ar X iv : 0 80 3 . 31 41 v 2 [ m at h . A C ] 1 3 O ct 2 00 8 FINITE SETS OF d - PLANES IN AFFINE SPACE

Abstract

Let A be a subvariety of affine space A whose irreducible components are d-dimensional linear or affine subspaces of A. Denote by D(A) ⊂ N 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 N 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 inD(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 A.

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Cite this paper

@inproceedings{Lederer2008arXI, title={ar X iv : 0 80 3 . 31 41 v 2 [ m at h . A C ] 1 3 O ct 2 00 8 FINITE SETS OF d - PLANES IN AFFINE SPACE}, author={Mathias Lederer}, year={2008} }