Symbolic generic initial systems of star configurations

Abstract

The purpose of this note is to describe limiting shapes (as introduced by Mayes) of symbolic generic initial systems of star configurations in projective spaces over a field of characteristic 0.Comment: 8 page

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].…”

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

“…In the sequel we will need the notion of a limiting shape and some results from [6] and [7]. Let us start with recalling the notion of limiting shapes.…”

“…Consider s generic crosses in P 3 with the ideal I s . Take the polynomial Λ s , see (5) and (6): without the help of a computer. We were curious if we can get a better result using computer approach.…”

“…Recently the asymptotic invariants of ideals have aroused great interest and have been studied by many researchers, see, for example, [8], [9], [6], [7], [19], [18], [20], and others. One of these asymptotic invariants is the so-called Waldschmidt constant of an ideal I ⊂ K[P n ] denoted by α(I), see, for example, [3], [4], [2].…”