Some special cases of Bobadilla's conjecture

Hepler, Brian; Massey, David B.

doi:10.1016/j.topol.2016.12.011

Cited by 4 publications

(9 citation statements)

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“…Hence, in this example µ(f [2] ) > λ 0 f,z . For this example, the inequality in Theorem 4.1 gives us: [2] ) − µ(f [1] ) + 1 λ 1 f,z µ(f [2] ) ≥ µ(f [2] ) µ(f [1] ) = 4 2 = 2.…”

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 71%

“…(Teissier, [11]) Suppose that s = 0. Then the following inequalities hold: [2] ) µ(f [1] ) ≥ µ(f [1] ) µ(f [0] ) .…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…There we found λ 1 f,z = 3 and λ 0 f,z = 2. Now, as f is homogeneous degree 3 with a 1-dimensional critical locus, f [2] and f [1] have isolated critical points and are also homogeneous of degree 3; thus µ(f [2] ) = (3 − 1) 2 = 4 and µ(f [1] ) = (3 − 1) = 2. Hence, in this example µ(f [2] ) > λ 0 f,z .…”

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 99%

“…Now, as f is homogeneous degree 3 with a 1-dimensional critical locus, f [2] and f [1] have isolated critical points and are also homogeneous of degree 3; thus µ(f [2] ) = (3 − 1) 2 = 4 and µ(f [1] ) = (3 − 1) = 2. Hence, in this example µ(f [2] ) > λ 0 f,z . For this example, the inequality in Theorem 4.1 gives us: [2] ) − µ(f [1] ) + 1 λ 1 f,z µ(f [2] ) ≥ µ(f [2] ) µ(f [1] ) = 4 2 = 2.…”

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 99%

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Non-isolated hypersurface singularities and Lê cycles

Massey¹

2016

Real and Complex Singularities

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Abstract. In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities.In Lecture 1, I will review basic definitions and results on complex hypersurfaces, and then present classical material on the Milnor fiber and fibration. In Lecture 2, I will present basic results from Morse theory, and use them to prove some results about complex hypersurfaces, including a proof of Lê's attaching result for Milnor fibers of non-isolated hypersurface singularities. This will include defining the relative polar curve. Lecture 3 will begin with a discussion of intersection cycles for proper intersections inside a complex manifold, and then move on to definitions and basic results on Lê cycles and Lê numbers of non-isolated hypersurface singularities. Lecture 4 will explain the topological importance of Lê cycles and numbers, and then I will explain, informally, the relationship between the Lê cycles and the complex of sheaves of vanishing cycles.

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“…Hence, in this example µ(f [2] ) > λ 0 f,z . For this example, the inequality in Theorem 4.1 gives us: [2] ) − µ(f [1] ) + 1 λ 1 f,z µ(f [2] ) ≥ µ(f [2] ) µ(f [1] ) = 4 2 = 2.…”

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 71%

“…(Teissier, [11]) Suppose that s = 0. Then the following inequalities hold: [2] ) µ(f [1] ) ≥ µ(f [1] ) µ(f [0] ) .…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 99%

Section: Minkowski Inequalities: the 1-dimensional Casementioning

confidence: 99%

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Non-isolated hypersurface singularities and Lê cycles

Massey¹

2016

Real and Complex Singularities

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“…Question 6.2. One notes that the formula for the dimension of the vector space [2]) 0 in Theorem 5.1 is very similar to the beta invariant, β f , of a hypersurface V (f ) with one-dimensional singular locus (defined by David Massey in [12], and further explored by the author and Massey in [10]).…”

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confidence: 93%

The Weight Filtration on the Constant Sheaf on a Parameterized Space

Hepler

2018

Preprint

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On an n-dimensional locally reduced complex analytic space X on which the shifted constant sheaf Qunderlies a mixed Hodge module of weight ≤ n on X, with weight n graded piece isomorphic to the intersection cohomology complex IC • X with constant Q coefficients. In this paper, we identify the weight n − 1 graded piece Grin the case where X is a "parameterized space", using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted constant sheaf Q • X [n] is perverse. In the case where X is a parameterized surface, we can completely determine the remaining terms in the weight filtration on Q, where we also show that the weight filtration is a local topological invariant of X. These examples arise naturally as affine toric surfaces in C 3 , images of finitely-determined maps from C 2 to C 3 , as well as in a well-known conjecture of Lê Dũng Tráng regarding the equisingularity of parameterized surfaces in C 3 .

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Topological Equisingularity: Old Problems from a New Perspective (with an Appendix by G.-M. Greuel and G. Pfister on SINGULAR)

Bobadilla

2022

Handbook of Geometry and Topology of Singularities III

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Some special cases of Bobadilla's conjecture

Cited by 4 publications

References 10 publications

Non-isolated hypersurface singularities and Lê cycles

Non-isolated hypersurface singularities and Lê cycles

The Weight Filtration on the Constant Sheaf on a Parameterized Space

Topological Equisingularity: Old Problems from a New Perspective (with an Appendix by G.-M. Greuel and G. Pfister on SINGULAR)

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