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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.

CONFIGURATION SPACES OF NON-SINGULAR CUBIC SURFACES WITH ECKARDT POINTS