## Osaka Journal of Mathematics

### A note on Todorov surfaces

Carlos Rito

#### 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].

#### Article information

Source
Osaka J. Math., Volume 46, Number 3 (2009), 685-693.

Dates
First available in Project Euclid: 26 October 2009

https://projecteuclid.org/euclid.ojm/1256564201

Mathematical Reviews number (MathSciNet)
MR2583324

Zentralblatt MATH identifier
1204.14018

#### Citation

Rito, Carlos. A note on Todorov surfaces. Osaka J. Math. 46 (2009), no. 3, 685--693. https://projecteuclid.org/euclid.ojm/1256564201

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