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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[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.

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 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.

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.

added 35 characters in body
Source Link
quim
quim
  • 1.8k
  • 13
  • 17

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 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.

Your nine points must be general, 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 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.

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 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.

Source Link
quim
quim
  • 1.8k
  • 13
  • 17

Your nine points must be general, 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 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.