Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 71, Number 3 (2019), 805-829.
Finite formal model of toric singularities
We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 805-829.
Received: 4 October 2017
Revised: 6 February 2018
First available in Project Euclid: 10 May 2019
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BOURQUI, David; SEBAG, Julien. Finite formal model of toric singularities. J. Math. Soc. Japan 71 (2019), no. 3, 805--829. doi:10.2969/jmsj/78927892. https://projecteuclid.org/euclid.jmsj/1557475349