A note on the projective varieties of almost general type

Details A note on the projective varieties of almost general type A note on the projective varieties of almost general type

2005-06-08

arxiv.org/abs/math/0506132v5

Rocky Mountain J. Math. 40, No. 2, 501-512 (2010)

Mathematics

Algebraic Geometry

11 pages, LaTeX2e; The published version

Scientific paper

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial canonical divisors, under the minimal model conjecture. Furthermore we prove that, for a projective variety $X$ with only terminal singularities, $X$ is of almost general type if and only if the canonical divisor $K_X$ is almost nup, under the minimal model conjecture.

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