Abhyankar’s Local Conjecture on Fundamental Groups

David Harbater and Katherine F. Stevenson

A chapter in Algebra, Arithmetic and Geometry with Applications, 2004, pp 473-485 from Springer

Abstract: Abstract This paper proves the remaining open case of Abhyankar’s higher dimensional conjecture on local fundamental groups in characteristic p ([Ab2], [Ab3]). This conjecture, which is analogous to Abhyankar’s conjectures on global fundamental groups, proposed that a finite group G is a Galois group over k[[x 1,…, x n]][(x 1… x r)-1] if and only if its maximal prime-to-p quotient is, provided n≥ 2 and 1 ≤ r ≤ n. For r > 1, this conjecture was disproven in [HP]. Here we prove that the conjecture is true in the case r = 1. So the Galois groups over k[[x 1,…, x n]][x 1 -11] are precisely the cyclic-by-quasi-p groups.

Keywords: Generic Point; Fundamental Group; Galois Group; Exceptional Divisor; Galois Cover (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-18487-1_26

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DOI: 10.1007/978-3-642-18487-1_26

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