## Journal of the Mathematical Society of Japan

### Finite formal model of toric singularities

#### Abstract

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

Source
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 805-829.

Dates
Revised: 6 February 2018
First available in Project Euclid: 10 May 2019

https://projecteuclid.org/euclid.jmsj/1557475349

Digital Object Identifier
doi:10.2969/jmsj/78927892

Mathematical Reviews number (MathSciNet)
MR3984243

#### Citation

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

#### References

• [1] D. Bourqui and J. Sebag, The Drinfeld–Grinberg–Kazhdan theorem for formal schemes and singularity theory, Confluentes Math., 9 (2017), 29–64.
• [2] D. Bourqui and J. Sebag, The local structure of arc schemes, to appear in the proceedings of the conference, Arc schemes and singularities, World Scientific.
• [3] D. Bourqui and J. Sebag, The minimal formal models of curve singularities, Internat. J. Math., 28 (2017), 1750081, 23pp.
• [4] D. Bourqui and J. Sebag, Smooth arcs on algebraic varieties, J. Singul., 16 (2017), 130–140.
• [5] A. Bouthier and D. Kazhdan, Faisceaux pervers sur les espaces d'arcs, preprint, arXiv:1509.02203v5, 2017.
• [6] A. Bouthier, B. C. Ngô and Y. Sakellaridis, On the formal arc space of a reductive monoid, Amer. J. Math., 138 (2016), 81–108.
• [7] C. Bouvier, Diviseurs essentiels, composantes essentielles des variétés toriques singulières, Duke Math. J., 91 (1998), 609–620.
• [8] C. Bouvier and G. Gonzalez-Sprinberg, Système générateur minimal, diviseurs essentiels et $G$-désingularisations de variétés toriques, Tohoku Math. J. (2), 47 (1995), 125–149.
• [9] A. Campillo and J. Castellanos, Curve singularities, an algebraic and geometric approach, Actualités mathématiques, Hermann, 2005.
• [10] D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, 124, Amer. Math. Soc., Providence, RI, 2011.
• [11] T. de Fernex, L. Ein and S. Ishii, Divisorial valuations via arcs, Publ. Res. Inst. Math. Sci., 44 (2008), 425–448.
• [12] V. Drinfeld, On the Grinberg–Kazhdan formal arc theorem, to appear in the proceedings of the conference, Arc schemes and singularities, World Scientific.
• [13] L. Ein and S. Ishii, Singularities with respect to Mather–Jacobian discrepancies, Commutative algebra and noncommutative algebraic geometry, II, Math. Sci. Res. Inst. Publ., 68, Cambridge Univ. Press, New York, 2015, 125–168.
• [14] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton Univ. Press, Princeton, NJ, 1993.
• [15] M. Grinberg and D. Kazhdan, Versal deformations of formal arcs, Geom. Funct. Anal., 10 (2000), 543–555.
• [16] S. Ishii, The arc space of a toric variety, J. Algebra, 278 (2004), 666–683.
• [17] S. Ishii, Mather discrepancy and the arc spaces, Ann. Inst. Fourier (Grenoble), 63 (2013), 89–111.
• [18] S. Ishii and J. Kollár, The Nash problem on arc families of singularities, Duke Math. J., 120 (2003), 601–620.
• [19] S. Ishii and A. J. Reguera, Singularities with the highest Mather minimal log discrepancy, Math. Z., 275 (2013), 1255–1274.
• [20] J. M. Johnson and J. Kollár, Arc spaces of $cA$-type singularities, J. Singul., 7 (2013), 238–252.
• [21] S. Lang, Algebra, third ed., Grad. Texts in Math., 211, Springer-Verlag, New York, 2002.
• [22] H. Mourtada and A. J. Reguera, Mather discrepancy as an embedding dimension in the space of arcs, Publ. Res. Inst. Math. Sci., 54 (2018), 105–139.
• [23] J. F. Nash, Jr., Arc structure of singularities, Duke Math. J., 81 (1995), 31–38.
• [24] B. C. Ngô, Weierstrass preparation theorem and singularities in the space of non-degenerate arcs, preprint, arXiv:1706.05926, 2017.
• [25] A. J. Reguera, Towards the singular locus of the space of arcs, Amer. J. Math., 131 (2009), 313–350.
• [26] J. Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France, 132 (2004), 1–54.
• [27] B. Teissier, Complex curve singularities: a biased introduction, Singularities in geometry and topology, World Sci. Publ., Hackensack, NJ, 2007, 825–887.