Bounds for smooth Fano weighted complete intersections

TL;DR

This paper establishes bounds on weights for smooth Fano complete intersections in weighted projective spaces and classifies such varieties in dimensions 4 and 5, including their invariants.

Contribution

It provides a new bound on weights for smooth varieties in weighted projective spaces and classifies all smooth Fano complete intersections in dimensions 4 and 5.

Findings

Weights of the ambient space do not exceed dimension + 1 for certain smooth varieties.

Complete classification of smooth Fano complete intersections in dimensions 4 and 5.

Computed invariants for these classified varieties.

Abstract

We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.