Curves on Heisenberg invariant quartic surfaces in projective 3-space

TL;DR

This paper studies Heisenberg invariant quartic surfaces in projective 3-space, revealing their Picard group structure, Picard number, and the role of smooth conics in generating their Picard groups.

Contribution

It determines the Picard number and Picard group of very general Heisenberg invariant quartic surfaces, and shows that smooth conics generate the Picard group in these cases.

Findings

Picard number of very general surfaces is 16

Contains 320 smooth conics on a general surface

Conics generate the Picard group in the very general case

Abstract

This paper is about the family of smooth quartic surfaces $X \subset \mathbb{P}^3$ that are invariant under the Heisenberg group $H_{2,2}$. For a very general such surface $X$, we show that the Picard number of $X$ is 16 and determine its Picard group. It turns out that the general Heisenberg invariant quartic contains 320 smooth conics and that in the very general case, this collection of conics generates the Picard group.