A note on Todorov surfaces

Abstract

Let $S$ be a Todorov surface, i.e., a minimal smooth surface of general type with $q=0$ and $p_{g}=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i$. The main result of this paper is that, if $P$ is the minimal smooth model of $S/i$, then $P$ is the minimal desingularization of a double cover of $\mathbb{P}^{2}$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_{j}$ with $K^{2}=1,\ldots,K_{S}^{2}-1$ and such that $P$ is also the smooth minimal model of $S_{j}/i_{j}$, where $i_{j}$ is the involution of $S_{j}$. Some examples are also given, namely an example different from the examples presented by Todorov in [9].