On K-stability of Fano weighted hypersurfaces

Abstract

Let $X\subset \mathbb{P}(a_0,\dots ,a_n)$ be a quasi-smooth weighted Fano hypersurface of degree $d$ and index $I_X$ such that $a_i|d$ for all $i$, with $a_0\le \dots \le a_n$. 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 $\dim X\le 3$. 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.

arXiv commentMinor changes. To appear in Algebraic Geometry. 26 pages

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