Ulrich bundles on non-special surfaces with pg=0p_g=0 and q=1q=1

Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\vert h\vert$ is at least $2$.Comment: 14 pages. A gap in the proof of Theorem 1.3 has been fixed. arXiv admin note: text overlap with arXiv:1609.0791

Special Ulrich bundles on non-special surfaces with pg=q=0p_g=q=0

Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for linearly normal non-special surface in $\mathbb P^4$ of degree at least $4$, Enriques surface and anticanonical rational surface.Comment: 17 pages, to appear in the International Journal of Mathematic

Quadratic sheaves and self-linkage

Generalizing an old result proved by P. Rao [see MR 83i:14025] for arithmetically Cohen-Macaulay, self-linked subschemes of codimension 2 in the projective n-space P, we give a characterization of self-linked pure subschemes of codimension 2 in P satisfying a necessary parity condition, in characteristic different from 2. We make use of the theory of quadratic sheaves described in a previous paper of the authors [see MR 99h:14044].Comment: AmS-ppt, 12 pages, no figure

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

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

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