Measuring Singularities with Frobenius: The Basics

2013/09/18 by Benito, Angélica, Faber, Eleonore, Smith, Karen E.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.4814

Abstract

Consider a polynomial f defined over a field k, the multiplicity is perhaps the most naive measurement of the singularities of f. This paper describes the first steps toward understanding a much more subtle measure of singularities which arises naturally in three different contexts-- analytic, algebro-geometric, and finally, algebraic. Miraculously, all three approaches lead to essentially the same measurement of singularities: the log canonical threshold (in characteristic zero) and the closely related F-pure threshold (in characteristic p). In this paper we present only the first steps in understanding these invariants, with an emphasis on the prime characteristic setting.

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