Dolgachev's Torelli conjecture for logarithmic bundles of hyperplane arrangements

Let H={H1,,H}\mathcal{H}=\{H_{1},\ldots,H_{\ell}\} be an arrangement of n+2\ell\geq n+2 hyperplanes on Pn\mathbf{P}^{n}. Assume that the logarithmic bundle Ω~Pn1(logH)\widetilde{\Omega}_{\mathbf{P}^{n}}^{1}(\log\mathcal{H}) is Gieseker-semistable, meaning that its reduced Hilbert polynomial is not smaller than that of any proper coherent subsheaf of positive rank. A stable normal rational curve of degree nn in Pn\mathbf{P}^{n} is a connected arithmetic-genus-zero curve whose irreducible components are smooth rational curves of degrees did_i spanning Pdi\mathbf{P}^{d_i}, with idi=n\sum_i d_i=n, and whose component spans together span Pn\mathbf{P}^{n}. Dolgachev's Torelli conjecture. The arrangement H\mathcal{H} is a Torelli arrangement if and only if the hyperplanes H1,,HH_{1},\ldots,H_{\ell} do not osculate a stable normal rational curve of degree nn in Pn\mathbf{P}^{n}. This gives a geometric criterion for whether the arrangement can be recovered from its logarithmic bundle; the source attributes the statement to Dolgachev (2007), but the supplied material gives no evidence that it has been proved or disproved.

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Primary source

Elena Angelini, “The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space”, arXiv:1506.01931 (2015).

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