The asymptotic behaviour of symbolic generic initial systems of generic points

Abstract

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) )

Asymptotic Hilbert polynomials and limiting shapes

“…2 Corollary 8. Combining the above theorem with Theorem 5 and Lemma 6 we obtain The convex set Δ(I) is called the limiting shape of I (compare [12,4]). Observe that Γ(I) is the closure of the complement of Δ(I).…”

“…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]. Let Γ(I) be the closure of the complement of Δ(I) in R n ≥0 .…”

“…(see Mayes [12]). 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]).…”

“…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]). …”

Symbolic generic initial systems of star configurations

“…[17,Section 2.4.B]. For a homogeneous ideal I, Mayes introduces in [19] symbolic generic initial systems gin(I (m) ) m . Here gin(J) denotes the reverse lexicographic generic initial ideal of a homogeneous ideal J and J (m) denotes the mth symbolic power of J.…”