BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Let X be an arbitrary smooth irreducible complex projective curve, E→X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1, D2, f},where D1, D2 are two effective divisors with d =deg(D1) + deg(D2), and f∈H^0(X, [D1]｜D2)is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1+D2, the author constructs a 2g×d matrix WE whose zero space gives Im{H^0(X, [D1])→ H^0(X, [D1]｜D1)}+Im{H^0(X, E)→H^0(X, [D2])→H^0(X, [D2]｜D2)}. From this and H^0(X, E) = H^0(X, [D1])+Im{H^0(X, E)→H^0(X, [D2])},it is got in particular that dimH^0(X, E) = deg(E)-rank(WE) + 2.

Let X be an arbitrary smooth irreducible complex projective curve, E (?)X a rank two vector bundle generated by its sections. The author first represents ?as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H?X, [D1] |D2) is a. collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using / and the Brill-Noether matrix of D1+D2, the author constructs a 2g × d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?) Im{H?X, E) (?) H0(X,[D2]) (?) H0(X, [D2]|D2)} From this and H0(X,E) = H0(X,[D1]) (?)Im{H0(X,E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.