Parametrized monomial surfaces in 4-space

Hernandes, M. E.; Ruas, María Aparecida Soares

doi:10.1093/qmath/hay052

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…In [13] the authors consider the invariant δ f := dim C On f * (Op) called the delta invariant of f . Notice that L e cod(f ) = p • δ f .…”

Section: Basic Notations Of Singularitiesmentioning

confidence: 99%

“…Parameterized singular surfaces in R 4 appear in [2], where Birbrair, Mendes and Nuño-Ballesteros study relations between topological and metric properties, and also in [1] where Benedini, Ruas and Sinha study their second order geometry. In [13], the authors classify A-finitely determined monomial surfaces in C 4 and so many others.…”

Section: Introductionmentioning

confidence: 99%

“…As a corollary of Theorem 4.4 we obtain infinitely many families of A-finite map-germs, not necessarily monomial. Two fundamental tools to obtain the previous results were the invariants: the semigroup associated to a parameterized curve and the delta invariant δ f := dim C On f * (Op) , defined in [13]. In Section 5, our focus is to study some A-invariants of map-germs f : (C n , 0) → (C p , 0) for p ≥ 2n.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Join operation and ${\mathcal A}$-finite map-germs

Hernandes¹,

Ruas²

2021

Preprint

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“…In [13] the authors consider the invariant δ f := dim C On f * (Op) called the delta invariant of f . Notice that L e cod(f ) = p • δ f .…”

Section: Basic Notations Of Singularitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Join operation and ${\mathcal A}$-finite map-germs

Hernandes¹,

Ruas²

2021

Preprint

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“…The classification of A-finite monomial map-germs (C 2 , 0) → (C 4 , 0) has been carried out by Rodrigues Hernandes and Ruas [18]. The list consists entirely of reflection maps of the form (x, y) → (x, x λ , y n , y m ).…”

Section: Final Remarksmentioning

confidence: 99%

Reflection maps

Sanchis

2020

Math. Ann.

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Given a reflection group G acting on a complex vector space V , a reflection map is the composition of an embedding X → V with the quotient map V → C p of G. We show how these maps, which can highly singular, may be studied in terms of the group action. We give obstructions to A-stability and A-finiteness of reflection maps and produce, in the unobstructed cases, infinite families of finitely determined map-germs of any corank. We relate these maps to conjectures of Lê, Mond and Ruas.with f 0 = f . Un unfolding F is called A-trivial if it is A-equivalent to id C r ×f as an unfolding (that is, A-equivalent via unfoldings of the identity maps on (C n , S) and (C p , 0)).Since no equivalence relation other than A-equivalence is used and we do not study global stability, here local A-stability is just called stability.

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“…We note that the surface (X, 0) of Example 5.4 (c) can also be seen as a toric surface parameterized by a monomial finitely determined map germ from C 2 to C 4 (see [11,Theorem 1 ]). Thus, using the results in [11], one can produce other examples of this kind. For the computations in the examples we have made use of the software Singular [14].…”

Section: Strong Simultaneous Resolutionmentioning

confidence: 99%

Whitney equisingularity in families of generically reduced curves

Silva

Snoussi

2019

manuscripta math.

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In this work we study equisingularity in a one-parameter flat family of generically reduced curves. We consider some equisingular criteria as topological triviality, Whitney equisingularity and strong simultaneous resolution. In this context, we prove that Whitney equisingularity is equivalent to strong simultaneous resolution and it is also equivalent to the constancy of the Milnor number and the multiplicity of the fibers. These results are extensions to the case of flat deformations of generically reduced curves, of known results on reduced curves. When the family (X, 0) is topologically trivial, we also characterize Whitney equisingularity through Cohen-Macaulay property of a certain local ring associated to the parameter space of the family.

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Parametrized monomial surfaces in 4-space

Abstract: In this work, we classify parametrized monomial surfaces f:(ℂ2,0)→(ℂ4,0) that are A-finitely determined. We study invariants that can be obtained in terms of invariants of a parametrized curve.

Cited by 4 publications

References 8 publications

Join operation and ${\mathcal A}$-finite map-germs

Join operation and ${\mathcal A}$-finite map-germs

Reflection maps

Whitney equisingularity in families of generically reduced curves

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