Quintic surfaces with 18 cusps

Abstract

We construct quintic surfaces in the three-dimensional projective space ${\mathbb{P}}^3$ with $18$ ordinary cusps. Our starting point is the Barth--Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$ cusps, and examples with $18$ cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth--Rams decompositions. The corresponding sets of $12$ cusps meet in $7$ points, and we prove that the locus of quintics admitting two such decompositions contains a $6$-dimensional component in the moduli space whose general member has $17$ cusps. This makes it possible to find members with $18$ cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton--Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree $22$. We verify that this surface has $18$ ordinary cusps and no other singularities.