Dolgachev's Torelli conjecture for logarithmic bundles of hyperplane arrangements
Dolgachev's Torelli conjecture for logarithmic bundles of hyperplane arrangements
Let be an arrangement of hyperplanes on . Assume that the logarithmic bundle 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 in is a connected arithmetic-genus-zero curve whose irreducible components are smooth rational curves of degrees spanning , with , and whose component spans together span . Dolgachev's Torelli conjecture. The arrangement is a Torelli arrangement if and only if the hyperplanes do not osculate a stable normal rational curve of degree in . 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.
Sources & referencesView supporting material
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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