On K-stability of Fano weighted hypersurfaces

Let X subset of P(a(0), ... , a(n)) be a quasi-smooth weighted Fano hypersurface of degree d and index I-X such that a(i) vertical bar d for all i. If I-X = 1, we show that, under a suitable condition, the alpha-invariant of X is greater than or equal to dim X/(dim X + 1) and X is K-stable. This can be applied in particular to any X as above such that dim X <= 3. If X is general and I-X < dim X, then we show that X is K-stable. We also give a sufficient condition for the finiteness of automorphism groups of quasi-smooth Fano weighted complete intersections.