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| BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES |
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Citation： |
TAN Xiaojiang.BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES[J].Chinese Annals of Mathematics B,2002,23(4):531~538 |
| Page view： 1133
Net amount： 640 |
Authors： |
TAN Xiaojiang; |
Foundation： |
Project supported by the National Natural Science Foundation of China. |
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| Abstract： |
Let $X$ be an arbitrary smooth irreducible complex projective curve, $E\mapsto X$ a rank two vector bundle generated by its sections. The author first represents $E$ as a triple $\{D_1,D_2,f \}$, where $D_1$, $D_2$ are two effective divisors with $d=\deg(D_1)+\deg(D_2),$ and $f \in H^0(X,[D_1]\mid_{D_2})$ is a collection of polynomials. $E$ is the extension of $[D_2]$ by $[D_1]$ which is determined by $f$. By using $f$ and the Brill-Noether matrix of $D_1+D_2$, the author constructs a $2g\times d$ matrix $W_{E}$ whose zero space gives $\Im\{H^{0}(X,[D_{1}]) \mapsto H^{0}(X,[D_1] \mid_{D_{1}}) \} \oplus \Im\{H^{0}(X,E) \mapsto H^{0}(X,[D_2]) \mapsto H^{0}(X,[D_2] \mid_{D_{2}}) \}$. From this and $H^{0}(X,E)=H^{0}(X,[D_{1}]) \oplus \Im\{H^{0}(X,E) \mapsto$ $H^{0}(X,[D_{2}]) \}$, it is got in particular that $\dim H^{0}(X,E)=\deg(E) -\rank(W_{E}) + 2$. |
Keywords： |
Brill-Noether matrix, Vector bundle, Effective divisor |
Classification： |
14H20 |
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