On the Gorenstein locus of some punctual Hilbert schemes

Let $k$ be an algebraically closed field and let \Hilb_{d}^{G}(\p{N}) be the open locus of the Hilbert scheme \Hilb_{d}(\p{N}) corresponding to Gorenstein subschemes. We prove that \Hilb_{d}^{G}(\p{N}) is irreducible for $d\le9$, we characterize geometrically its singularities for $d\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in \Hilb_{d}^{G}(\p{N}).Comment: The exposition has been improved and some of the main results have been extended to degree $d\le 9

A structure theorem for 2-stretched Gorenstein algebras

In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of Macaulay's inverse system generalizing a recent result by J. Elias and M.E. Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of the punctual Hilbert scheme of degree 11.Comment: 24 pages. We removed lemma 2.1 because it was false and we modified the proof of proposition 3.2 accordingly inserting some new due reference

Examples of rank two aCM bundles on smooth quartic surfaces in P3\mathbb{P}^3

Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let $\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that $c_1(\mathcal{E})=\mathcal{O}_F(2h)$, $c_2(\mathcal{E})=8$ and $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$. We also deal with their stability.Comment: 22 pages. Exposition improve

TDOA--based localization in two dimensions: the bifurcation curve

In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane, by computing the Cartesian equation of the bifurcation curve in terms of the positions of the receivers. From that equation, we can compute its real asymptotic lines. The present manuscript completes the analysis of [Inverse Problems, Vol. 30, Number 3, Pages 035004]. Our result is useful to check if a source belongs or is closed to the bifurcation curve, where the localization in a noisy scenario is ambiguous.Comment: 11 pages, 3 figures, to appear in Fundamenta Informatica

Canonical curves with low apolarity

Let $k$ be an algebraically closed field and let $C$ be a non--hyperelliptic smooth projective curve of genus $g$ defined over $k$. Since the canonical model of $C$ is arithmetically Gorenstein, Macaulay's theory of inverse systems allows to associate to $C$ a cubic form $f$ in the divided power $k$--algebra $R$ in $g-2$ variables. The apolarity of $C$ is the minimal number $t$ of linear form in $R$ needed to write $f$ as sum of their divided power cubes. It is easy to see that the apolarity of $C$ is at least $g-2$ and P. De Poi and F. Zucconi classified curves with apolarity $g-2$ when $k$ is the complex field. In this paper, we give a complete, characteristic free, classification of curves $C$ with apolarity $g-1$ (and $g-2$)

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $\mathbb{P}^n$ for $d\geq 4$.Comment: v4: final. v2: expanded proof of Theorems A and B. 33 pages, comments welcome