Reflection maps

Abstract

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.

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

“…More precisely, in Theorem 4.4, we prove that if f is a corank q monomial map-germ, then f is A-finite if and only if f is A-equivalent to the join of the identity in C n−q with q parameterized curves. These monomial normal forms can be seen as A-finite reflection maps, as defined in [12]. As a corollary of Theorem 4.4 we obtain infinitely many families of A-finite map-germs, not necessarily monomial.…”

On the topology of the transversal slice of a quasi-homogeneous map germ

“…Each g i is a homogeneous finitely determined map germ of corank 2, of same type, (2, 3, 5; 1, 1) (see [22,Ex. 16] and [23,Ex.…”

Superfícies com singularidades não isoladas

“…Uma aplicação desta forma é conhecida como aplicação de reflexão. Estas aplicações foram estudadas recentemente por Peñafort-Sanchis no interessante trabalho [48]. Nesse artigo, são obtidos novos importantes resultados sobre determinação finita e estabilidade.…”

“…Nesse artigo, são obtidos novos importantes resultados sobre determinação finita e estabilidade. A seguir, vamos usar um método diferente do apresentado em [48] para estudar as aplicações da forma f (x, y) = (x n , y m , p(x, y)), seguindo de perto as ideias do trabalho de Marar e Nuño-Ballesteros em [33].…”

“…Observação 4.32. Em [48], Peñafort-Sanchis mostrou que se n, m, k são coprimos em pares, então o germe de aplicação…”