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

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.