IJPAM: Volume 27, No. 2 (2006)

BRILL-NOETHER LOCI OF $\mathcal {M}_g$

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

Abstract.Fix integers $g$, $t$, $r$ and $d$ such that $r>0$, $g \ge t+2 \ge 3$, $2 \le d \le 2g-4$ and $\rho (g,r,d)=-t$. For all integers $u>0,v>0$ set $\mathcal {M}_g(u,v):= \{C\in \mathcal {M}_g: C$ has a $g^u_v\}$. Here we prove the existence of $X\in \mathcal {M}_g$ with the following properties:

(i) $X\in \mathcal {M}_g(r,d)$ and $\mathcal {M}_g(r,d)$ is smooth and of dimension $3g-3-t$ at $X$;

(ii) $X$ has no $g^y_x$ such that $\rho (g,y,x) < 0$ and $0 < x < d$;

(iii) $X$ has no $g^u_v$ such that $v < 2d$ and $\rho (g,u,v) \le -t-1$.

Received: February 7, 2006

AMS Subject Classification: 14H10, 14H51

Key Words and Phrases: Brill-Noether locus, moduli scheme of curves, limit linear series, stable curves

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