# zbMATH — the first resource for mathematics

Smoothness and jet schemes. (English) Zbl 1207.14023
Brasselet, Jean-Paul (ed.) et al., Singularities, Niigata-Toyama 2007. Proceedings of the 4th Franco-Japanese symposium, Niigata, Toyama, Japan, August 27–31, 2007. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-55-6/hbk). Advanced Studies in Pure Mathematics 56, 187-199 (2009).
Jet schemes over a variety encode information about its singularities. Given a smooth variety $$X$$, it is a well-known fact that the jet-schemes $$X_m$$ are non-singular for every $$m\in {\mathbb N}$$ and that every truncation morphism $$X_{m'} \longrightarrow X_m$$, for $$m'>m$$, is smooth. In this article, the author answers the question, whether these properties characterize smoothness of $$X$$, and obtains results which are even a bit stronger:
A morphism of $$k$$-schemes $$f:X \longrightarrow Y$$ is smooth if and only if there is one $$m \in {\mathbb N}$$ such that $$f_m:X_m \longrightarrow Y_m$$ is smooth. Hence a scheme $$X$$ of finite type over $$k$$ is smooth if and only if one $$X_m$$ is smooth. Using truncation morphisms, a similiar criterion is formulated for an algebraically closed field $$k$$ of characteristic zero: A scheme of finite type over $$k$$ is non-singular iff one truncation morphism $$X_{m'} \longrightarrow X_m$$ is a flat morphism. In positive characteristic, this criterion does not hold as is illustrated in an example; but with the additional condition that $$X$$ is reduced, an analogous statement is proved.
For the entire collection see [Zbl 1181.00034].

##### MSC:
 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J17 Singularities of surfaces or higher-dimensional varieties
##### Keywords:
arc spaces; jet schemes; smoothness
Full Text: