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

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 \(X\).


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


Zbl 1052.14011
Full Text: DOI arXiv


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