IJPAM: Volume 27, No. 2 (2006)

STABLE COHERENT SYSTEMS ON INTEGRAL
PROJECTIVE CURVES: AN ASYMPTOTIC
EXISTENCE THEOREM

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]


Abstract.Fix integers $k > n \ge 2$, an integral projective curve $X$, a rank $n$ vector bundle $E$ on $X$ and an ample line bundle $H$ on $X$. Here we prove the existence of an integer $t_0$ (depending only on $k, n, X, H,E$) such that for all integers $t \ge t_0$ a general $k$-dimensional linear subspace $V$ of $E\otimes H^{\otimes t}$ spans $E$ and the coherent system $(E,V)$ is $\alpha$-stable for every $\alpha \gg 0$. To prove this result we prove the existence of an integer $t_0$ (depending only on $k, n, X, H,E$) such that for all integers $t \ge t_0$ a general $k$-dimensional linear subspace $V$ of $E\otimes H^{\otimes t}$ spans $E$ and the natural map $\bigwedge ^n(V) \to H^0(C,\mbox{\rm det}(E))$ is injective.

Received: October 21, 2005

AMS Subject Classification: 14H60

Key Words and Phrases: coherent systems on curves, vector bundles on curves, stable vector bundles, integral curve

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 27
Issue: 2