Some Results on Asymptotic Regularity of Ideal Sheaves

2011/06/14 by Wenbo Niu, Niu, Wenbo
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG

paper · pdf · doi:10.48550/arxiv.1106.2585

15 pages, all comments welcome

arxiv created 2011/06/14 · openalex publication_date 2011/06/14 · arxiv updated 2011/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathscrI be an ideal sheaf on Pⁿ defining a subscheme X. Associated to \mathscrI there are two elementary invariants: the invariant s which measures the positivity of \mathscrI, and the minimal number d such that \mathscrI(d) is generated by its global sections. It is now clear that the asymptotic behavior of \reg \mathscrI^t is governed by s but usually not linear. In this paper, we first describe the linear behavior of the asymptotic regularity by showing that if s=d, i.e., s reaches its maximal value, then for t large enough \reg \mathscrI^t=dt+e for some positive constant e. We then turn to concrete geometric settings to study the asymptotic regularity of \mathscrI in the case that X is a nonsingular variety embedded by a very ample adjoint line bundle. Our approach also gives regularity bounds for \mathscrI^t once we know \reg \mathscrI and assume that X is a local complete intersection.

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