Star points on cubic surfaces

Abstract

A cubic surface in P 3 is given by a non-zero cubic homogeneous polynomial in 4 variables. Fixing an ordering of monomials of degree 3 in the polynomial ring k[x0; x1; x2; x3 ], each cubic surface denes a point in P 19 . The locus P 19 of singular cubic surfaces is a closed subset of codimension 1. A non-singular cubic surface X contains twenty-seven lines. There exist at most 3 lines among these twenty-seven lines through a given point of X. A star point (also called Eckardt point) on anon-singular cubic surface is the intersection point of three lines on the surface. In this Ph.D. thesis, we denote by Hk the subset of P 19 consisting of points corresponding to non-singular cubic surfaces with at least k star points. We study these Hk as subvarieties of P 19 and their images in a compactication M of the moduli space M of non-singular cubic surfaces. Chapter 1 deals with some basis facts on cubic surfaces and actions of group varieties. We study non-singular cubic surfaces with star points in Chapter 2. We describe the specic congurations of six points in general position corresponding to non-singular cubic surfaces with a given number of star points. We consider the irreducibility, the local closedness and the dimension of Hk . Moreover, we determine the inclusion relationship between the irreducible components of these Hk . In Chapter 3, we study the boundaries of the subsets Hk inside P 19 and the boundaries of their images in the compactication M. To do so, we describe in Section 3.1 a classication of singular cubic surfaces and compute the number of singular points, the number of lines on each singular cubic surface with their conguration. Moreover, we compute the codimension of these classes and determine the relationship between their closures. In Section 3.2, we give a proof for the well-known basic fact that semi-stable cubic surfaces are those containing at most A2 singularities and stable cubic surfaces are those containing at most A1 singularities. In Section 3.3, we study semi-stable cubic surfaces. We prove that for any semi-stable cubic surface X, there exists a 6-point scheme P such that the linear system L P of cubic forms through P has dimension 4; furthermore, for any basis of L P, the closure of the image of the rational map from P 2 to P 3 dened by the basis is isomorphic to X. We also dene and compute the multiplicities of lines and triple intersections on semi-stable cubic surfaces. Section 3.4 contains several results on the boundaries of Hk and the boundaries of the images of Hk in M.