The circuit ideal of a vector configuration

JOURNAL OF ALGEBRA

卷:309

Article

Bogart, Tristram ; Jensen, Anders N. ; Thomas, Rekha R.

关键词: toric ideal;    circuit ideal;    initial ideal;    primary decomposition;    associated primes;    fans;   

DOI  :  10.1016/j.jalgebra.2006.07.025

来源: Elsevier

【 摘 要 】

Given a configuration A= {a(1),...,a(n)} subset of Z(d), a basis ideal of A is an ideal J(B) = x(u+) -x(u-): u is an element of B) subset of k[x(1),..., x(n)] where b spans the lattice L-A = {u is an element of Z(n): Sigma.a(i)u(i) = 0}. Our main interest is to understand when the toric ideal, I-A, of A equals a basis ideal J(B) with radical I-A. The circuit ideal, J(CA), of A is an example of such a basis ideal. We study such a J(B) in relation to I-A from various algebraic and combinatorial perspectives with a special focus on J(CA). We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of J(B) and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that the embedded primes of J(B) are indexed by certain faces of the cone spanned by A. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to I-A and J(CA). The Gr6bner fan of J(CA) is shown to refine that of (I)A when the codimension of the ideals is at most two. (c) 2006 Elsevier Inc. All rights reserved.

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