Finite sets of d-planes in affine space

Let A be a subvariety of affine space An whose irreducible components are d-dimensional linear or affine subspaces of An. Denote by D(A)⊂Nn the set of exponents of standard monomials of A. We show that the combinatorial object D(A) reflects the geometry of A in a very direct way. More precisely, we define a d-plane in Nn as being a set γ+⊕j∈JNej, where #J=d and γj=0 for all j∈J. We call the d-plane thus defined to be parallel to ⊕j∈JNej. We show that the number of d-planes in D(A) equals the number of components of A. This generalises a classical result, the finiteness algorithm, which holds in the case d=0. In addition to that, we determine the number of all d-planes in D(A) parallel to ⊕j∈JNej, for all J. Furthermore, we describe D(A) in terms of the standard sets of the intersections A∩{X1=λ}, where λ runs through A1.