Plücker varieties and higher secants of Sato's Grassmannian

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

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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)

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

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