Geometry of the moduli of parabolic bundles on elliptic curves

Abstract

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 . We also showcase a special curve Γ isomorphic to C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P 1 via a modular map which turns out to be the 2:1 cover ramified in Γ. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles. IRMAR, Univ. Rennes

A Map Between Moduli Spaces of Connections

“…The present work relies strongly on several results from [7,17], which we discuss in Section 6. Finally, we remark the following for the 2-punctured elliptic curve case.…”

“…The present work relies heavily on the constructions, results and ideas that appear in [7,17]. For the sake of the reader's convenience, we provide in Section 6 a brief survey of the results needed from these papers.…”

“…Usually the determinant of the bundle is fixed to be either O C in the even case (as in the present paper), or O C (w ∞ ), where w ∞ ∈ C is the identity element for the group structure of C. The moduli space of connections on C with two poles and fixed determinant O C (w ∞ ) has already been described in detail in [6], together with its symplectic structure and apparent map. As pointed out in [7], it is possible to pass from the moduli space in the even degree case to that in the odd degree case. This is done by one elementary transformation followed by a twist by a rank 1 connection of degree zero.…”

“…We remark that a typical element of Bunμ P 1 , D has a trivial underlying bundle (as will be discussed in Section 5). On the other hand, a typical element of Bunμ(C, T ) (for example aμ-stable parabolic bundle) is such that its underlying bundle can be written as E = L ⊕ L −1 , where L is a rank 1 bundle on C of degree zero [7,Proposition 4.5]. For such case, Fig.…”

“…The moduli spaces Conμ ν P 1 , D and Bunμ P 1 , D have been explicitly described in [17], as well as the fibration Bun between them. The moduli space of parabolic bundles Bunμ(C, T ) was later studied in [7]. Moreover, the latter paper also describes geometrically and in coordinates the map φ : Bunμ P 1 , D → Bunμ(C, T ).…”