Unirational Fano threefolds

We use the relative Abel–Jacobi construction of Cheltsov–Loginov–Orlov–Prokhorov to study unirational Fano threefolds. Our main result is a $50$-dimensional family of smooth unirational del Pezzo threefolds of degree one. We also construct explicit one-parameter families of terminal unirational sextic double solids and quartic threefolds of Picard rank one. Finally, we reinterpret the Roth–Iskovskikh unirationality construction for smooth double quadric threefolds in terms of a correspondence induced by the relative Abel–Jacobi map.