The index of an algebraic variety

By Ofer Gabber, Qing Liu, Dino J. Lorenzini. 2012

71 citations recorded by the source corpus.

Let K be the field of fractions of a Henselian discrete valuation ring ${{\mathcal {O}}_{K}}$. Let XK/K be a smooth proper geometrically connected scheme admitting a regular model $X/{{\mathcal {O}}_{K}}$. We show that the index δ(XK/K) of XK/K can be explicitly computed using data pertaining only to the special fiber Xk/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 $\operatorname {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 γ(A) of a singular local ring $(A, {\mathfrak {m}})$: the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all ${\mathfrak {m}}$-primary ideals Q in ${\mathfrak {m}}$. We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of $\operatorname {Spec}A$, and we give a new way of computing the index of a smooth subvariety X/K of ${\mathbb{P}}^{n}_{K}$ over any field K, using the invariant γ of the local ring at the vertex of a cone over X.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b676800b91dd496497afdb832ad2a939457a06f1

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Hypersurfaces in projective schemes and a moving lemma (2014, methodology): “3, where the result is then used to prove a formula for the index of an algebraic variety over a Henselian field ([21],”