Rationality of Fano threefolds with terminal Gorenstein singularities. II (Q6159156)

A projective variety over a field \(k\) is said to be rational if it is birational to the projective space \(\mathbb{P}^n_k\). The rationality problem for smooth Fano threefolds is mostly solved but it is still widely open for singular ones. In this paper, the author continues his previous work [\textit{Y. G. Prokhorov}, Proc. Steklov Inst. Math. 307, 210--231 (2019; Zbl 1471.14032); translation from Tr. Mat. Inst. Steklova 307, 230--253 (2019)] and studies the rationality of a terminal Gorenstein Fano threefold \(X\) of Picard number 1. Via Sarkisov links, the author classifies such \(X\) which satisfies \((-K_X)^3\ge 8\) and \(\mathrm{rk}\,\mathrm{Cl}(X)\le 2\) (see Theorems 1.1 and 1.2). The author also proves that if such \(X\) satisfies \((-K_X)^3\ge 8\) and \(\mathrm{rk}\,\mathrm{Cl}(X)>2\), and assuming that for any small \(\mathbb{Q}\)-factorization \(X'\to X\), the variety \(X'\) has no birational Mori contractions, then \((-K_X)^3=8\), \(\mathrm{rk}\,\mathrm{Cl}(X)=3\) and \(X=X_8\subseteq\mathbb{P}^6\) is a special intersection of three quadrics (see Theorem 1.3 and Proposition 6.1).