The arc space of a toric variety. (English) Zbl 1073.14066

Ishii, Shihoko

J. Algebra 278, No. 2, 666-683 (2004).

The main objects of the paper under review are jet schemes \(X_m\) and the arc space \(X_{\infty}\) of a toric variety \(X\) defined over an algebraically closed field \(k\) of arbitrary characteristic. The author proves that \(X_{\infty}\) is irreducible and contains the arc space \(T_{\infty}\) as an open orbit. Every component of \(\pi^{-1}(\text{Sing} X)\), where \(\pi\colon X_{\infty}\to X\) stands for the natural projection, is a good component in sense of S. Ishii and J. Kollár [Duke Math. J. 120, 601–620 (2003; Zbl 1052.14011)]. Each \(T_{\infty}\)-orbit on \(X_{\infty}\) is associated to a lattice point, and a description of arising dominant relation of two orbits is given in terms of the corresponding lattice points. As a corollary, the author obtains an answer to the embedded version of the Nash problem for an invariant ideal of the coordinate ring of \(X\).

Reviewer: Boris Kunyavskii (Ramat Gan)

MSC:

14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14L15	Group schemes

Keywords:

Nash problem; jet schemes

Citations:

Zbl 1052.14011

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