Generalized Plücker Formulas

Thorup, Anders

doi:10.1007/978-1-4612-1316-1_11

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Further generalisations, for arbitrary projective varieties with isolated singularities, and then without conditions on singularities, were found notably by Kleiman, Pohl and respectively Thorup, see e.g. [15].…”

Section: A General Plücker-type Formula For Affine Hypersurfacesmentioning

confidence: 97%

The Gauss–Bonnet defect of complex affine hypersurfaces

Siersma

Tibăr²

2006

Bulletin des Sciences Mathématiques

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Section: A General Plücker-type Formula For Affine Hypersurfacesmentioning

confidence: 97%

The Gauss–Bonnet defect of complex affine hypersurfaces

Siersma

Tibăr²

2006

Bulletin des Sciences Mathématiques

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“…(See [10] and [19].) Let X be a subvariety (or closed, reduced and irreducible subscheme) of dimension d of the projective m-space P m over an algebraically closed ground field of arbitrary characteristic.…”

Section: Projective Geometrymentioning

confidence: 99%

“…Assume that x is an isolated singular point of X (see [19] for a more general discussion). Let F be the fiber of C X over x.…”

Section: Projective Geometrymentioning

confidence: 99%

Conormal Geometry of Maximal Minors

Kleiman¹,

Thorup

2000

Journal of Algebra

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Let A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A N be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A M be the subalgebra generated by M. Set C = Proj A M and r = dim C. Form the (closed) subset W of Spec A of primes p where A N p is not a finitely generated module over A M p , and denote the preimage of W in C by E. We prove this: (1) dim E = r − 1 if either (a) N is free and A N is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thom's A f -condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of Böger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A N is a finitely generated module over A M .

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Formulas for the multiplicity of graded algebras

Xie

2012

Trans. Amer. Math. Soc.

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Let R be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of R by means of local j-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion A ⊆ B of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of A in terms of that of B and of local j-multiplicities of hyperplane sections along Proj (B). Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier's Plücker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

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Generalized Plücker Formulas

Cited by 3 publications

References 9 publications

The Gauss–Bonnet defect of complex affine hypersurfaces

The Gauss–Bonnet defect of complex affine hypersurfaces

Conormal Geometry of Maximal Minors

Formulas for the multiplicity of graded algebras

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