The original Fujita
approximation theorem states that the volume of a big divisor D on a projective
variety X can always be approximated arbitrarily closely by the self-intersection
number of an ample divisor on a birational modification of X. One can also formulate
it in terms of graded linear series as follows: Let W∙= {Wk} be the complete graded
linear series associated to a big divisor D, where
For each fixed positive integer p, define W∙(p) to be the graded linear subseries of
W∙ generated by Wp:
Then the volume of W∙(p) approaches the volume of W∙ as p →∞. We will show
that, under this formulation, the Fujita approximation theorem can be generalized to
the case of multigraded linear series.
Keywords
Fujita approximation, multigraded linear series, Okounkov
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