Abstract
We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking the dual, and Hecke transforms (using the filtrations given by the parabolic structure). A Torelli theorem for parabolic bundles with arbitrary rank and generic weights is also obtained. These results are extended to the classification of birational equivalences which are defined over “big” open subsets (3-birational maps, i.e. birational maps giving an isomorphism between open subsets with complement of codimension at least 3).Finally, an analysis of the stability chambers for the parabolic weights is performed in order to determine precisely when two moduli spaces of parabolic vector bundles with different parameters (curve, rank, determinant and weights) can be isomorphic.
Highlights
It is proven that such an automorphism exists if and only if E is obtained from E by one of the previously described transformations
For r = 2, the dual of a parabolic vector bundle can be rewritten in terms of a tensor product by a certain line bundle, so every isomorphism Ψ comes from a basic transformation that does not involve dualization
In Theorem 7.25 we describe the group of automorphisms of the moduli space M(r, α, ξ) as a subgroup of the group of basic transformations T described in Section 5, which varies depending on α and ξ
Summary
Let X be an irreducible smooth complex projective curve. Let D = {x1, . . . , xn} be a set of n ≥ 1 different points of X and let us denote U = X\D. Let M(X, r, α, d), or just M(r, α, d) be the moduli of semistable parabolic vector bundles (E, E) on (X, D) of rank r with system of weights α and deg(E) = d It has dimension dim(M(r, α, d)) = r2(g − 1) + 1 + r2tn,n. Let us prove that under that genus condition, the (l, m) stable parabolic vector bundles form a nonempty Zariski open subset of M(r, α, d) whose complement has codimension at least k. The proof of this first part is completely analogous to the proof of [3, Proposition 2.7]. As this holds for every subbundle F , (E, E) is stable
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