Two algorithms for computing the general component of jet scheme and applications (Q2133916)

Let \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\). If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\). Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7. The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9).