IJPAM: Volume 47, No. 2 (2008)

ULRICH SHEAVES AND ARITHMETICALLY
COHEN-MACAULAY VECTOR BUNDLES
ON RULED SURFACES

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Fix an integral $n$-dimensional projective variety and $A \subseteq B \subseteq \eta _+$, where $\eta _+$ is the ample cone of $X$. An Ulrich master for $(A,B)$ is a coherent sheaf $E$ on $X$ such that $H^i(E\otimes L) = H^i(E\otimes L^\ast )=0$ for all $L\in B$, and all $1 \le i \le n-1$, $H^i(E) =0$ for all $1 \le i \le n-1$, and $H^n(E\otimes L^\ast )=0$ for all $L\in A$. Let $C$ be a smooth genus $g$ curve and $G$ a rank $2$ unstable vector bundle on $C$. Set $X:= \mathbb {P}(G)$. Here for all integers $r \ge 1$ we prove the existence of rank $r$ Ulrich masters on $X$ for $(A,\eta _+)$ with $A$ very large, and give several necessary conditions for their numerical invariants.

Received: July 28, 2008

AMS Subject Classification: 14J60

Key Words and Phrases: Ulrich sheaf, ACM vector bundle, arithmetically Cohen-Macaulay vector bundle, ruled surface, unstable vector bundle

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 47
Issue: 2