# Boundedness results for singular Fano varieties and applications to Cremona groups

## TL;DR

This survey discusses Birkar's proof that Fano varieties with mild singularities are bounded in fixed dimension and explores its implications for the Jordan property of birational automorphism groups of projective spaces.

## Contribution

It explains how Birkar's boundedness result leads to the proof that these automorphism groups satisfy the Jordan property, resolving a longstanding question.

## Key findings

- Fano varieties with mild singularities are bounded in fixed dimension
- Birational automorphism groups of projective spaces satisfy the Jordan property
- The work confirms a long-standing conjecture and answers Serre's question

## Abstract

This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive.