K-polystability of two smooth Fano threefolds

TL;DR

This paper provides new proofs for the K-polystability of two specific smooth Fano threefolds, one being a divisor in a product of projective spaces and the other a blow-up of a particular complete intersection.

Contribution

It introduces novel proofs establishing K-polystability for two particular smooth Fano threefolds, expanding understanding of their geometric stability properties.

Findings

Proved K-polystability of a smooth divisor in P^1×P^1×P^2.

Established K-polystability of a blow-up of a specific complete intersection.

Enhanced methods for verifying stability of Fano threefolds.

Abstract

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+\omega x_1^2+\omega^2x_2^2+\big(x_3^2+\omega x_4^2+\omega^2x_5^2\big)+\big(x_0x_3+\omega x_1x_4+\omega^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=x_1=x_2=0$, where $\omega$ is a~primitive cube root of unity.