On complete intersections of three quadrics in $\mathbb{P}^7$

Abstract

We describe explicit birational maps from some rational complete intersections of three quadrics in $\mathbb{P}^7$ to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the cohomologically associated Castelnuovo surface of general type with $K^2=2$ and $χ=4$ (the double cover of $\mathbb{P}^2$ ramified along the discriminant curve of the net of quadrics defining the complete intersection) as the minimal model of the non ruled irreducible component of the base locus of the inverse maps. In passing we also revisit and reformulate the results in [arXiv:1706.01371] about the existence of infinitely many loci of rational complete intersection of three quadrics in $\mathbb{P}^7$ to produce explicitly some of these loci of low codimension together with many other irreducible unirational components of the Noether--Lefschetz locus.