AbstractLet $$(S,{\mathcal {L}})$$
(
S
,
L
)
be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$
L
of degree $$d > 25$$
d
>
25
. In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$
χ
(
O
S
)
≥
-
1
8
d
(
d
-
6
)
. The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$
χ
(
O
S
)
=
-
1
8
d
(
d
-
6
)
if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$
|
H
0
(
S
,
L
)
|
embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$
T
⊂
P
5
of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$
d
2
Q
, where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$
H
∈
|
H
0
(
S
,
L
)
|
of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$
V
⊆
P
5
, from a point $$x\in V\backslash C$$
x
∈
V
\
C
.
Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology
