Families of higher dimensional germs with bijective Nash map

Garside families and Garside germs

Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various criteria for establishing the existence of normal decompositions of a certain type. In 1969, F.A.Garside...

A Classification of 1-parameter Families of Map Germs U\ 0 -> 1r,0 with Applications to Condensation Problems

In his seminal paper [1] Arnold discusses, amongst other things, the problem of the evolution of galaxies. The basic model he considers is the following. Consider a medium of non-interacting particles in IR with an initial velocity distribution v = v(x) and a density distribution p = p(x), where v and p are a smooth vector field and a smooth function, respectively, on IR. The inertial motion of...

Flag Higher Nash Blowups

In his previous paper [5], the author has defined a higher version of the Nash blowup and considered it a possible candidate for the one-step resolution. In this paper, we will introduce another higher version of the Nash blowup and prove that it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions.

Higher Nash Blowups

For each non-negative integer n, we define the n-th Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n = 1, it coincides with the classical Nash blowup. We prove that sufficiently high Nash blowups separate analytic branches. We also determine for a monomial curve in characteristic zero when its higher Nash blowups are smooth.

Linearization of families of germs of hyperbolic vector fields