$\ell$-away ACM Bundles on Fano Surfaces

Abstract

We propose the definition of $\ell$-away ACM bundle on a polarized variety $(X,{\mathcal{O}}_X(h))$. Then we give constructions of $\ell$-away ACM bundles on $({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$, $({\mathbb{P}}^1\times {\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(1,1))$ and the anticanonically polarized blow up of ${\mathbb{P}}^2$ up to three non collinear points. Also, we give the complete classification of $\ell$-away ACM bundles $\mathcal{E}$ of rank 2 for values $1\le \ell \le 2$ on $({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$. Similarly, on $({\mathbb{P}}^1\times {\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(1,1))$, we give such a classification if $\det (\mathcal{E})={\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(a,a)$ for some $a\in \mathbb{Z}$. Moreover, we prove that the corresponding graded module ${\mathrm{H}}_{\ast }^1(\mathcal{E})=\underset{t\in \mathbb{Z}}{⨁}{\mathrm{H}}^1(\mathcal{E}(th))$ is connected, extending the similar result for bundles on ${\mathbb{P}}^2$.

24 pages. The title has been changed from "l-away ACM Bundles on DelPezzo Surfaces" to "$\ell$-away ACM Bundles on Fano Surfaces". Also, several typos have been fixed and the exposition improved. The final version appears in Bollettino dell'Unione Matematica Italiana

References in corpus (1)

• Even and odd instanton bundles on Fano threefolds