The Multiplicity Polar Theorem, Collections of 1-forms and Chern Numbers

Gaffney, Terence; Grulha, Nivaldo G.

doi:10.5427/jsing.2013.7d

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…(Cf. [15] for all of the necessary hypotheses for this number to be defined.) If the local ring of X is Cohen Macaulay, then we can hope to calculate this number as a length.…”

Section: The Theory Of the Integral Closure Of Modules Polar Varietie...mentioning

confidence: 99%

“…The two candidates are e(M 1 , O C(M 2 ),x ) and e(M 2 , O C(M 1 ),x ). If the local ring of X is Cohen Macaulay, then these multiplicities are the colength of the ideal of maximal minors of each module ( [15] cor 2.4). In [15] Theorem 2.3 and Corollary 2.5 it is shown that these numbers are equal and both are equal to Serre's intersection number.…”

Section: The Theory Of the Integral Closure Of Modules Polar Varietie...mentioning

confidence: 99%

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Pairs of modules and determinantal isolated singularities

Gaffney,

Rangachev

2015

Preprint

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“…(Cf. [15] for all of the necessary hypotheses for this number to be defined.) If the local ring of X is Cohen Macaulay, then we can hope to calculate this number as a length.…”

Section: The Theory Of the Integral Closure Of Modules Polar Varietie...mentioning

confidence: 99%

Section: The Theory Of the Integral Closure Of Modules Polar Varietie...mentioning

confidence: 99%

Pairs of modules and determinantal isolated singularities

Gaffney,

Rangachev

2015

Preprint

Self Cite

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“…(b) An important problem not addressed in this work is the determination of an algebraic formula for the polar multiplicity as in Lê-Greuel's formula for ICIS. See [9], for an algebraic approach characterizing the d-th polar multiplicity of d-dimensional singular spaces.…”

Section: Index Of 1-forms On Determinantal Varietiesmentioning

confidence: 99%

Codimension Two Determinantal Varieties with Isolated Singularities

Ruas

Pereira

2014

MATH. SCAND.

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We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in $\mathsf{C}^4$, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

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“…The Chern number of collections of forms has a foudantion similar to the Euler obstruction of maps; however it has traveled a different trajectory "avoiding" cellular decompotitions, which are, in some sense, covered by the study of the loci of collections of forms. That is another way to present a generalization of the local Euler obstruction and, in [11], Gaffney and Grulha present an algebraic treatment to study the Chern number.…”

Section: Introductionmentioning

confidence: 99%

The geometrical information encoded by the Euler obstruction of a map

Grulha¹,

Ruiz²,

Santana³

2020

Preprint

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In this work we investigate the topological information captured by the Euler obstruction of a map, f : (X, 0) → (C 2 , 0), with (X, 0) a germ of a complex d-equidimensional singular space, with d > 2, and its relation with the local Euler obstruction of the coordinate functions and, consequently, with the Brasselet number. Moreover, under some technical conditions on the departure variety we relate the Chern number of a special collection related to the mapgerm f at the origin with the number of cusps of a generic perturbation of f on a stabilization of (X, f).

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The Multiplicity Polar Theorem, Collections of 1-forms and Chern Numbers

Abstract: Abstract. In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for collection of 1-forms.

Cited by 6 publications

References 13 publications

Pairs of modules and determinantal isolated singularities

Pairs of modules and determinantal isolated singularities

Codimension Two Determinantal Varieties with Isolated Singularities

The geometrical information encoded by the Euler obstruction of a map

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