Tangent cones to generalised theta divisors and generic injectivity of the theta map

Let C be a Petri general curve of genus g and E a general stable vector bundle of rank r and slope g - 1 over C with h(0)(C, E) = r + 1. For g >= (2r + 2)(2r + 1), we show how the bundle E can be recovered from the tangent cone to the generalised theta divisor Theta(E) at O-C. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map SU (C)( r) - - -> |r Theta| is generically injective for large values of g.