On rational maps between moduli spaces of curves and of vector bundles

Abstract

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.

25 pages, new version. Added a couple of new Prop.s and lots of references, corrected inaccuracies and typos

References in corpus (2)

• The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian
• Heisenberg invariant quartics and SU_C(2) for a curve of genus four