BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Abstract

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 ∈ 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 × 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.