The minimal degree of plane models of double covers of smooth curves

Details The minimal degree of plane models of double covers of smooth curves The minimal degree of plane models of double covers of smooth curves

2008-09-10

arxiv.org/abs/0809.1853v2

Mathematics

Algebraic Geometry

13 pages; Sharpened the main result (Theorem 3.8); Corrected some errors (Theorem 4.1); Final version to appear in JPAA

Scientific paper

10.1016/j.jpaa.2009.09.006

If $X$ is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then $X$ has been known to be a double cover of another smooth curve $Y$ under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover $X$ in terms of the gonality of the base curve $Y$ and the genera of $X$ and $Y$. In particular, the upper bound equals to the lower bound in case $Y$ is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.

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