Uniform bounds and symbolic powers on smooth varieties

By L. Ein, R. Lazarsfeld, Karen E. Smith. 2000

284 citations recorded by the source corpus.

The purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety. In particular, weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety.Let

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1e5a7abbf7da1ba211b13f3c131748ca9123b9c4

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

A primer on Seshadri constants (2008, background): “Swanson [Swa00] showed that an upper bound exists for many ideals I. This was the inspiration for the papers [ EiLaSm01 ] and [HocHun02], whose results imply that (I) 6 N for any nontrivial homogeneous ideal I. In fact, if we dene codim( to be the maximum height among associated primes of I other th”
Symbolic Powers of Monomial Ideals (2013, background): “Among the known results is a celebrated containment of Ein-Lazarsfeld-Smith [9] and Hochster-Huneke [12].”
Jumping coefficients of multiplier ideals (2003, unknown): “At least when f has isolated singularities, Loeser and Vaquié [25] observe that results of Varchenko [36] establish a connection between jumping numbers and the spectrum of f : this in turn implies the Theorem in the case of isolated singularities.”
A generalization of tight closure and multiplier ideals (2002, background): “Recently, the theory of multiplier ideals with ”rational coefficients” has been developed and applied successfully to various problems in algebraic geometry and commutative algebra ([ ELS ], [La]).”
On a non-vanishing conjecture of Kawamata and the core of an ideal (2003, background): “Recently, multiplier ideals have been used to prove surprising new results in commutative algebra as well; see [10] and [11].”
Symbolic Powers of Ideals (2015, background): “s’s proof of Fermat’s Last Theorem [Wil95]. An important property of symbolic powers over regular rings containing ﬁelds, which was surprising at the time, is the theorem of Ein, Lazarfeld, and Smith [ELS01] in the geometric case in characteristic zero, and Hochster and Huneke [HH02] in general for ”
The resurgence of ideals of points and the containment problem (2009, background): “I(m) and indeed, for 0 6= I ( R, Ir ⊆ I(m) holds if and only if r ≥ m [PSC, Lemma 8.1.4]. A much more diﬃcult problem is to determine when there are containments of the form I(m) ⊆ Ir. The results of [ELS] and [HoH] show that I(m) ⊆ Ir holds whenever m ≥ Nr. The second author has proposed the follow”
Uniform approximation of Abhyankar valuation ideals in smooth function fields (2002, unknown): “0, any very large p will do; see [8], Remark following Definition 1.4.”