Enumeration of surfaces containing an elliptic quartic curve

TL;DR

This paper calculates the degrees of loci of surfaces in projective space that contain elliptic quartic curves, providing formulas for general cases and specific quartic surfaces, advancing understanding of algebraic surface configurations.

Contribution

It derives a formula for the degree of the locus of surfaces of degree ≥5 containing elliptic quartic curves and computes the degree for quartic surfaces specifically.

Findings

Formula for the degree of the locus of surfaces with elliptic quartic curves

Degree computation for quartic surfaces containing elliptic quartic curves

Extension of known results to specific quartic cases

Abstract

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.