Log Canonical Thresholds and Generalized Eckardt Points

Abstract

Let $X$ be a smooth hypersurface of degree $n\ge 3$ in ${\mathbb{P}}^n$. We prove that the log canonical threshold of $H\in |-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a hyperplane section $H$ of $X$ is a cone in ${\mathbb{P}}^{n-1}$ over a smooth hypersurface of degree $n$ in ${\mathbb{P}}^{n-2}$ if and only if the log canonical threshold of $H$ is $\frac{n-1}{n}$.

Extended version, 14 pages, latex