Familles d'espaces de m-jets et d'espaces d'arcs

Abstract

Families of m-jet spaces and arc spaces. Let V be an algebraic variety defined over an algebraically closed field of characteristic zero. The m-jet spaces and the arc space provide the information on the geometry of the variety V, therefore it is natural to ask the following questions: When will a deformation of V induce a deformation of the m-jet spaces and the arc space? If one considers a deformation of V which admits a flat simultaneous resolution, how will the image of the Nash application vary? In section 3 some partial answers to these questions can be found. In section 4 two families of hypersurfaces of the 4-dimensional affine space are shown in which the Nash application is bijective. What's more, in each case a desingularization in which all the irreducible components of the exceptional fiber are essential divisors is explicitly given. It is important to note that very few examples of this type are found in the literature.