  This paper is a sequel to the paper \cite{refGH}. 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
$n\le8$ 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 $-K_X$ 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 \cite{refBW}. 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 $Z=m_1p_1+...+m_6p_6$ such
that the points $p_i$ are essentially distinct and $-K_X$ is nef, given only
the configuration type of the points $p_1,...,p_6$ and the coefficients $m_i$.
