Configuration types and cubic surfaces

This paper is a sequel to the paper [E. Guardo, B. Harbourne, Resolutions of ideals of six fat points in P2, J. Algebra 318 (2) (2007) 619–640]. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at n⩽8 essentially distinct points of the projective plane. Each type gives rise to a surface X obtained by blowing up the points. We classify those types such that n=6 and −KX is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well known in characteristic 0 [J.W. Bruce, C.T.C. Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19 (2) (1979) 245–256]. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme Z=m1p1+⋯+m6p6 such that the points pi are essentially distinct and −KX is nef, given only the configuration type of the points p1,…,p6 and the coefficients mi.