-PURITY VERSUS LOG CANONICITY FOR POLYNOMIALS

Abstract

In this article, we consider the conjectured relationship between $F$-purity and log canonicity for polynomials over $\mathbb{C}$. In particular, we show that log canonicity corresponds to dense $F$-pure type for all polynomials whose supporting monomials satisfy a certain nondegeneracy condition. We also show that log canonicity corresponds to dense $F$-pure type for very general polynomials over $\mathbb{C}$. Our methods rely on showing that the $F$-pure and log canonical thresholds agree for infinitely many primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial term ideals.