A remark on du Val linear systems

2023/10/19 by Arbarello, Enrico
#14H51 #14J27 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.12930

Abstract

Let |L_g|, be the genus g du Val linear system on a Halphen surface Y of index k. We prove that the Clifford index cliff(C) is constant on smooth curves C∈ |L_g|. Let γ(C) be the gonality of C. When cliff(C)<\lfloor(g-1)/(2)\rfloor (the relevant case), we show that γ(C)=cliff(C)+2=k, and that the gonality is realized by the Weierstrass linear series |-kK_Y|C|, which is totally ramified at one point. The proof of the first statement follows closely the path indicated by Green and Lazarsfeld for a similar statement regarding K3 surfaces.

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