$$A$$ and the vanishing topology of discriminants

Damon, James; Mond, David

doi:10.1007/bf01243911

Cited by 69 publications

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“…In this section we will show that all singular A e -codimension-1 germs R n , 0 → R p , 0 of minimal corank have an M-deformation. This result is best possible, because for dimensions (4,5) there is a corank-1 map-germ of A e -codimension 2 that fails to have an M-deformation [18]. For n ≥ p the claim follows from the results in [17] (because all the rank p − 1 germs of positive A-modality have A e -codimension greater than one).…”

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confidence: 50%

“…We also show that in dimensions (4,5) the open Aorbit in A 3 is A-simple and consists of germs that do not have an M-deformation and also do not have a good real perturbation. This was the first example of an A-simple singular germ of minimal corank without an M-deformation (more examples will be constructed in Section 7 of the present paper).…”

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confidence: 99%

“…Then for dimensions (4,5) and also for (n, p) = (4 + 3k, 5 + 4k), all k ≥ 1, the open A-orbit in A 3 (which is not of "low multiplicity") does not have an M-deformation. The remaining results can be found in the following articles: A.3 for n ≥ p is in [17], A.3 for p = 2n is in [1,7] for n = 1 and in [10] for n ≥ 2 (and note that for p > 2n there are no 0-stable invariants, hence all germs have a trivial M-deformation), and finally A.3 for p = n + 1 and B for k = 0 are from [18].…”

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confidence: 99%

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Real deformations and invariants of map-germs

Rieger

Ruas

Atique

2008

Banach Center Publications

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Abstract. A stable deformation f t of a real map-germ f : R n , 0 → R p , 0 is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification f t C are real. A related notion is that of a good real perturbation f t of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of f t coincides with that of f t C . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good real perturbation. We show that all singular map-germs of minimal corank (i.e. of corank max(n − p + 1, 1)) and A e -codimension 1 have an M-deformation. More generally, there is the question whether all A-simple singular map-germs of minimal corank have an M-deformation. The answer is "yes" for the following three dimension ranges (n, p): n ≥ p, p ≥ 2n and p = n + 1, n = 4. We describe some new techniques for obtaining these results, which lead to simpler proofs and also to new results in the dimension range n + 2 ≤ p ≤ 2n − 1.

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Real deformations and invariants of map-germs

Rieger

Ruas

Atique

2008

Banach Center Publications

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“…where ∼ is the identification of all the origins of the intervals in the copies of [0,1]. This construction is such that X + is homotopically equivalent to a wedge of all the U j .…”

Section: The Homotopy Type Of the Disentanglementmentioning

confidence: 99%

“…It is well known that such a Milnor fibre is homotopically equivalent to a wedge of spheres. In the case of disentanglements it is known that for n ≥ p − 1 that the disentanglement is homotopically a wedge of spheres of dimension p − 1, see [1,7]. (These references give the statements and proofs for mono-germs but the multi-germ proof is practically the same.…”

Section: Introductionmentioning

confidence: 99%

Homotopy type of disentanglements of multi-germs

Houston

2009

Math. Proc. Camb. Phil. Soc.

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For a complex analytic map f from n-space to p-space with n < p and with an isolated instability at the origin, the disentanglement of f is a local stabilization of f that is analogous to the Milnor fibre for functions.For mono-germs it is known that the disentanglement is a wedge of spheres of possibly varying dimensions. In this paper we give a condition that allows us to deduce that the same is true for a large class of multi-germs.

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