The Limiting Shape of the Generic Initial System of a Complete Intersection

Abstract

Consider a complete intersection I of type (d1, . . . , dr) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I n )}n obtained by taking the reverse lexicographic generic initial ideals of the powers of I and describe its asymptotic behavior. This behavior is nicely captured by the limiting polytope which is shown to depend only on the type of the complete intersection. arXiv:1202.1317v1 [math.AC] 6 Feb 2012 a definition only for this special case. See [BL04] for an introduction to multiplier ideals or [Laz04] for a more general treatment. Definition 2.8 ([How01]). Let J ⊂ R be a monomial ideal and let P be its Newton polytope. The multiplier ideal ofThe asymptotic definition requires the following lemma.Lemma 2.9 (Lemma 1.3 of [ELS01]). Let J • be a graded system of ideals and fix a rational number c > 0. Then for p 0 the multiplier ideals J ( c p · J p ) coincide. Definition 2.10. Let J • = {J k } k∈N be a graded system of ideals on R. Given c > 0 the asymptotic multiplier ideal of J • with coefficient c isfor any sufficiently large p (guaranteed by Lemma 2.9). When J • is a graded system of monomial ideals with limiting polytope P , J (c · J • ) = {x λ : λ + 1 ∈ Int(cP ) ∩ N m } for p 0.

Asymptotic Hilbert polynomials and limiting shapes

“…So far, these limiting shapes have been found for complete intersections (Mayes [13]), points in P 2 (Mayes [12] assuming Segre-Hirschowitz-Gimigliano-Harbourne conjecture) and star configurations in P n (the authors with T. Szemberg, [4]). The crucial result, which allows all the above computations, has been observed by Mustaţă [14, Theorem 1.7 and Lemma 2.13] and Mayes [12,Proposition 2.14].…”

“…[11,Section 2.4.B]. An example of such an invariant is the limiting shape, defined by S. Mayes [13]. This invariant carries information about the geometry of V (I), computing e.g.…”

“…This invariant carries information about the geometry of V (I), computing e.g. its asymptotic initial degree (called also the Waldschmidt constant) or asymptotic regularity (for more details see [13,12]). …”

The generic initial ideals of powers of a 2-complete intersection

“…Consider the collection of ideals {gin(I n )} n obtained by taking the generic initial ideals of powers of a fixed ideal I in a polynomial ring. Our study of such families of monomial ideals was initially motivated by the desire to understand their asymptotic behaviour (see [May12]). It soon became clear, however, that the individual ideals within such families are interesting in their own right.…”

“…The complexity of this result even in this small case, however, gives further evidence that finding generators of the generic initial ideals of powers of larger complete intersections may be optimistic and provides motivation to instead study the asymptotic behaviour of generic initial systems {gin(I n )} n . One consequence of Theorem 4.1 is a different proof for the case of a 2-complete intersection of the main result of [May12] on the asymptotic behaviour of the generic initial system.…”