On large configurations of lines on quartic surfaces

2025/06/28 by Alex Degtyarev, Degtyarev, Alex, Sławomir Rams +1
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2506.22733

Abstract

We estimate the number of lines on a non-K3 quartic surface. Such a surface with only isolated double point(s) contains at most twenty lines; this bound is attained by a unique configuration of lines and by a surface with a certain limited set of singularities. We have similar itemized bounds for other types of non-simple singularities, which culminate in at most 31 lines on a non-K3 quartic not ruled by lines; this bound is only attained on the quartic monoids described by K.~Rohn.

Citations

• Lines on K3–Quartics Via Triangular Sets
• Quartic monoid surfaces with maximum number of lines
• The maximum number of lines lying on a K3 quartic surface
• Ruled quartic surfaces, models and classification
• On the Classification of Cubic Surfaces

Related

• Lines on K3–Quartics Via Triangular Sets
• Counting lines on surfaces
• Quartic monoid surfaces with maximum number of lines
• On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic
• The number of singular points of quartic surfaces (char=2)