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Mathematics > Algebraic Geometry

arXiv:1812.00288 (math)
[Submitted on 1 Dec 2018]

Title:On the Nash problem for surfaces in positive characteristic

Authors:Augusto Nobile
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Abstract:This paper seeks to prove the bijectivity of the "Nash mapping" from the set of irreducible components of the scheme parametrizing analytic arcs on an algebraic surface $X$ whose origin is a singular point, into the set of irreducible components of the exceptional locus of a minimal desingularization $X'$ of $X$ when the base field has positive characteristic. The idea is to view the surface as a specialization of another defined over a field of characteristic zero. A number of related results are proved. Among them, the construction of a scheme of arcs for a suitable one parameter family of surfaces which, by using a theorem of M. Artin on lifting of normal surface singularities to characteristic zero, seems a reasonable candidate to be the desired tool. But there are some points, necessary for a complete proof, which are not verified yet.
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14E18, 14E15, 14B05, 14B07, 14E99, 14J99
Cite as: arXiv:1812.00288 [math.AG]
  (or arXiv:1812.00288v1 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1812.00288

Submission history

From: Augusto Nobile [view email]
[v1] Sat, 1 Dec 2018 23:32:04 UTC (18 KB)
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