F-purity versus log canonicity for polynomials

Abstract

In this article, we consider the conjectured relationship between F-purity and log canonicity for polynomials over the complex numbers. We associate to a collection M of n monomials a rational polytope P contained in [0,1]^n. Using P and the Newton polyhedron associated to M, we define a non-degeneracy condition under which log canonicity and dense F-pure type are equivalent for all linear combinations of the monomials in M. We also show that log canonicity corresponds to F-purity for very general polynomials. Our methods rely on showing that the F-pure and log canonical threshold agree for infinitely primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial ideals.

20 pages, 1 figure

References in corpus (1)

• F-thresholds and Bernstein-Sato polynomials

Cited by in corpus (6)

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• F-pure thresholds of binomial hypersurfaces
• On the F-purity of isolated log canonical singularities
• F-purity of hypersurfaces
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms