Birational geometry and the canonical ring of a family of determinantal 3-folds

TL;DR

This paper explores a specific family of determinantal 3-folds in a product of projective spaces, providing insights into their canonical rings and birational geometry, which are generally difficult to compute explicitly.

Contribution

It introduces a new family of determinantal 3-folds where the canonical ring and birational geometry are explicitly accessible, advancing understanding in this area.

Findings

Explicit computation of the canonical ring for the family

Analysis of the birational properties of these 3-folds

Identification of non-trivial birational geometric features

Abstract

Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.