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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].

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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