Intersection numbers, Segre numbers and generalized Samuel multiplicities

Achilles, Rüdiger; Rams, Sławomir

doi:10.1007/pl00000509

Cited by 23 publications

(30 citation statements)

References 12 publications

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“…His construction is based on a pointwise defined intersection multiplicity g( [5]) has recently proved that g(x) = e(G I (A)) and thatg(X) is composed of the generalized Samuel multiplicities c 0 (I, A), . .…”

Section: Remark 37 (Analytic Case)mentioning

confidence: 99%

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Self-Intersections of Surfaces and Whitney Stratifications

Achilles

Manaresi²

2003

Proceedings of the Edinburgh Mathematical Society

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Let X be a surface in C n or P n and let C X (X × X) be the normal cone to X in X × X (diagonally embedded). For a point x ∈ X, denote by g( is a former result of the authors that g(x) is the degree at x of the Stückrad-Vogel cycle v(X, X) = C j(X, X; C)[C] of the self-intersection of X, that is, g(x) = C j(X, X; C)ex(C).We prove that the stratification of X by the multiplicity g(x) is a Whitney stratification, the canonical one if n = 3. The corresponding result for hypersurfaces in A n or P n , diagonally embedded in a multiple product with itself, was conjectured by van Gastel. This is also discussed, but remains open.

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Section: Remark 37 (Analytic Case)mentioning

confidence: 99%

“…In the analytic case, the Segre numbers of an ideal I introduced by Gaffney and Gassler [10] are also special cases of the generalized Samuel multiplicities c k (I, A) (see [5]). …”

Section: Remark 37 (Analytic Case)mentioning

confidence: 99%

Self-Intersections of Surfaces and Whitney Stratifications

Achilles

Manaresi²

2003

Proceedings of the Edinburgh Mathematical Society

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“…In [8,9] we provided sufficient conditions for such a good collection, namely, certain linear conditions on the tangent spaces at P to the hypersurfaces of the collection. The generalized intersection index of V and S at P coincides with the bidegree sequence of an algebraic bicone B, and the intersection multiplicity is the multiplicity at P of the normal cone C := C V ∩S V (see [7,8,9] and also [1,2]). Those formulae can be established by the method of deformation to the normal cone, applied to the analytic intersection algorithm.…”

Section: Deformation To An Algebraic Bicone Bmentioning

confidence: 99%

Remarks on the generalized index of an analytic improper intersection

Nowak

2003

Ann. Polon. Math.

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Abstract. This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7,8,9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be admissible with respect to B are of geometric nature: it is not necessary to analyse the embedded components of the intersections involved, but only the supports of the intersections of B with successive hyperplanes. Introduction.The generalized index at a point P of improper intersection of a purely dimensional analytic set V with a submanifold S was defined by Tworzewski [10] as the extended multiplicity at P of the total result of the analytic intersection algorithm performed for a generic admissible collection of smooth divisors. It is, of course, possible to confine oneself to the case where S is a linear subspace (flattening of S).In the papers [8,9] we reduced the computation of the generalized index at a point P of improper intersection of a purely dimensional analytic set V with a submanifold S

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“…Note that, in this particular case, the coherent O P n (C) -ideal sheaf I Z is globally generated by (m − 1)(n + 1) holomorphic sections σ l of the same line bundle O P n (C) (1). Moreover the zero sets σ −1 l (0), l = 1, ..., (m − 1)(n + 1), define here Z as a complete intersection in the ambient n-dimensional manifold X = P n (C).…”

mentioning

confidence: 99%

“…It is indeed necessary to overcome the difficulty which is inherent to the fact that averaging such Vogel residue currents in order to get suitable Bochner-Martinelli currents (for control of the degree in effectivity questions) does not preserve the arithmetic structure of the data (which would be necessary in order to get in parallel control on the heights). It seems also opportune to mention that the initial approach to Theorem 4 by F. Amoroso in [2] relies on the Northcott-Rees notion of superficial elements in ideals, that is also present in the construction of Vogel sequences (more specifically of filtered sequences, see [1]). …”

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confidence: 99%

Averaging Residue Currents and the Stückrad–Vogel Algorithm

Yger

2012

The Mathematical Legacy of Leon Ehrenpreis

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Trace formulas (Lagrange, Jacobi-Kronecker, Bergman-Weil) play a key role in division problems in analytic or algebraic geometry (including arithmetic aspects, see for example [10]). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows to carry local or semi global analytic problems to a global geometric setting (think about Crofton's formula), averaging the Cauchy kernel (from C n \{z 1 . . . z n = 0} ⊂ P n (C)), in order to get the Bochner-Martinelli kernel (in C n+1 \ {0} ⊂ P n+1 (C) = C n+1 ∪ P n (C)), leads to the construction of explicit candidates for the realization of Grothendieck's duality, namely BM residue currents ([27, 3, 6]), extending thus the cohomological incarnation of duality which appears in the complete intersection or Cohen-Macaulay cases. We will recall here such constructions and, in parallel, suggest how far one could take advantage of the multiplicative inductive construction introduced in [13] by N. Coleff and M. Herrera, by relating it to the Stückrad-Vogel algorithm developed in ([30], [31], [8]) towards improper intersection theory. Results presented here were initiated all along my long term collaboration with Carlos Berenstein. To both of us, the mathematical work of Leon Ehrenpreis certainly remained a constant and how much stimulating source of inspiration. This presentation relies also deeply on my collaboration with M. Andersson, H. Samuelsson and E. Wulcan in Göteborg, through the past years.

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Intersection numbers, Segre numbers and generalized Samuel multiplicities

Cited by 23 publications

References 12 publications

Self-Intersections of Surfaces and Whitney Stratifications

Self-Intersections of Surfaces and Whitney Stratifications

Remarks on the generalized index of an analytic improper intersection

Averaging Residue Currents and the Stückrad–Vogel Algorithm

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