Plücker varieties and higher secants of Sato’s Grassmannian
| Autor: | Rob H. Eggermont, Jan Draisma |
|---|---|
| Přispěvatelé: | Discrete Mathematics, Discrete Algebra and Geometry |
| Rok vydání: | 2015 |
| Předmět: |
Noetherian
Discrete mathematics Pure mathematics Computer Science::Information Retrieval Applied Mathematics General Mathematics 010102 general mathematics 010103 numerical & computational mathematics Plücker embedding 01 natural sciences 510 Mathematics Secant variety Grassmannian Symmetric space Bounded function 0101 mathematics Plucker Vector space Mathematics |
| Zdroj: | Journal für die reine und angewandte Mathematik, 189-215. Walter de Gruyter GmbH ISSUE=737;STARTPAGE=189;ENDPAGE=215;ISSN=0075-4102;TITLE=Journal für die reine und angewandte Mathematik |
| ISSN: | 1435-5345 0075-4102 |
| Popis: | 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 auxiliary 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. |
| Databáze: | OpenAIRE |
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