A note on Todorov surfaces

TL;DR

This paper characterizes Todorov surfaces as double covers of the projective plane ramified over two cubics and explores their involutions, minimal models, and constructions of related surfaces with varying invariants.

Contribution

It provides a geometric description of Todorov surfaces via double covers and demonstrates how to construct related surfaces with different invariants.

Findings

Minimal model of S/i is a double cover of P^2 ramified over two cubics.

Construction of Todorov surfaces with K^2 from 1 to K_S^2 - 1.

An example of a Todorov surface different from previous known examples.

Abstract

Let $S$ be a {\em Todorov surface}, {\it 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 $K^2=1,...,K_S^2-1$ 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 \cite{To2}.