Lines on K3 quartic surfaces in characteristic 3

Abstract

We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

One example removed. Some examples and some proofs now with more details

References in corpus (4)

• Lines on quartic surfaces
• At most 64 lines on smooth quartic surfaces (characteristic 2)
• Lines on K3 quartic surfaces in characteristic 2
• Symmetries and equations of smooth quartic surfaces with many lines

Cited by in corpus (1)

• Lines on K3-quartics via triangular sets