K-polystability of the First Secant Varieties of Rational Normal Curves

Abstract The first secant variety $\Sigma $ of a rational normal curve of degree $d \geq 3$ is known to be a $\textbf{Q}$-Fano threefold. In this paper, we prove that $\Sigma $ is K-polystable, and hence, $\Sigma $ admits a weak Kähler–Einstein metric. We also show that there exists a $(-K_{\Sigma })$-polar cylinder in $\Sigma $.