Rationality of weighted hypersurfaces of special degree

Abstract

Let $X\subset P(w_0,w_1,w_2,w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L=\mathrm{lcm}(w_0,w_1,w_2,w_3)$. We characterize when X is rational under the assumption that L divides $\mathrm{deg}\left(X\right)$. Furthermore, we give a new family of normal rational weighted projective hypersurfaces with ample canonical divisor, valid in all dimensions, adding to the list of examples discovered by Kollár. Finally, we determine precisely which affine Pham-Brieskorn threefolds are rational, answering a question of Rajendra V. Gurjar.