Let
P19 be the parametrizing space of cubic surfaces in
P3. The subset corresponds to non-singular cubic surfaces open in
P19. We denote by
Mk ⊂
P19 the subset of points corresponding to non-singular cubic surfaces in
P3 with at least
k Eckardt points. For every
k, we determine the dimension and the number of irreducible components of
Mk. A nonsingular cubic surface can be viewed as the blowing-up of
P2 at six points in general position. A close study of the configuration of six points in
P2 enables us to describe the configuration space of points in
P19 corresponding to non-singular cubic surfaces with a given number of Eckardt points. This study also provides an easy method to obtain the classification of nonsingular cubic surfaces according to the number of Eckardt points, which is a well-known result.
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