Giovanna Ilardi

PUISEUX POWER SERIES SOLUTIONS FOR SYSTEMS OF EQUATIONS

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newtons method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

A Family of Algebraically Closed Fields Containing Polynomials in Several Variables

We introduce a family of fields of series with support in strongly convex rational cones. All these fields contain polynomials in several variables. We prove that they are algebraically closed with a construction that is analogous to the Newton polygon for algebraic curves. As a corollary, we show the existence of fractional power solutions with support in cones for systems of equations.

Laplace equations, Lefschetz properties and line arrangements

Abstract In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6] . The starting point was an example in [3] . Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3] .

Linear Systems on ℙ1 with Syzygies

Consider the k-dimensional linear systems of homogeneous polynomials in two variables of degree n with a given number j of syzygies of degree d. This is in some way equivalent to study hypersurfaces in ℙ1 of degree n with the same d-polar system. All these linear systems fill up a subvariety X k, j, d of the Grassmannian 𝔾r(k, n). We compute the dimension of this variety and give some geometric description of it by means of a suitable fibration. We examine the blow-up of X k,j+1,d in X k,j,d .

Linear systems on

Consider the k-dimensional linear systems of homogeneous polynomials in two variables of degree n with a given number j of syzygies of degree d. This is in some way equivalent to study hypersurfaces in ℙ1 of degree n with the same d-polar system. All these linear systems fill up a subvariety X k, j, d of the Grassmannian 𝔾r(k, n). We compute the dimension of this variety and give some geometric description of it by means of a suitable fibration. We examine the blow-up of X k,j+1,d in X k,j,d .

Bbb Psp 1

ABSTRACT Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ≅ R 𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭). Then there exists a bijection γ𝔪:Proj(G 𝔭(A) ⊗ A/𝔭 k) → Proj(G(A 𝔭) ⊗ k(𝔭)K) such that for every subset Σ of Proj(G 𝔭(A) ⊗ A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.