Removing non-basepoint from linear sytem. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T07:47:07Z http://mathoverflow.net/feeds/question/70027 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70027/removing-non-basepoint-from-linear-sytem Removing non-basepoint from linear sytem. phil 2011-07-11T16:35:01Z 2011-07-11T16:48:58Z <p>Can you tell me, why the following is true: Let $C$ be a smooth, complete curve over an algebraically closed field. Let $D=P_1+...+P_n$ be an effective divisor that is linear combination of (not necessarily different) points $P_1,..., P_n \in C$. Let $P_n$ be not a basepoint of the linear system of $D$. Can you tell me why the dimension of $H^0(O_C(D))$ is smaller by at least $1$ than the dimension of $H^0(O_C(D-P_n))$, i.e. the dimension of the global sections drops by $1$ when removing $P_n$.</p> http://mathoverflow.net/questions/70027/removing-non-basepoint-from-linear-sytem/70030#70030 Answer by Mike Skirvin for Removing non-basepoint from linear sytem. Mike Skirvin 2011-07-11T16:48:58Z 2011-07-11T16:48:58Z <p>This is exactly Proposition 3.1(a) in Chapter IV of Hartshorne. The proof, in my opinion, provides a good explanation of why the result is true.</p>