A New Form of Smooth Cubic Surfaces with 9 Lines

Abstract

A smooth cubic surface has at most 27 lines, with equality if and only if the underlying field is algebraically closed. Only a few cases are possible regarding the number of lines over fields that are not algebraically closed. The next two cases of interest are smooth cubic surfaces with 15 or 9 lines. The author has recently settled the case of 15 lines. In this paper, we address the case of smooth cubic surfaces with 9 lines. We describe a way to create some cubic surfaces with 9 or more lines based on a set of six field elements. Conditions on the six parameters are given under which the surface has exactly 9, 15, or 27 lines. However, the problem of generating all cubic surfaces with 9 lines remains open.