Explicit Deligne pairing

TL;DR

This paper provides an explicit formula for the Deligne pairing in algebraic geometry, relating it to the determinant of cohomology and drawing an analogy with intersection theory on smooth projective varieties.

Contribution

It introduces a concrete formula for the Deligne pairing using determinant of cohomology, enhancing understanding of its explicit computation.

Findings

Explicit formula for Deligne pairing in terms of determinant of cohomology

Justification via analogy with intersection theory on smooth projective varieties

Clarification of the construction for proper, flat morphisms

Abstract

We give an explicit formula for the Deligne pairing for a proper and flat morphisms $f:X\to S$ of schemes, in terms of the determinant of cohomology. The whole construction is justified by an analogy with the intersection theory on non-singular projective algebraic varieties.