Two algorithms for computing the general component of jet scheme and applications

Abstract

Let k be a perfect field. Let X be an integral k-variety. Let m∈N. In this article, we study, from the theoretical and computational points of view, the component Gm(X) of the jet scheme Lm(X) defined to be the Zariski closure of the set of truncated arcs with a regular base-point. We prove that Gm(X) can be described from any smooth birational model of X. When X is supposed to be affine embedded in AkN, this description allows us to provide an algorithm, valid in arbitrary characteristic, which computes a Gröbner basis of a presentation of Gm(X) in Ak(m+1)N from the datum of a given explicit smooth affine birational model of X. Then, we show how to use the datum of a presentation of Gm(X) (in the suitable affine space) to deduce algebraic and geometric properties in two contexts from the theoretical and computational points of view. Firstly, we apply our main result to obtain bases for a class of homogeneous differential operators logarithmic along plane curves in arbitrary characteristics. Secondly, we introduce a motivic power series which encodes the geometry of all the Gm(X) and prove its rationality in the specific case of homogeneous plane curve singularities thanks to our description of Gm(X)via the normalization of X.