Let $S$ the blow up of $P^2$ in nine points. Why is the anticanonical divisor $-K\_S$ not semiample?



Anticanonical divisor of the blow up of P^2 in 9 points
Your nine points must be [**EDIT: very**] general [see MP's answer], otherwise it *can* be semiample.


The only effective anticanonical divisor is then (the strict transform of) the cubic C through the nine points. Since there is no other cubic curve cutting C in your nine points, the restriction of -K\_S to C is a noneffective divisor of degree 0 (C has genus 1). So the restriction of -mK\_S is also noneffective for all m (the points are [**very**] general in C! [I guess] Torsion points can make a difference) which means the only effective divisor in -mK\_S is mC. Thus -mK\_S is never base point free.


