## Fontanari: Birational geometry of defective varieties (2003)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Ballico03fontanari:birational,

author = {Edoardo Ballico and Claudio Fontanari},

title = {Fontanari: Birational geometry of defective varieties},

institution = {},

year = {2003}

}

### OpenURL

### Abstract

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality. AMS Subject Classification: 14N05.

### Citations

30 |
Weakly defective varieties
- Chiantini, Ciliberto
(Show Context)
Citation Context ... up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered 1and generalized by Chiantini and Ciliberto in =-=[1]-=- and [2]. It turns out that one of the main tools for understanding defective varieties is provided by the technique of tangential projections. Namely, assume that X is not (k − 1)defective and let p1... |

16 |
Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti
- Severi
- 1901
(Show Context)
Citation Context ... ≥ 1. It seems reasonable to regard defective varieties as exceptional and try to classify them. The first result in this direction, stated by Del Pezzo in 1887 and proved by Severi in 1901 (see [5], =-=[10]-=-, and also [4] for a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]),... |

13 |
Sulle varietá algebriche per le quali sono minori dell’ ordinario, senza riempire lo spazio ambiente, una o alcune delle varietá formate da spazi seganti, Atti Accad. Torino 44
- Palatini
- 1909
(Show Context)
Citation Context ... [10], and also [4] for a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and =-=[7]-=-), Scorza ([8] and [9]), and Terracini ([11] and [12]) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently rec... |

12 |
Sulle Vk per cui la varietà degli Sh-h+1 seganti ha dimensione minore dell’ordinario
- Terracini
- 1911
(Show Context)
Citation Context ...aracterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza ([8] and [9]), and Terracini (=-=[11]-=- and [12]) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered 1and generalized by Chiantini and... |

10 |
determinazione delle varietá a tre dimensioni di Sr (r ≥ 7) i cui S3 tangenti si tagliano a due a
- Scorza, Sulla
- 1908
(Show Context)
Citation Context ...o [4] for a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza (=-=[8]-=- and [9]), and Terracini ([11] and [12]) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered 1an... |

8 |
Threefolds with degenerate secant variety: on a theorem of
- Chiantini, Ciliberto
(Show Context)
Citation Context ...taining it, in particular it is a plane curve. Moreover, if Σ had degree d > 2, then the general secant line to X would be a multisecant, which is a contradiction (the above argument is borrowed from =-=[2]-=-, proof of Proposition 4.2). Hence X is uniruled and if Σ is irreducible then X is rationally connected. More precisely, if d = 1 then two general points on X would be joined by a straight line, a con... |

8 |
Sulle varietá a quattro dimensioni di Sr (r ≥ 9) i cui S4 tangenti si tagliano a due a
- Scorza
- 1909
(Show Context)
Citation Context ...r a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza ([8] and =-=[9]-=-), and Terracini ([11] and [12]) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered 1and genera... |

7 |
due problemi concernenti la determinazione di alcune classi di superficie, considerate da
- Terracini, Su
(Show Context)
Citation Context ...es the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza ([8] and [9]), and Terracini ([11] and =-=[12]-=-) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered 1and generalized by Chiantini and Cilibert... |

6 |
Terracini’s lemma and the secant variety of a curve
- Dale
- 1984
(Show Context)
Citation Context ...eneral points on X ⊂ P r . The general ktangential projection τX,k is the projection of X from the linear span of its tangent spaces at p1, . . .,pk. By the classical Lemma of Terracini (see [11] and =-=[3]-=- for a modern version), Xk := τX,k(X) is lower dimensional than X if and only if X is k-defective. Therefore, the classification of defective varieties reduces to the classification of varieties which... |

3 |
Sulle superficie algebriche i cui Sh (h+1)-seganti non riempiono lo spazio ambiente, Atti. Accad
- Palatini
- 1906
(Show Context)
Citation Context ...see [5], [10], and also [4] for a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini (=-=[6]-=- and [7]), Scorza ([8] and [9]), and Terracini ([11] and [12]) set up the classification of k-defective surfaces and of 1-defective varieties in dimension up to four. This classical work has been rece... |

1 |
Severi’s theorem on the Veronese-surface
- Dale
- 1985
(Show Context)
Citation Context ...reasonable to regard defective varieties as exceptional and try to classify them. The first result in this direction, stated by Del Pezzo in 1887 and proved by Severi in 1901 (see [5], [10], and also =-=[4]-=- for a modern proof), characterizes the 2-Veronese of P 2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza ([8] a... |