On the rationality of certain Fano threefolds

Abstract In this paper we study the rationality problem for Fano threefolds $$X\subset {\mathbb P}^{p+1}$$ X ⊂ P p + 1 of genus p , that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$p\geqslant 8$$ p ⩾ 8 (this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$p\geqslant 7$$ p ⩾ 7 containing a plane is rational; (3) any Fano threefold of genus $$p\geqslant 17$$ p ⩾ 17 is rational; (4) a Fano threefold of genus $$p\geqslant 12$$ p ⩾ 12 containing an ordinary line $$\ell $$ ℓ in its smooth locus is rational.