Let $ X \subset \mathbb{P}^r_K$ denote a variety of almost minimal degree other than a normal del Pezzo variety. Then $ X$ is the projection of a rational normal scroll $ \tilde X \subset {\mathbb{P}}^{r+1}_K$ from a point $ p \in {\mathbb{P}}^{r+1}_K \setminus \tilde X.$ We show that the arithmetic depth of $ X$ can be expressed in terms of the rank of the matrix $ M'(p),$ where $ M'$ is the matrix of linear forms whose $ 3\times 3$ minors define the secant variety of $ \tilde X.$

Brodmann, M; Park, E; Schenzel, P (2011). *On varieties of almost minimal degree II: A rank-depth formula.* Proceedings of the American Mathematical Society, 139(6):2025-2032.

## Abstract

Let $ X \subset \mathbb{P}^r_K$ denote a variety of almost minimal degree other than a normal del Pezzo variety. Then $ X$ is the projection of a rational normal scroll $ \tilde X \subset {\mathbb{P}}^{r+1}_K$ from a point $ p \in {\mathbb{P}}^{r+1}_K \setminus \tilde X.$ We show that the arithmetic depth of $ X$ can be expressed in terms of the rank of the matrix $ M'(p),$ where $ M'$ is the matrix of linear forms whose $ 3\times 3$ minors define the secant variety of $ \tilde X.$

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2011 |

Deposited On: | 08 Jan 2012 19:47 |

Last Modified: | 05 Apr 2016 15:17 |

Publisher: | American Mathematical Society |

ISSN: | 0002-9939 |

Additional Information: | First published in Proceedings of the American Mathematical Society in vol. 139 (2011)published by the American Mathematical Society |

Publisher DOI: | https://doi.org/10.1090/S0002-9939-2010-10667-6 |

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