Positivity of mixed multiplicities of filtrations

Cutkosky, Steven Dale; Srinivasan, Hema; Verma, J. K.

doi:10.1112/blms.12328

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…The multiplicities and mixed multiplicities of powers of m R -primary ideals are always positive ( [37] or [36,Corollary 17.4.7]). The multiplicities and mixed multiplicities of m R -filtrations are always nonnegative, as is clear for multiplicities, and is established for mixed multiplicities in [16,Proposition 1.3]. However, they can be zero.…”

Section: Introductionmentioning

confidence: 85%

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The Minkowski Equality for Filtrations

Cutkosky¹

2020

Preprint

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Suppose that R is an analytically irreducible or excellent local domain with maximal ideal mR. We consider multiplicities and mixed multiplicities of R by filtrations of mR-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of mR-primary ideals.

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Section: Introductionmentioning

confidence: 85%

“…However, they can be zero. If R is analytically irreducible, then all mixed multiplicities are positive if and only if the multiplicities e R (I(j); R) are positive for 1 ≤ j ≤ r. This is established in [16,Theorem 1.4].…”

Section: Introductionmentioning

confidence: 87%

The Minkowski Equality for Filtrations

Cutkosky¹

2020

Preprint

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“…We now prove 2). Statement 1) implies that e R (I(j); R) > 0 for 1 ≤ j ≤ r. Thus all mixed multiplicities are positive by [15,Theorem 1.4].…”

Section: First Properties Of Mixed Multiplicities Of Divisorial Filtrmentioning

confidence: 92%

“…For each i with 1 ≤ i ≤ r choose a flag (15) with Y 1 = E i and p a closed point such that p is nonsingular on X and E i and p ∈ E j for j = i. Let π 1 : R d+1 → R be the projection onto the first factor.…”

Section: Rees's Theorem For Divisorial Filtrationsmentioning

confidence: 99%

“…We have that The multiplicities and mixed multiplicities of m R -primary ideals are always positive ( [41] or [40,Corollary 17.4.7]). The multiplicities and mixed multiplicities of filtrations are always nonnegative, as is established in [15,Proposition 1.3], but can be zero. If R is analytically irreducible, then all mixed multiplicities are positive if and only if the multiplicities e R (I(j); R) are positive for 1 ≤ j ≤ r. This is established in [15,Theorem 1.4].…”

Section: Introductionmentioning

confidence: 98%

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Mixed multiplicities of divisorial filtrations

Cutkosky

2019

Advances in Mathematics

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Suppose that R is an excellent local domain with maximal ideal mR. The theory of multiplicities and mixed multiplicities of mR-primary ideals extends to (possibly non Noetherian) filtrations of R by mR-primary ideals, and many of the classical theorems for mR-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for mR-primary ideals do not hold for filtrations.In this article, we consider multiplicities and mixed multiplicities of R by mR-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of mR-primary ideals, which are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of mR-primary ideals I ⊂ I ′ with the same multiplicity then I and I ′ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. The classical Minkowski inequalities for mR-primary ideals I1 and I2 hold quite generally for filtrations. If R has dimension two and there is equality in the Minkowski inequalities, then Teissier and Rees and Sharp have shown that there are powers I a 1 and I b 2 which have the same integral closure. This theorem does not hold for arbitrary filtrations. The Teissier Rees Sharp theorem has been extended by Katz to mR-primary ideals in arbitrary dimension. We show that the Teissier Rees Sharp theorem does hold for divisorial filtrations in an excellent domain of dimension two.We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme X birationally dominating R, when R is an algebraic local domain.2010 Mathematics Subject Classification. 13H15, 13A18, 14C17.

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Multiplicities and Mixed Multiplicities of Filtrations

Cutkosky

Srinivasan

2021

Commutative Algebra

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Positivity of mixed multiplicities of filtrations

Cited by 6 publications

References 21 publications

The Minkowski Equality for Filtrations

The Minkowski Equality for Filtrations

Mixed multiplicities of divisorial filtrations

Multiplicities and Mixed Multiplicities of Filtrations

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