Quarto-quartic birational maps of $\mathbb{P}_3(\mathbb{C})$

TL;DR

This paper constructs a family of complex projective space transformations using trigonal curves and analyzes the structure of the variety of such birational maps, revealing multiple irreducible components.

Contribution

It introduces a determinantal family of quarto-quartic transformations of  and describes the irreducible components of the variety of 4-birational maps.

Findings

Constructed a determinantal family of transformations from trigonal curves.

Showed the variety of (4,4)-birational maps has at least four irreducible components.

Described three of the irreducible components.

Abstract

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of $\mathbb{P}_3(\mathbb{C})$ has at least four irreducible components and describe three of them.