IJPAM: Volume 50, No. 4 (2009)

BRILL-NOETHER THEORY OF RANK $R$
SHEAVES ON STABLE CURVES: AN EXTREMAL CASE

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


Abstract.Let $X$ be a stable curve. Fix an integer $r>0$. Here we give a bijection between the set of all disconnecting nodes of $X$ and the depth $1$ sheaves with pure rank $r$ on $X$ satisfying cetain properties (among them $F$ spanned, $\deg (F) =r$, $h^0(X,F) \ge 2r$ and $F$ `` maximally non-locally free ''). Let $X_1$ and $X_2$ be the closures in $X$ of $X\backslash \}P\}$. If $F$ is $\omega _X$-semistable, then $p_a(X) = 2\cdot p_a(X_1) = 2\cdot p_a(X_2)$. The converse is true if $X_1$ and $X_2$ are ireducible.

Received: December 20, 2008

AMS Subject Classification: 14H60, 14H10, 14H51

Key Words and Phrases: stable curve, reducible curve, Brill-Noether theory, Brill-Noether theoy for vector bundles

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 4