The minimal log discrepancies on a smooth surface in positive characteristic

TL;DR

This paper proves Mustata-Nakamura's conjecture for smooth surfaces with multiideals in positive characteristic, establishing key properties of minimal log discrepancies and log canonical thresholds.

Contribution

It demonstrates the conjecture's validity in positive characteristic for smooth surfaces and derives related finiteness and chain condition results.

Findings

Mustata-Nakamura's conjecture holds for smooth surfaces in positive characteristic.

Established the ascending chain condition for minimal log discrepancies and log canonical thresholds.

Proved finiteness of minimal log discrepancies set for fixed real exponents.

Abstract

This paper shows that Mustata-Nakamura's conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.