Cuspidal quintics and surfaces with pg=0,p_g=0, K2=3K^2=3 and 5-torsion

If $S$ is a quintic surface in $\mathbb P^3$ with singular set $15$ $3$-divisible ordinary cusps, then there is a Galois triple cover $\phi:X\to S$ branched only at the cusps such that $p_g(X)=4,$ $q(X)=0,$ $K_X^2=15$ and $\phi$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb Z_5$, so that $X/{\mathbb Z_5}$ is a smooth minimal surface of general type with $p_g=0$ and $K^2=3$. We find two different quintics, one of which is the Van der Geer--Zagier quintic, the other is new. We also construct a quintic threefold passing through the $15$ singular lines of the Igusa quartic, with $15$ cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular set $17\mathsf A_2$, $16\mathsf A_2$, $15\mathsf A_2+\mathsf A_3$ and $15\mathsf A_2+\mathsf D_4$.Comment: Exposition improved according to the Referee suggestions. Final versio

A note on Todorov surfaces

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}.Comment: 9 page

A surface with canonical map of degree 2424

We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto $\mathbb P^2$.Comment: Minor changes, according to the Referee comments. Final versio

On surfaces with pg=q=1p_g=q=1 and non-ruled bicanonical involution

This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.Comment: revised version, correction in main theorem, to appear in Ann. Scuola Norm. Sup. Pis