$\ell$-away ACM Bundles on Fano Surfaces

TL;DR

This paper introduces the concept of $ ext{ extlangle} ext{ extlangle} ext{away} ext{ extlangle} ext{ extlangle} $-away ACM bundles on polarized varieties, constructs examples on specific surfaces, classifies certain rank 2 bundles, and proves connectedness of associated graded modules.

Contribution

It defines $ ext{ extlangle} ext{ extlangle} ext{away} ext{ extlangle} ext{ extlangle} $-away ACM bundles, provides explicit constructions and classifications on Fano surfaces, and extends known results on the connectedness of their cohomology modules.

Findings

Constructed $ ext{ extlangle} ext{ extlangle} ext{away} ext{ extlangle} ext{ extlangle} $-away ACM bundles on projective planes and blow-ups.

Classified rank 2 $ ext{ extlangle} ext{ extlangle} ext{away} ext{ extlangle} ext{ extlangle} $-away ACM bundles on specific surfaces.

Proved the graded module $ ext{H}_*^1( ext{E})$ is connected for these bundles.

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 $1 \leq \ell \leq 2$ 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 $\mathrm{det}(\mathcal{E}) = \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,a)$ for some $a \in \mathbb{Z}$. Moreover, we prove that the corresponding graded module $\mathrm{H}_*^1 ( \mathcal{E}) = \underset{{t \in \mathbb{Z} }}{\bigoplus} \mathrm{H}^1 (\mathcal{E} (th))$ is connected, extending the similar result for bundles on $\mathbb{P}^2$.