2010/05/04 by Shin-Yao Jow, Jow, Shin-Yao
Mathematics · #14C20 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C20
paper · pdf · doi:10.48550/arxiv.1005.0432
6 pages; minor changes; to appear in Pacific Journal of Mathematics
arxiv created 2011/01/04 · arxiv updated 2011/01/05
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\bullet = \Wk \ be the complete graded linear series associated to a big divisor D: Wk = H0(X,OX(kD)). For each fixed positive integer p, define W(p)\bullet to be the graded linear subseries of W\bullet generated by Wp: W(p)m=cases 0, if p\nmid m; Image (Sk Wp → Wkp ), if m=kp. cases Then the volume of W(p)\bullet approaches the volume of W\bullet as p→∞. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.