IJPAM: Volume 20, No. 2 (2005)

VECTOR SUBSPACES OF SECTIONS AND
STABLE COHERENT SYSTEMS ON CURVES

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


Abstract.Let $X$ be a smooth and connected projective curve, $E$ a non-trivial rank $n$ vector bundle on $X$ and $V$ an $m$-dimensional linear subspace of $H^0(X,E)$ spanning $E$. Assume that $\phi _{E,V} : X \to G(n,m)$ is unramified. Fix a general $n$-dimensional linear subspace of $W$. Then:

(a) for a general $(n-1)$-dimensional linear subspace $A$ of $W$ the evaluation map $u_A: \mathcal {O}_X\otimes W \to E$ is injective and with locally free cokernel;

(b) for every $(n-1)$-dimensional linear subspace $B$ of $W$ the evaluation map $u_B: \mathcal {O}_X\otimes W \to E$ is injective as a map of sheaves;

(c) there is a non-empty family $S(W)$ of the hyperplanes of $W$ such that the evaluation map $u_B: \mathcal {O}_X\otimes W \to E$ is injective as a map of sheaves, but it has a non-locally free cokernel if and only if $B\in S(W)$; at each $P\in X$ the fiber at $P$ of $u_B$ has rank at least $n-2$; the saturation of $u_B(\mathcal {O}_X\otimes B)$ in $E$ has degree one; $S(W)$ has pure dimension $n-2$.

Received: March 15, 2005

AMS Subject Classification: 14H60

Key Words and Phrases: coherent system, stable vector bundle, stable coherent system, vector bundles on curves, Grassmannian

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 2