Consider the ideal I ⊆ K[x, y, z] corresponding to six points of P 2 . We study the limiting behaviour of the symbolic generic initial system, {gin(I (m) }m of I obtained by taking the reverse lexicographic generic initial ideals of the uniform fat point ideals I (m) . The main result of this paper is a theorem describing the limiting shape of {gin(I (m) }m for each of the eleven possible configuration types of six points. m P gin(I (m) ) , where P gin(I (m) )

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.

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We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Söderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from [ES09]. In higher codimension, obstructions arise that inspire our work on an alternative algorithm.