The jet schemes have been introduced by J. Nash. They play important role in many results about the singularities of an algebraic variety and their invariants. One could ask the following question: having a property satisfied by a jet scheme \(X_n\), does it hold for the base scheme \(X\)?
In the article under review are considered some questions of this kind with properties taken to be locally complete intersection, \(\mathbb{Q}\)-factorial, \(\mathbb{Q}\)-Gorenstein, canonical, or terminal. After some preparatory material on jet schemes and their canonical sheaves (in the smooth case), it is proved for a jet scheme \(X_n\) having any one of these properties, that the base scheme will inherit it as well. Continuing in that direction it is shown also that in \(\text{char\,}k= 0\), if \(X_n\) has log-canonical singularities at worst, then \(X\) has at worst log-terminal singularities. At the end is cosidered the same kind of question about morphisms, and is shown for a morphism of schemes \(f\) that if the induced morphism of jet schemes \(f_n\) is flat, then \(f\) itself is flat, but the converse does not hold in general. Everywhere in the article are given useful and illuminating examples.