论文标题
$ \ ell $ - ACM捆绑在Fano表面上
$\ell$-away ACM Bundles on Fano Surfaces
论文作者
论文摘要
我们提出了$ \ ell $ -AWAY ACM BUNDLE在两极分化$（x，\ Mathcal {o} _ {x}（h））$上的定义。然后，我们在$（\ Mathbb {p}^2，\ Mathcal {o} _ {\ Mathbb {p}^2}（1）$，$（\ Mathbb {p}^p}^1 \ times \ Mathbb {p p}^p p}^1， \ Mathcal {o} _ {\ Mathbb {p}^1 \ times \ times \ Mathbb {p}^1}（1,1））$和对$ \ Mathbb {p}^2 $的抗态极化吹出。另外，我们将$ \ ell $ -AWAY ACM捆绑的完整分类为$ \ MATHCAL {E} $等级2的$ $ 1 \ leq \ leq \ ell \ ell \ leq 2 $ on $（\ Mathbb {p}^2，\ Mathcal {\ Mathcal {o}同样，在$（\ mathbb {p}^1 \ times \ mathbb {p}^1，\ mathcal {o} _ {\ mathbb {\ mathbb {p}^1 \ times \ times \ times \ mathbb {p}^1}^1}^1}（1,1）） \ Mathcal {o} _ {\ Mathbb {p}^1 \ times \ times \ Mathbb {p}^1}（a，a，a）$，用于某些$ a \ in \ Mathbb {z} $。此外，我们证明了相应的分级模块$ \ MATHRM {h} _*^1（\ MATHCAL {e}）= \ unterSet {{t \ in \ Mathbb {z}}}}}}}} {\ bigoplus} {\ bigoplus} \ bigRm} \ Mathrm { $ \ mathbb {p}^2 $上的捆绑包。
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 $\mathbb{P}^2$.
