Abstract
AbstractWe propose the definition of $$\ell $$
ℓ
-away ACM bundle on a polarized variety $$(X, {\mathcal O}_{X}(h))$$
(
X
,
O
X
(
h
)
)
. Then we give constructions of $$\ell $$
ℓ
-away ACM bundles on $$({\mathbb {P}^{2}}, {\mathcal O}_{{\mathbb {P}^{2}}}(1))$$
(
P
2
,
O
P
2
(
1
)
)
, $$({\mathbb {P}^{1}} \times {\mathbb {P}^{1}}, {\mathcal O}_{{\mathbb {P}^{1}} \times {\mathbb {P}^{1}}}(1,1))$$
(
P
1
×
P
1
,
O
P
1
×
P
1
(
1
,
1
)
)
and the anticanonically polarized blow up of $${\mathbb {P}^{2}}$$
P
2
up to three non collinear points. Also, we give the complete classification of $$\ell $$
ℓ
-away ACM bundles $${\mathcal E}$$
E
of rank 2 for values $$1 \le \ell \le 2$$
1
≤
ℓ
≤
2
on $$({\mathbb {P}^{2}}, {\mathcal O}_{{\mathbb {P}^{2}}}(1))$$
(
P
2
,
O
P
2
(
1
)
)
. Similarly, on $$({\mathbb {P}^{1}} \times {\mathbb {P}^{1}}, {\mathcal O}_{{\mathbb {P}^{1}} \times {\mathbb {P}^{1}}}(1,1))$$
(
P
1
×
P
1
,
O
P
1
×
P
1
(
1
,
1
)
)
, we give such a classification if $$\textrm{det}({\mathcal E}) = {\mathcal O}_{{\mathbb {P}^{1}} \times {\mathbb {P}^{1}}}(a,a)$$
det
(
E
)
=
O
P
1
×
P
1
(
a
,
a
)
for some $$a \in \mathbb {Z}$$
a
∈
Z
. Moreover, we prove that the corresponding graded module $$\textrm{H}_*^1 ( {\mathcal E}) = {\bigoplus _{t \in \mathbb {Z}}} \textrm{H}^{1} ({\mathcal E}(th))$$
H
∗
1
(
E
)
=
⨁
t
∈
Z
H
1
(
E
(
t
h
)
)
is connected, extending the similar result for bundles on $${\mathbb {P}^{2}}$$
P
2
.
Publisher
Springer Science and Business Media LLC