Abstract
We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of \({\Omega_{\mathbb{P}^{n-1}}^{}(2)}\), the twisted cotangent bundle on \({\mathbb{P}^{n-1}}\), is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n = 4.
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F. Tanturri, On the Hilbert scheme of degeneracy loci of twisted differential forms, to appear in Trans. Amer. Math. Soc.
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Partially supported by the PRIN 2010/2011 “Geometria delle varietà algebriche”.
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Tanturri, F. Degeneracy loci of twisted differential forms and linear line complexes. Arch. Math. 105, 109–118 (2015). https://doi.org/10.1007/s00013-015-0768-z
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DOI: https://doi.org/10.1007/s00013-015-0768-z