BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei 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 ∈ H0(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 X d matrix WE whose zero space gives Im{H0(X,D1]) (?) H0(X, D1] |D1)}(?)Im{H0(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.

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 ∈ Ho (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 29 × d matrix WE whose zero space gives Im{Ho(X, D1]) → Ho(X, D1] |D1 )} Im{Ho(X, E) → Ho(X, D2]) →H0(X, D2] |D2)}. From this and Ho(X, E) = Ho(X, D1]) Im{Ho(X, E) → Ho(X, D2])},it is got in particular that dimHo(X, E) = deg(E) - rank(WE) + 2.