IJPAM: Volume 20, No. 4 (2005)

STABLE VECTOR BUNDLES ON REDUCIBLE
CURVES: PRELIMINARY REMARKS

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


Abstract.Here is one of our results. Let $Y$ be a reduced, but not irreducible projective curve such that all irreducible components of $Y$ are smooth and rational. Let $E$ be a rank $r \ge 1$ vector bundle on $Y$. There is at least a polarization on $Y$ such that $E$ is not stable for this polarization. $E$ is semistable for all polarizations, if and only if $E\vert T \cong {\mathcal {O}_T}^{\oplus r}$ for every irreducible component $T$ of $Y$. $E$ is semistable with respect to a polarization $H$ if and only if there is an integer $c$ such that $E \cong (H^{\otimes c})^{\oplus r}$.

Received: March 6, 2005

AMS Subject Classification: 14H60

Key Words and Phrases: vector bundles on curves, stable vector bundles, reducible curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 4