Borel Ideals in Three Variables

i=1 gijXi, g = (gij) ∈ Gl(n,k)), given any term-ordering< and homogeneous ideal a ⊆ P(n), there exists a non-empty open subset U of Gl(n,k) such that as g ranges in U, gin(a) := in(g(a)) is constant. Moreover, gin(a) is fixed by the group B of upper-triangular invertible matrices, if X1 > · · · > Xn, while gin(a) is fixed by the group B′ of lower-triangular invertible matrices if X1 < · · · < Xn. Monomial ideals a ⊆ P(n), can be studied via the associated order-ideal N (a) consisting of all the terms (= monic monomials) ‘outside’ a and called sous-éscalier of a ([6], [8] and [10]). For a Borel ideal b ⊆ P(n), N (b) is fixed by B′ if X1 > · · · > Xn, and by B if X1 < · · · < Xn. Studying Borel ideals through their sous-éscaliers, following A. Galligo ([7]), we consider X1 < · · · < Xn. In Section 2 we fix our notation. In Section 3 we introduce the Borel subsets of the multiplicative semigroup of terms in P(n), illustrating some of their features and giving a ‘general construction’ to produce Borel subsets of assigned cardinality in each degree. In Section 4 we describe the Borel ideals b ⊂ P(n); in particular, basing on the combinatorics of N (b), we associate to every 0-dimensional b ⊆ P(n), generated in degrees ≤ s+1, an n by s+1 matrix M̃(b) with non-negative integral entries m̃i,j(b). Since on M̃(b)′s rows one reads the Hilbert functions of sections of P(n)/b with linear spaces (see Definition 4.10 and Remark 4.12 a)),