# A classification of terminal quartic 3-folds and applications to rationality questions

## TL;DR

This paper classifies terminal quartic 3-folds based on their birational geometry and demonstrates their rationality in many cases, especially when the divisor class group has high rank.

## Contribution

It provides a detailed classification of terminal quartic 3-folds and applies MMP techniques to establish rationality results for various families.

## Key findings

- Non-factorial terminal Gorenstein Fano 3-folds are often rational.
- Examples of rational quartic hypersurfaces with Cl Y of rank 2.
- Y is always rational when Cl Y has rank greater than 6.

## Abstract

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces with Cl Y of rank 2, and show that when Cl Y has rank greater than 6, Y is always rational.