The Vanishing Topology of Non Isolated Singularities

Siersma, Dirk

doi:10.1007/978-94-010-0834-1_18

Cited by 41 publications

(43 citation statements)

References 38 publications

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“…The analysis of the vanishing cycles in the case of 1-dimensional singular locus is relatively well understood (see [44], [26,II.8.10]). Specifically, we stratify C D † 1 [ † 0 such that the vanishing cycles form a local system over † 1 .…”

Section: The Monodromy Around Semistable Cubic Fourfoldsmentioning

confidence: 99%

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The moduli space of cubic fourfolds via the period map

Laza¹

2010

Ann. Math.

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We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the geometric invariant theory (GIT) compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga compactification associated to this arrangement. This paper builds on and is a natural continuation of our previous work on the GIT compactification of the moduli space of cubic fourfolds.

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Section: The Monodromy Around Semistable Cubic Fourfoldsmentioning

confidence: 99%

“…Since the vertical monodromy, given by the natural action of 1 . V C i / is nontrivial on each component (see [44,Ch. 3]), there are no nonzero sections of the local system over † 1 .…”

Section: The Monodromy Around Semistable Cubic Fourfoldsmentioning

confidence: 99%

The moduli space of cubic fourfolds via the period map

Laza¹

2010

Ann. Math.

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“…One can find examples of this in the literature (see, for instance, [24], [19], [26]); perhaps the easiest is the following:…”

Section: Comments Questions and Counterexamplesmentioning

confidence: 99%

“…It is well-known (see [9]) that the reduced integral homology, H k (F f ), of F f can be non-zero only for n − s ≤ k ≤ n, and is free Abelian in degree n. Cohomologically, this means that H k (F f ) can be non-zero only for n − s ≤ k ≤ n, and is free Abelian in degree n − s. For a general reference to non-isolated hypersurface singularities, see [26].…”

Section: Introductionmentioning

confidence: 99%

Hypersurface Singularities and Milnor Equisingularity

Massey¹,

Tráng²

2006

Pure and Applied Mathematics Quarterly

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“…Specifically, we consider round functions on surfaces (two-dimensional smooth manifolds) and 3-manifolds (three-dimensional smooth manifolds) and their local behaviour near critical submanifolds. Some of these results are rather simple and we do not exclude that they may be known for the experts or even belong to "mathematical folklore", but we have good evidence to believe that in any case our presentation contains certain novelties arising from the treatment of round functions from the singularity theory viewpoint in the spirit of isolated line singularities [27], [28].…”

Section: Introductionmentioning

confidence: 98%

Remarks on minimal round functions

Khimshiashvili

Siersma

2003

Geometry and Topology of Caustics – Caustics '02

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Abstract. We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions. Introduction.A differentiable function on a differentiable manifold is called a round function if its critical set is a union of embedded one-dimensional submanifolds. In general it is not assumed that the critical submanifolds are non-degenerate critical submanifolds in the sense of R. Bott [3] so this is a straightforward extension of the notion of round Morse function introduced by W. Thurston [30].Our main concern in this paper are round functions on a given compact closed (i.e., without boundary) smooth manifold with the minimal possible number of critical submanifolds (critical loops). By analogy with the terminology of [29] they will be called minimal round functions as in [21]. We are especially interested in describing possible changes of the topology of their Lebesgue sets (preimages of semi-infinite intervals) and typical local models of their singular behaviour near critical loops. Our setting and approach are much in the spirit of F. Takens' paper [29] which contains a comprehensive

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The Vanishing Topology of Non Isolated Singularities

Cited by 41 publications

References 38 publications

The moduli space of cubic fourfolds via the period map

The moduli space of cubic fourfolds via the period map

Hypersurface Singularities and Milnor Equisingularity

Remarks on minimal round functions

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