Journal of the Mathematical Society of Japan

Finite formal model of toric singularities

David BOURQUI and Julien SEBAG

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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.

Article information

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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Mathematical Reviews number (MathSciNet)

Primary: 14B20: Formal neighborhoods 14E18: Arcs and motivic integration 14M25: Toric varieties, Newton polyhedra [See also 52B20]

arc schemes formal neighborhoods toric varieties


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.

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