Two rational nodal quartic threefolds

Abstract

We prove that the quartic threefolds defined by $$\sum _{i=0}^5{x}_i=\sum _{i=0}^5{x}_i^4-t{\left(\sum _{i=0}^5{x}_i^2\right)}^2=0$$ in ${\mathbb{P}}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.

28 pages, Remark 6.15 and one reference are removed due to redundancy