Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space

Secant bundles, relative secant bundles, and the natural maps among themDefinition.We say that M separates k points if M e Pic^C) has the property that h°(C, M) = h°(C, M(-D)) + k for all DeC k .Thus M separates one point if and only if M is base point free, and M separates two points if and only if M is very ample.Let i^+ 1 = CxC k c CxC k+ι be the "universal" divisor, embedded via (p,D)i-+(p 9 p + D), and let π c : C x C k+ι -» C and π k+ι : C x C k+ι -> C k+ι be the projections.