Singularities of pluri-fundamental divisors on Gorenstein Fano varieties of coindex $4$

Abstract

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X,H)\ge n-2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m\ge 1$. When $X$ has terminal singularities and $n\ge 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m\ge 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.

arXiv comment11 pages, contributed to Moscow-Shanghai-Pohang conference proceedings