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Published online by Cambridge University Press: 29 May 2025
In this survey article we explore various limits and asymptotic properties in commutative algebra and algebraic geometry. We show that several important invariants have good asymptotic behavior. We develop this and give some examples of pathological behavior for topics such as multiplicity of graded families of ideals, volumes of line bundles on schemes and regularity of powers of ideals.
This article is on the general theme of limits arising in commutative algebra and algebraic geometry, in asymptotic multiplicity and rgularity. The four sections of this article are based on the four talks that I gave during the Spring semester of the special year on Commutative Algebra, held at MSRI during the 2012–2013 academic year. The first section is based on an Evans lecture I gave at Berkeley.
Multiplicity and projection from a point. We begin by discussing a formula involving the multiplicity of a point on a variety, which evolved classically. Proofs of the formula (1) can be found in Theorem 5.11 of [Mumford 1976] (over k = ℂ), in Section 11 of [Abhyankar 1998], Section 12 of [Lipman 1975] and Theorem 12.1 [Cutkosky 2009].
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