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

Abstract

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 $15\mathsf A_2+\mathsf D_4$.