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Involutions on surfaces of general type with $p_{g}=0$

초록/요약

Let $S$ be a minimal surface of general type with $p_g=0$ having an involution $\sigma$ over the field of complex numbers. It is well known that if the bicanonical map $\varphi_{|2K_{S}|}$ of $S$ is composed with $\sigma$ (i.e.\ $\varphi_{|2K_{S}|}$ factors through $\sigma$) then the quotient $S/\sigma$ is rational or birational to an Enriques surface. We prove that for $K_{S}^{2}=5,6,7,8$ and an involution $\sigma$ for which $\varphi_{|2K_{S}|}$ is composed with $\sigma$, the quotient $S/\sigma$ is rational. This result applies in part to surfaces $S$ with $K_{S}^{2}=5$ for which $\varphi_{|2K_{S}|}$ has degree $4$ and is composed with an involution $\sigma$. Moreover when $\varphi_{|2K_{S}|}$ is not composed with $\sigma$ we give a list of the possible models of the quotient $S/\sigma$ and their branch divisors induced by $\sigma$. Surfaces $S$ with $K_{S}^{2}=7$ for which the quotient $S/\sigma$ is birational to an Enriques surface are treated in detail. Also we list the examples available in the literature.

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