Characteristic cycle of a rank one sheaf and ramification theory

TL;DR

This paper computes the characteristic cycle of a rank one sheaf on a smooth surface in positive characteristic, linking it with ramification theory and Kato's logarithmic characteristic cycle.

Contribution

It introduces a canonical lifting of the characteristic cycle on the cotangent bundle and proves its equality with the characteristic cycle, advancing ramification theory in algebraic geometry.

Findings

Computed the characteristic cycle of a rank one sheaf on a smooth surface.

Established the equality between the characteristic cycle and the canonical lifting.

Connected the singular support with ramification theory.

Abstract

We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato's logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory.