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If $X$ is a smooth curve such that the minimal degree of its plane models isnot too small compared with its genus, then $X$ has been known to be a doublecover 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 ofdouble covers except some special cases. In this paper, we give upper and lowerbounds for the minimal degree of plane models of the double cover $X$ in termsof the gonality of the base curve $Y$ and the genera of $X$ and $Y$. Inparticular, the upper bound equals to the lower bound in case $Y$ ishyperelliptic. We give an example of a double cover which has plane models ofdegree equal to the lower bound.

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