The Mukai conjecture via Cox rings for special toric ambient embeddings

TL;DR

This paper proves the Mukai conjecture for a specific class of locally factorial Fano varieties characterized by Cox rings and toric embeddings, extending the conjecture via Mori dream space theory.

Contribution

It establishes the Mukai conjecture for certain Fano varieties using Cox rings and toric ambient embeddings, linking it to Mori dream space theory.

Findings

Characterization of Fano varieties via Cox rings and toric embeddings

Extension of the Mukai conjecture to this class of varieties

Use of log Mukai conjecture for toric ambient spaces

Abstract

We prove the Mukai conjecture on the characterisation of products of projective spaces among Fano varieties for a class of locally factorial Fano varieties defined in terms of their Cox rings. The Fano varieties of this class are characterised in terms of the property that they admit an embedding into a smooth projective toric variety via the bunched ring theory of Mori dream spaces. Our approach inherits the Mukai conjecture for this class from a log version of the Mukai conjecture on the toric ambient embedding.