The index of an algebraic variety

TL;DR

This paper provides explicit methods to compute the index of a smooth proper algebraic variety over a field using data from its special fiber, introducing new invariants and proofs involving moving lemmas and local algebra.

Contribution

It introduces a new invariant b3(A) for singular local rings and demonstrates how to compute the index of algebraic varieties using this invariant and data from special fibers.

Findings

The index of a variety can be computed from the special fiber data.

A new invariant b3(A) relates to the index of exceptional divisors.

Two proofs of the main theorem using different moving lemmas.

Abstract

Let K be the field of fractions of a Henselian discrete valuation ring O_K. Let X_K/K be a smooth proper geometrically connected scheme admitting a regular model X/O_K. We show that the index \delta(X_K/K) of X_K/K can be explicitly computed using data pertaining only to the special fiber X_k/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an FA-scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant \gamma(A) of a singular local ring (A, \m): the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all \m-primary ideals Q in \m. We relate this invariant \gamma(A) to the index of the exceptional divisor in a resolution of the singularity of Spec(A), and we give a new way of computing the index of a smooth subvariety X_K/K of P^n_K over any field K, using the invariant \gamma of the local ring at the vertex of a cone over X.