Boden–Hu conjecture on small resolutions of parabolic bundle moduli spaces

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Let NN and ss be integers with 0<s<N0<s<N, and let

W(N,s)∘:={α∈RN:0<α1<⋯<αN<1 and ∑n=1Nαn=s}.W(N,s)^{\circ}:=\{\alpha\in\mathbb{R}^N:0<\alpha_1<\cdots<\alpha_N<1\text{ and }\sum_{n=1}^N\alpha_n=s\}.

For α∈W(N,s)∘\alpha\in W(N,s)^{\circ}, let M(1‾)αM(\underline{\mathbf{1}})^\alpha be the moduli scheme of quasiparabolic bundles with multiplicity vector 1‾=(N,−s,1,…,1)\underline{\mathbf{1}}=(N,-s,1,\ldots,1), and for a generic weight vector β\beta near α\alpha let

ϕβ:M(1‾)β⟶M(1‾)α\phi_\beta:M(\underline{\mathbf{1}})^\beta\longrightarrow M(\underline{\mathbf{1}})^\alpha

be the canonical morphism induced by the identity functor. A morphism is small if the locus where its fibers have dimension at least kk has codimension greater than 2k2k for every positive integer kk. Boden–Hu conjecture. Near every α∈W(N,s)∘\alpha\in W(N,s)^{\circ}, there is a generic β∈W(N,s)∘\beta\in W(N,s)^{\circ} such that ϕβ\phi_\beta is a small map. This predicts that every moduli space arising from an arbitrary weight vector admits, locally in weight space, a generic-weight resolution with no divisorial exceptional contribution. The supplied text does not state whether the conjecture is open or resolved.

References

Primary source

Norbert Hoffmann, “The Boden-Hu conjecture holds precisely up to rank eight”, arXiv:math/0310160 (2003).

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