The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{ P}^{ 55}=\mathbb{ P}(H^0(\mathcal{ O}_{ \mathbb{ P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{ 14}$. We use congruences of 5-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14, 26, 38$ in Hassett's notation.
Categories: math.AG