Specifically, for all N ≥ 2, we study fat points schemes whose supports are n distinct points on a line in PN and three nonlinear points in PN respectively.
16.00 - 16.30
coffee break
16.30 - 17.30
The containment problem for some monomial ideals and connection with 0-dimensional schemes in P2
Giancarlo Rinaldo (U. di Trento),
joint project with G. Favacchio (U. di Catania) and E. Guardo (U. di Catania).
We study monomial ideals defining certain 0-dimensional schemes in P2. In particular we study ideals of two distinct points P1, P2 and P3 infinitely near to P2. We compute the Waldschmidt constant (and the resurgence) of such ideals.
Saturday 24th
9.00 - 10.00
Regularity of bicyclic graphs
Yairon Cid Ruiz (U. de Barcelona),
joint project with S. Jafari (U. di Genova).
We will describe the relations between the regularity of edge ideals and induced matching number in the case of bicyclic graphs. In fact, we will show a combinatorial characterization of the regularity in terms of the induced matching number for bicyclic graphs.
10.00 - 11.00
Regularity of powers of dumbbell graphs
Navid Nemati (Pierre and Marie Curie U. - Paris 6),
joint project with B. Picone (U. di Catania).
In this talk we will present a formula for the regularity of all powers of dumbbell graphs. In particular, we will show that regularity becomes a linear function from the first power.
11.00 - 11.30
coffee break
11.30 - 12.30
Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals
Lorenzo Guerrieri (U. di Catania),
joint project with B. Drabkin (U. Nebraska-Lincoln).
Let I be an ideal in a polynomial ring whose symbolic Rees algebra is Noetherian. For m ≥ 1, the m-th symbolic defect sdefect(I,m) of I is defined to be the minimal number of generators of the module I(m)/ Im. We prove that sdefect(I,m) is an asymptotically quasipolynomial function in m. Then, after giving a decomposition of the symbolic powers of I in terms of smaller symbolic powers we compute the Waldschimdt constant of I in the monomial case, and we also compute explicit formulas for the symbolic defect of cover ideals of some graphs.
Sponsors:
