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On Fano varieties of index one that are linear sections of Grassmannians of lines

  • Michele Bolognesi
  • Ciro Ciliberto
  • Alessandro Verra
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Abstract

In this paper we consider the general nβˆ’2 dimensional linear section X of the Grassmannian 𝔾(1,n) of lines in β„™n, that is a Fano variety of index 1. We first prove that for n odd X is birational to a hypersurface defined by a pfaffian determinant of order n+1 of linear forms in β„™nβˆ’1, and we study this birational transformation in some detail. Then for n even, we prove that X is unirational and birational to a hypersurface defined by a pfaffian determinant of order n of linear forms in β„™nβˆ’1. Moreover we prove that the hypersurface of β„™n described by the lines corresponding to the points in X has degree nβˆ’1 and is rational, and we find a rational parametrization of it.

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Cite

BibTeXRIS
BibTeX
@misc{bolognesi2026fano,
  title = {{On Fano varieties of index one that are linear sections of Grassmannians of lines}},
  author = {Michele Bolognesi and Ciro Ciliberto and Alessandro Verra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21503},
  eprint = {2609.21503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21503},
}
RIS
TY  - UNPB
TI  - On Fano varieties of index one that are linear sections of Grassmannians of lines
AU  - Michele Bolognesi
AU  - Ciro Ciliberto
AU  - Alessandro Verra
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21503
UR  - https://arxiv.org/abs/2609.21503
AB  - In this paper we consider the general nβˆ’2 dimensional linear section X of the Grassmannian 𝔾(1,n) of lines in β„™^n, that is a Fano variety of index 1. We first prove that for n odd X is birational to a hypersurface defined by a pfaffian determinant of order n+1 of linear forms in β„™^nβˆ’1, and we study this birational transformation in some detail. Then for n even, we prove that X is unirational and birational to a hypersurface defined by a pfaffian determinant of order n of linear forms in β„™^nβˆ’1. Moreover we prove that the hypersurface of β„™^n described by the lines corresponding to the points in X has degree nβˆ’1 and is rational, and we find a rational parametrization of it.
ER  -