Borel Ideals in Three and (four) Variables

Fixed any term-ordering < on the set T(n) of terms in n variables x1, . . . , xn, we study homogeneous ideal a ⊆ P(n) of the ring of polynomials in n variables over a field k, via the associated order-ideal N (a), consisting of all the terms which are not maximal terms of elements of a and called sous-éscalier of a [7, 12, 14]. In particular we will focus our attention on the following combinatorial properties of subsets B ⊂ T(n) (considered by R. Hartshorne [11] and first by N. Gunther [10]): (1) for every 1 ≤ j < k ≤ n such that ak > 0 x1 1 . . . x aj j . . . x ak k . . . x an n ∈ B ⇒ x a1 1 . . . x aj+1 j . . . x ak−1 k . . . x an n ∈ B; (2) for every 1 ≤ j < k ≤ n such that aj > 0 x1 1 . . . x aj j . . . x ak k . . . x an n ∈ B ⇒ x a1 1 . . . x aj−1 j . . . x ak+1 k . . . x an n ∈ B. Recalling that, if chark = 0, then for a monomial ideal a ⊂ P(n) TFAE: (I) choosing x1 > . . . > xn, a satisfies (1); a is Borel fixed (i.e. g(a) = a, ∀ g ∈ B, the Borel group of upper-triangular invertible matrices); N (a) satisfies (2). (II) choosing x1 < . . . < xn, a satisfies (2); a is fixed by the subgroup B′ of lower-triangular invertible matrices; N (a) satisfies (1). It seems natural to call Borel subset of T(n) any B satisfying (1), thus, as we are dealing with N (a), we will consider term-orderings with x1 < . . . < xn and call Borel ideals the monomial ideals b ⊂ P(n) whose N (b) is a Borel subset of T(n). Borel ideals are the special monomial ideals occurring (given a term-ordering <) as initial ideals in<(a) of homogeneous ideals a ⊆ P(n), in generic coordinates by the fundamental Galligo’s and Bayer-Stillman’s results [7, 1]. This initial ideal, denoted gin<(a) and called generic initial ideal with respect to <, has been widely studied together with the algebraic and geometric information it brings on, in particular when the fixed term-ordering is the lexicographical (lex) one or the degree reverse lexicographical (drl) one.