IJPAM: Volume 16, No. 4 (2004)

STABLE VECTOR BUNDLES ON CURVES:
SYMMETRIC AND ANTI-SYMMETRIC MORPHISMS

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


Abstract.Let $C$ be a smooth curve of genus $g \ge 2$. For any vector bundle $E$ on $C$ and any $L\in \mbox{Pic}(C)$ the involution $\sigma : A\otimes A \to A\otimes A$ defined by $\sigma (u\otimes v) = v\otimes u$ defines an involution on the vector space $W:= H^0(C,{{\rm Hom}}(E\otimes E,L))$. Let $W_+$ (resp. $W_-$) its invariant (resp. anti-invariant) subspace. Here we compute $\mbox{\rm dim}(W_+)$ and $\mbox{\rm dim}(W_-)$ for all integers $r, d, x$ such that $r \ge 2$, $x \ge g + 2d/r$ and $d \equiv 0 \pmod{r}$ when $E$ is a general stable vector bundle with rank $r$ and degree $d$ and $L$ is a general line bundle of degree $x$.

Received: September 14, 2004

AMS Subject Classification: 14H60

Key Words and Phrases: stable vector bundle, vector bundles on curves, anti-symmetric morphism, symmetric morphism

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 16
Issue: 4