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

TL;DR

This paper investigates the singularities of divisors on Gorenstein Fano varieties of coindex 4, establishing conditions under which general divisors have at worst canonical or terminal singularities, with specific examples in four dimensions.

Contribution

It proves that general elements of certain linear systems on Gorenstein canonical Fano varieties have at worst canonical or terminal singularities, extending understanding of their geometric properties.

Findings

General elements of |mH| have at worst canonical singularities for varieties with certain conditions.

For n ≥ 5, these divisors have at worst terminal singularities.

Counterexample in dimension 4 shows not all divisors are terminal.

Abstract

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. 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.