Multigraded regularity: syzygies and fat points.
Noether-Lefschetz, space curves and mathematical instantons.
Numerical characters of graded algebras
This is the summary of the plenary talk I gave in Milan at the XVII Meeting of the Unione Matematica Italiana. We focus on some relevant numerical characters of the standard graded algebras and, in some case, we explain their geometric meaning.
On 0-dimensional complete intersections.
On Castelnuovo's regularity and Hilbert functions
On Hilbert polynomial of certain determinantal ideals.
On multipliciteis of local rings.
On some properties of partial intersection schemes.
Partial intersection subschemes of Pr of codimension c were used to furnish various graded Betti numbers which agree with a fixed Hilbert function. Here we study some further properties of such schemes; in particular, we show that they are not in general licci and we give a large class of them which are licci. Moreover, we show that all partial intersections are glicci. We also show that for partial intersections the first and the last Betti numbers, say m and p respectively, give bounds each other;...
On Stanley-Reisner rings
On the Abel-Jacobi map for divisors of higher rank on a curve.
On the Betti numbers of perfect ideals
On the Complexity of the Projective Classification of Surfaces.
On the computation of a-invariants.
On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces
Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P n with one point as support and with high regularity index.
On the homogeneous ideal of unreduced projective schemes.
On the Jacobian ideal of the binary discriminant.
Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...
On the length of the absolute Samuel stratum
On the minimal reduction and multiplicity of .
On the partial Euler-Poincare characteristics of certain systems of parameters in local rings.
On the Poincaré series for a plane divisorial valuation.