Algebraic ellipticity of double covers of $({\mathbf{P}}^1{)}^3$

Abstract

We show that every smooth complex double cover of $({\mathbf{P}}^1{)}^3$ branched along a smooth divisor of tridegree $(2,2,2)$ is algebraically Gromov elliptic. These irrational Fano threefolds form Mori--Mukai family 3-1 and have anticanonical genus seven and Picard rank three. Their three coordinate projections define degree-four del Pezzo fibrations. Each threefold also admits a dominant rational map of degree two from ${\mathbf{P}}^3$.

arXiv comment14 pages