### Impacto

Giraldo Suárez, Luis and Sols Lucía, Ignacio
(1997)
*Chow forms of congruences.*
Mathematical Proceedings of the Cambridge Philosophical Society, 121
.
pp. 31-37.
ISSN 0305-0041

Official URL: http://www.journals.cambridge.org/journal_MathematicalProceedingsoftheCambridgePhilosophicalSociety

## Abstract

For X PN an n-dimensional variety the set of linear spaces of dimension N − n − 1 meeting X defines a hypersurface, H, in the Grassmann variety G(N − n,N + 1).

The homogeneous form in the Pl¨ucker coordinates defining H or H itself is called the Chow form of X. This notion was defined by Cayley [A. Cayley, “On a new analytical representation of curves in space”, Q. J. Pure Appl. Math. 3, 225-236 (1860), and 5, 81-86 (1862); for a modern treatment see M. Green and I. Morrison, Duke Math. J.

53, 733-747 (1986; Zbl 0621.14028)].

In the present paper the authors study Chow forms of integral surfaces in G(2, 4) following the approach of M. Green and I. Morrison. Let V be a fixed 4-dimensional space and F P3 ×ˇP3, the flag variety parametrizing all chains V1 V3, where Vi is a subspace of V with dim Vi = i. F parametrizes the lines of G and to each integral surface Y in G there corresponds, in a natural way, an integral hypersurface X in F. The main result in this paper is a characterization of integral hypersurfaces X in F that are Chow forms of integral surfaces in G, in terms of some differential equations.

Item Type: | Article |
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Uncontrolled Keywords: | Grassmannian; Chow form; integral surfaces; flag variety |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 16806 |

Deposited On: | 23 Oct 2012 08:30 |

Last Modified: | 19 Mar 2013 19:06 |

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