Higher dimensional examples of manifolds whose adjoint bundles are not spanned

Abstract

Let $(X,L)$ be an $n$-dimensional polarized variety. Fujita's conjecture says that if $L^n>1$ then the adjoint bundle $K_X+nL$ is spanned and $K_X+(n+1)L$ is very ample. There are some examples such that $K_X+nL$ is not spanned or $K_X+(n+1)L$ is not very ample. These are $(\P^n,\O(1))$, hypersurface $M$ of degree $6$ in weighted projective space $\P(3,2,1,1,\cdots ,1)$ with $\O_M(1)$ and numerically Godeaux surface etc. Numerically Godeaux surface is the quotient space of a Fermat type hypersurface of degree $5$ in $\P^3$ by an action of order $5$. These examples are not so much. We construct new examples for any dimention.