arXiv:0903.5515v3 [math.AG] 14 Jun 2009 A STRUCTURE THEOREM FOR SU C (2) AND THE MODULI OF POINTED GENUS ZERO CURVES ALBERTO ALZATI AND MICHELE BOLOGNESI Abstract. Let SU C (2) be the moduli space of rank 2 semistable vector bun- dles with trivial determinant on a smooth complex curve C of genus g> 1,non- hyperellptic if g> 2. In this paper we prove a birational structure theorem for SU C (2) that generalizes that of [Bol07] for genus 2. Notably we give a birational description of SU C (2) as a fibration over P g , where the fibers are GIT compactifications of the moduli space M 0,2g of 2g-pointed genus zero curves. This is done by describing the classifying maps of extensions of the line bundles associated to some effective divisors. In particular, for g = 3 our construction shows that SU C (2) is birational to a fibration in Segre cubics over a P 3 . 1. Introduction The first ideas about moduli of vector bundles on curves date back some eighty years, when in [Wei38] for the first time the author suggested the idea that an analogue of the Picard variety could be provided by higher rank bundles. Then, in the second half of last century, a more complete construction of these moduli spaces was carried out, mainly by Mumford, Newstead [MN68] and the mathematicians of the Tata institute, e.g. [NR69a]. Let us denote SU C (r) the moduli space of semistable vector bundles of rank r and trivial determinant on a smooth complex curve C of genus g. If g = 2 we will also assume through all the paper that C is not hyperelliptic. Some spectacular results have been obtained on the projective structure of these moduli spaces in low genus and rank, especially thanks to the relation with the work on theta functions and classical algebraic geometry of A.B.Coble [Cob82]. This interplay has produced a flourishing of beautiful results (see [Pau02], [Bea03], [Ort05], [Ngu07], or [DO88] for a survey) where both classical algebraic geometry and modern moduli theory come into play. On the other hand, even if some important advances have been made in the Brill- Noether theory (for instance [Muk95],[TiB91] or [BV07]) and on the local structure of SU C (r) ([Ser07], [Las96]), the theory seem to lack general results in arbitrary genus about the structure of these moduli spaces and their birational geometry. For instance, even though it is known that SU C (r) is unirational, the only case where it is known whether it is rational is for r = 2 on a genus 2 curve, and the answer is postive, since SU C (2) ∼ = P 3 . This paper aims to start to fill this gap for rank 2 vector bundles. In fact our main theorem gives a description of the structure of SU C (2) for a curve C of g(C) > 1. Here it is. 1