The Enumerative Theory of Singularities

Kleiman, Steven L.

doi:10.1007/978-94-010-1289-8_10

Cited by 107 publications

(84 citation statements)

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“…[35], [14], [1], [12], [31]). Knowing the Thom polynomial of a singularity η, denoted T η , one can compute the cohomology class represented by the η-points of a map.…”

Section: Introductionmentioning

confidence: 99%

Thom polynomials and Schur functions: the singularities I_{2,2}(-)

Pragacz¹

2007

Ann. inst. Fourier

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“…[35], [14], [1], [12], [31]). Knowing the Thom polynomial of a singularity η, denoted T η , one can compute the cohomology class represented by the η-points of a map.…”

Section: Introductionmentioning

confidence: 99%

Thom polynomials and Schur functions: the singularities I_{2,2}(-)

Pragacz¹

2007

Ann. inst. Fourier

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“…Proof. As in other recent work on similar questions (see [3,4,5]), a key ingredient in the proof is an application of a "residual-intersection formula", of which we only require a relatively simple case, due to Fulton and MacPherson [2]. Consider the following cartesian diagram:…”

Section: Theorem Given a K-quasi-resolution As Above Assume That Ukmentioning

confidence: 99%

“…Our result yields new formulas even for surfaces in P 4 . For a modern account of these and related matters, see Kleiman's surveys [3, 5].Admittedly, the hypothesis of existence of a "resolution" is a severe restriction on the morphism ƒ. I am hopeful, however, that by pursuing further the same principles as in this paper, I will eventually obtain a united-set formula valid without such a restriction, and which would moreover be completely "intrinsic", in the sense of taking place on a suitable space associated solely to X (which is not the case with the present formula).We shall work in the category of complete (usually nonsingular) varieties over a field. Everything goes through with no change, however, in the category of compact complex manifolds.…”

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confidence: 99%

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A united-set formula

Ran¹

1983

Bull. Amer. Math. Soc.

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Let a morphism ƒ : X -• Y of algebraic varieties be given. A united set or united k-tuple 2 for ƒ is a fc-tuple x±,..., Xk of distinct points on (or "infinitely near") X, such that /(xi) = ••• = f(xk)> The purpose of this note is to announce an enumerative formula, valid under a restrictive hypothesis, for the united &-tuples of a map, i.e., a formula for the rational equivalence (or homology) class of a suitable cycle which parameterizes them. This yields as special cases formulas for the united /c-tuples which contain a fci-tuple, a /c2-tuple, etc. of mutually infinitely-near points. For our united-A>tuple cycle even to be defined, the morphism ƒ has to admit a certain kind of "resolution" (essentially it must factor through a "generic" map into a variety fibred by smooth curves over Y). Our result is sufficient, however, to yield formulas for the lines having prescribed contacts with a given projective variety having "generic" singularities and arbitrary dimension and codimension; these in turn yield formulas for the Thom-Boardman-Roberts singularity schemes [8] of a generic projection of such a variety. Classically such formulas were known for curves, for surfaces in P 3 , and in a few other cases, cf.[1]. Some recent results were obtained by Lascoux [6], Roberts [9] and LeBarz [7]. Our result yields new formulas even for surfaces in P 4 . For a modern account of these and related matters, see Kleiman's surveys [3, 5].Admittedly, the hypothesis of existence of a "resolution" is a severe restriction on the morphism ƒ. I am hopeful, however, that by pursuing further the same principles as in this paper, I will eventually obtain a united-set formula valid without such a restriction, and which would moreover be completely "intrinsic", in the sense of taking place on a suitable space associated solely to X (which is not the case with the present formula).We shall work in the category of complete (usually nonsingular) varieties over a field. Everything goes through with no change, however, in the category of compact complex manifolds.

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“…It is known (see [12]) that the dual variety V* of a nonsingular complete intersection V in P;v(C) is a hypersurface in the dual projective space PJv(C) of the hyperplanes; therefore the class of V, i.e. the degree of V*, will be obtained as the intersection number of V* with a general line of PJ(C).…”

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confidence: 99%