A New Conjecture, a New Invariant, and a New Non-splitting Result

Massey, David B.

doi:10.1007/978-3-319-39339-1_10

Cited by 7 publications

(5 citation statements)

References 8 publications

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“…The following numerical invariant, defined and discussed in [6], is crucial to the contents and goal of this paper.…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…Using Proposition 2.4, β f may be equivalently expressed as It is shown in [6] that β f ≥ 0. The interesting question is how strong the requirement that β f = 0 is.…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…We approach Conjecture 1.2 via the beta invariant of a hypersurface with a 1-dimensional critical locus, first defined and explored by the second author in [6]. The beta invariant, β f , of f is an invariant of the local ambient topological-type of the hypersurface V (f ).…”

Section: Introductionmentioning

confidence: 99%

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Some special cases of Bobadilla's conjecture

Hepler

Massey

2017

Topology and its Applications

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We prove two special cases of a conjecture of J. Fernández de Bobadilla for hypersurfaces with 1-dimensional critical loci.We do this via a new numerical invariant for such hypersurfaces, called the beta invariant, first defined and explored by the second author in 2014. The beta invariant is an algebraically calculable invariant of the local ambient topological-type of the hypersurface, and the vanishing of the beta invariant is equivalent to the hypotheses of Bobadilla's conjecture.

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“…The following numerical invariant, defined and discussed in [6], is crucial to the contents and goal of this paper.…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…Using Proposition 2.4, β f may be equivalently expressed as It is shown in [6] that β f ≥ 0. The interesting question is how strong the requirement that β f = 0 is.…”

Section: Notation and Known Resultsmentioning

confidence: 99%

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Some special cases of Bobadilla's conjecture

Hepler

Massey

2017

Topology and its Applications

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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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“…We have written about Bobadilla's Conjecture twice before, in [4] and [2]. The conjecture is: Conjecture 6.4.…”

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confidence: 99%

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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A New Conjecture, a New Invariant, and a New Non-splitting Result

Cited by 7 publications

References 8 publications

Some special cases of Bobadilla's conjecture

Some special cases of Bobadilla's conjecture

The Weight Filtration on the Constant Sheaf on a Parameterized Space

Non-isolated hypersurface singularities and Lê cycles

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