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Brill-Noether generality of binary curves

Part of: Curves

Published online by Cambridge University Press:  13 October 2020

Xiang He
Xiang He*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram. Jerusalem, 9190401, Israel
*
e-mail:xiang.he@mail.huji.ac.il
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Abstract

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.

Keywords

Algebraic curveslinear systemsBrill–Noether theory

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14H51: Special divisors (gonality, Brill-Noether theory) 14H60: Vector bundles on curves and their moduli
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 787 - 807
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Copyright
© Canadian Mathematical Society 2020

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