## 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 $$X$$.

### MSC:

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

### Keywords:

Nash problem; jet schemes

Zbl 1052.14011
Full Text:

### References:

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