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.

Minor changes. To appear in Algebraic Geometry. 26 pages

References in corpus (4)

• Optimal destabilizing centers and equivariant K-stability
• K-stability of log Fano hyperplane arrangements
• K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces
• Seshadri constants and K-stability of Fano manifolds