### Abstract

In this article we study non-linearly normal smooth projective varieties X ⊂ P^{r} of deg (X) = codim (X, P^{r}) + 2. We first give geometric characterizations for X (Theorem 1.1). Indeed X is the image of an isomorphic projection of smooth varieties over(X, ̃) ⊂ P^{r + 1} of minimal degree. Also if over(X, ̃) is not the Veronese surface, then there exists a smooth rational normal scroll Y ⊂ P^{r} which contains X as a divisor linearly equivalent to H + 2 F where H is the hyperplane section of Y and F is a fiber of the projection morphism π : Y → P^{1}. By using these characterizations, (1) we determine all the possible types of Y from the type of over(X, ̃) (Theorem 1.2), and (2) we investigate the relation between the Betti diagram of X and the type of Y (Theorem 1.3). In particular, we clarify the relation between the number of generators of the homogeneous ideal of X and the type of Y. As an application, we construct non-linearly normal examples where the converse to Theorem 1.1 in [D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restriction linear syzygies: Algebra and geometry, Compos. Math. 141 (2005) 1460-1478] fails to hold (Remark 2).

Original language | English |
---|---|

Pages (from-to) | 185-208 |

Number of pages | 24 |

Journal | Journal of Algebra |

Volume | 314 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Aug 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*314*(1), 185-208. https://doi.org/10.1016/j.jalgebra.2007.02.027

**Smooth varieties of almost minimal degree.** / Park, Euisung.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 314, no. 1, pp. 185-208. https://doi.org/10.1016/j.jalgebra.2007.02.027

}

TY - JOUR

T1 - Smooth varieties of almost minimal degree

AU - Park, Euisung

PY - 2007/8/1

Y1 - 2007/8/1

N2 - In this article we study non-linearly normal smooth projective varieties X ⊂ Pr of deg (X) = codim (X, Pr) + 2. We first give geometric characterizations for X (Theorem 1.1). Indeed X is the image of an isomorphic projection of smooth varieties over(X, ̃) ⊂ Pr + 1 of minimal degree. Also if over(X, ̃) is not the Veronese surface, then there exists a smooth rational normal scroll Y ⊂ Pr which contains X as a divisor linearly equivalent to H + 2 F where H is the hyperplane section of Y and F is a fiber of the projection morphism π : Y → P1. By using these characterizations, (1) we determine all the possible types of Y from the type of over(X, ̃) (Theorem 1.2), and (2) we investigate the relation between the Betti diagram of X and the type of Y (Theorem 1.3). In particular, we clarify the relation between the number of generators of the homogeneous ideal of X and the type of Y. As an application, we construct non-linearly normal examples where the converse to Theorem 1.1 in [D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restriction linear syzygies: Algebra and geometry, Compos. Math. 141 (2005) 1460-1478] fails to hold (Remark 2).

AB - In this article we study non-linearly normal smooth projective varieties X ⊂ Pr of deg (X) = codim (X, Pr) + 2. We first give geometric characterizations for X (Theorem 1.1). Indeed X is the image of an isomorphic projection of smooth varieties over(X, ̃) ⊂ Pr + 1 of minimal degree. Also if over(X, ̃) is not the Veronese surface, then there exists a smooth rational normal scroll Y ⊂ Pr which contains X as a divisor linearly equivalent to H + 2 F where H is the hyperplane section of Y and F is a fiber of the projection morphism π : Y → P1. By using these characterizations, (1) we determine all the possible types of Y from the type of over(X, ̃) (Theorem 1.2), and (2) we investigate the relation between the Betti diagram of X and the type of Y (Theorem 1.3). In particular, we clarify the relation between the number of generators of the homogeneous ideal of X and the type of Y. As an application, we construct non-linearly normal examples where the converse to Theorem 1.1 in [D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restriction linear syzygies: Algebra and geometry, Compos. Math. 141 (2005) 1460-1478] fails to hold (Remark 2).

UR - http://www.scopus.com/inward/record.url?scp=34249296356&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249296356&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2007.02.027

DO - 10.1016/j.jalgebra.2007.02.027

M3 - Article

AN - SCOPUS:34249296356

VL - 314

SP - 185

EP - 208

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -