Ishii, Shihoko The arc space of a toric variety. (English) Zbl 1073.14066 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) Cited in 1 ReviewCited in 22 Documents MSC: 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14L15 Group schemes Keywords:Nash problem; jet schemes Citations:Zbl 1052.14011 PDF BibTeX XML Cite \textit{S. Ishii}, J. Algebra 278, No. 2, 666--683 (2004; Zbl 1073.14066) Full Text: DOI arXiv OpenURL References: [1] Batyrev, V, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, (), 1-32 · Zbl 0963.14015 [2] Blickle, M, Multiplier ideals and modules on toric varieties, Math. Z., in press · Zbl 1061.14055 [3] Danilov, V.I, The geometry of toric varieties, Russian math. surveys, 33, 2, 97-154, (1978) · Zbl 0425.14013 [4] Denef, J; Loeser, F, Germs of arcs on singular varieties and motivic integration, Invent. math., 135, 201-232, (1999) · Zbl 0928.14004 [5] Ein, L; Mustaţǎ, M; Yasuda, T, Jet schemes, log discrepancies and inversion of adjunction, Invent. math., 153, 519-535, (2003), Preprint · Zbl 1049.14008 [6] Ein, L; Lazarsfeld, R; Mustaţǎ, M, Contact loci in arc spaces, Comput. Math., in press · Zbl 1060.14004 [7] Fulton, W, Introduction to toric varieties, Ann. of math. stud., 131, (1993) [8] Ishii, S; Kollár, J, The Nash problem on arc families of singularities, Duke math. J., 120, 601-620, (2003), Preprint · Zbl 1052.14011 [9] Kolchin, E.R, Differential algebra and algebraic groups, Pure and appl. math., vol. 54, (1973), Academic Press New York · Zbl 0264.12102 [10] Mustaţǎ, M, Jet schemes of locally complete intersection canonical singularities, Invent. math., 145, 397-424, (2001), with an appendix by David Eisenbud and Edward Frenkel · Zbl 1091.14004 [11] Mustaţǎ, M, Singularities of pairs via jet schemes, J. amer. math. soc., 15, 599-615, (2002) · Zbl 0998.14009 [12] Nash, J.F, Arc structure of singularities, Duke math. J., 81, 31-38, (1995) · Zbl 0880.14010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.