Birational Mori fiber structures of $\mathbb{Q}$-Fano $3$-fold weighted complete intersections, III

TL;DR

This paper continues the study of birational Mori fiber structures of certain $Q$-Fano 3-folds, establishing birational rigidity or birigidness for specific families of weighted complete intersections.

Contribution

It extends previous classifications by proving that 21 additional families of $Q$-Fano 3-folds are birationally birigid, completing a systematic analysis of these structures.

Findings

19 families are birationally rigid

14 families are birationally birigid

21 families are now proven to be birationally birigid

Abstract

This is a continuation of a series of papers studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consists of birationally rigid varieties and $14$ families consists of birationally birigid varieties. The aim of this paper is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.