Positivity and complexity of ideal sheaves

By Steven Dale Cutkosky, L. Ein, R. Lazarsfeld. 2000

60 citations recorded by the source corpus.

The problem of bounding the “complexity" of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years. Results in this direction go back at least as far as the classical work [17] of Hermann on the ideal membership problem, and the effective Nullstellensatz of Brownawell [4] and Kollar [21] marks a major recent advance.With the development of computational algebraic geometry the question has taken on increasing importance, and it came into particularly clear focus through the influential paper [3] of Bayer andMumford.More recently, the theorem of [8] and [20] concerning regularity of powers raises the question of bounding the complexity of powers of an ideal, and suggests that asymptotically the picture should become very clean. The aim of the present paper is to examine some of the results and questions of [3], [30] and [7] from a geometric perspective, in the spirit of [12]. Our thesis is that much of this material is clarified, and parts rendered transparent, when viewed through the lenses of vanishing theorems and intersection theory. Specifically,motivated by thework [25], [26] of Paolettiwe introduce an invariant s(J ) thatmeasures in effect howmuchonehas to twist an idealJ in order tomake

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

View Paper: https://www.semanticscholar.org/paper/26c420388c30dca7f9d11e3e9d70de81e1a08b4b

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In Algebra and Number Theory

1. Hsin-Ju Wang
2. Ngô Viêt Trung
3. Nero Budur
4. Huy Tài Hà

In Computational Theory and Mathematics

1. Ngô Viêt Trung
2. Nero Budur
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In Geometry and Topology

1. Hsin-Ju Wang
2. Ngô Viêt Trung
3. Nero Budur
4. Morihiko Saito
5. Huy Tài Hà

Citations in the corpus

• On Hodge spectrum and multiplier ideals (2002, background): “in algebraic geometry through vanishing theorems, see [EV92]. Since then, the multiplier ideals found numerous applications in algebraic geometry (see [De95], [Si98], [Ei97], [Ka99], [EL99], [ELS01], [CEL01]). Recently, the multiplier ideals turned out to be connected with the test ideals of commuta”
• On the asymptotic linearity of Castelnuovo–Mumford regularity (2002, background): “ Swanson [Swa97] proved this for all homogeneous ideals. Recently, Cutkosky et al [CuHT99] and Kodiyalam [Kod00] found out that reg(In) is asymptotically a linear function. See also [Cha97], [Cut00], [CuEL], [GGP95] for related results. The aim of this paper is to prove the above phenomenon for the ”