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arXiv · 2310.12930

A remark on du Val linear systems

Enrico Arbarello
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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 $C\in |L_g|$. Let $\gamma(C)$ be the gonality of $C$. When $cliff(C)<\lfloor{\frac{g-1}{2}}\rfloor$ (the relevant case), we show that $\gamma(C)=cliff(C)+2=k$, 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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Enrico Arbarello. 2023-10-19. A remark on du Val linear systems. https://arxiv.org/abs/2310.12930

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