Higher dimensional examples of manifolds whose adjoint bundles are not spanned

Details Higher dimensional examples of manifolds whose adjoint bundles are not spanned Higher dimensional examples of manifolds whose adjoint bundles are not spanned

1996-03-28

arxiv.org/abs/alg-geom/9603023v1

Mathematics

Algebraic Geometry

TeX-Type: AmS-TeX 2.1, 6 pages

Scientific paper

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.

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