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
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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.
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