Draisma, Jan; Eggermont, Rob H.
(2018).
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Plücker varieties and higher secants of Sato's Grassmannian.
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Journal für die reine und angewandte Mathematik, 2018(737), pp. 189-215.
de Gruyter
10.1515/crelle-2015-0035

Text
1402.1667.pdf - Submitted Version Restricted to registered users only Available under License Publisher holds Copyright. Download (527kB) | Request a copy |

Every Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we call Plücker varieties. A Plücker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural number k, the k-th secant variety of any Plücker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plücker variety in the dual of a highly symmetric space known as the infinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxilliary result that for every natural number p the space of p-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Plücker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties of p-tuples of matrices.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |

## UniBE Contributor: |
Draisma, Jan |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
0075-4102 |

## Publisher: |
de Gruyter |

## Language: |
English |

## Submitter: |
Olivier Bernard Mila |

## Date Deposited: |
14 May 2019 15:45 |

## Last Modified: |
24 Oct 2019 09:21 |

## Publisher DOI: |
10.1515/crelle-2015-0035 |

## ArXiv ID: |
1402.1667v3 |

## BORIS DOI: |
10.7892/boris.125493 |

## URI: |
https://boris.unibe.ch/id/eprint/125493 |