Jet schemes over auto-arc spaces and deformations of locally complete intersection varieties

Abstract

This paper concerns the study of certain reduced generalized jet schemes LZ(Y) which naturally live over another type of generalized jet scheme termed the auto-arc space of Z. Auto-arc spaces were originally introduced by H. Schoutens and later studied by the author. The structure of generalized jet schemes and in particular auto-arc spaces are difficult to understand even after applying reduction. The major advance in this work is obtained by considering reduced generalized jet schemes LZ(Y) along Z of a infinitesimal deformation Y of a complete intersection variety X over the same fat point Z. Some well-known results of Mustaţǎ on classical jet schemes of locally complete intersections carry over to this case. For example, as a consequence of miracle flatness and some generalizations of the aforementioned results of Mustaţǎ, it is shown that the reduced generalized jet space, denoted Ln(Xn) in this case, of a flat deformation Xn→Dn along the so-called linear jets ▪ can be viewed as a global flat deformation over Akn of the classical jet scheme of order n provided the base scheme X0 is a locally complete intersection. Also, the initiation of the so-called regulated defect of a formal deformation in analogy to log canonical threshold is introduced. In the end, the situation is studied when the base scheme X0 is an algebraic curve, and the notion of so-called strong/weak deformations are introduced in this context. Finally, a so-called deformed motivic volume is defined.