The classification of smooth well-formed Fano weighted complete intersections

TL;DR

This paper classifies smooth well-formed Fano weighted complete intersections based on their variance, providing a structured understanding and specific classifications for cases with small variance, and discusses properties of their anticanonical systems.

Contribution

It introduces a natural partition of Fano weighted complete intersections by variance and classifies those with small variance, advancing the understanding of their geometric structure.

Findings

Partition of families by variance

Classification of small variance cases

Anticanonical system properties

Abstract

We show that the set of families of smooth well-formed Fano weighted complete intersections admits a natural partition with respect to the variance $\mathrm{var}(X) = \mathrm{coind}(X) - \mathrm{codim}(X)$. Moreover, we obtain the classification of smooth well-formed Fano weighted complete intersections of small variance. We also prove that the anticanonical linear system on a smooth well-formed Fano weighted complete intersection of anticanonical degree one is never base-point free.