Brill-Noether generality of binary curves

He, Xiang

Canadian mathematical bulletin, Tome 64 (2021) no. 4, pp. 787-807

Résumé

We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$. This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$. In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$, where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

Détail

DOI : 10.4153/S0008439520000831

Mots-clés : Algebraic curves, linear systems, Brill–Noether theory