## Tangential projections and secant defective varieties (2008)

### BibTeX

@MISC{Muñoz08tangentialprojections,

author = {Roberto Muñoz and José Carlos Sierra and Luis E. Solá Conde},

title = {Tangential projections and secant defective varieties},

year = {2008}

}

### OpenURL

### Abstract

Abstract: Going one step further in Zak’s classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of P n given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of PN that can be projected isomorphically into P2n when N ≥ − 2.

### Citations

722 | Algebraic Geometry, Graduate Texts - Hartshorne - 1977 |

260 | Geometry of algebraic curves - Arbarello, Cornalba, et al. - 1985 |

30 |
Weakly defective varieties
- Chiantini, Ciliberto
(Show Context)
Citation Context ...tailed account. Let us remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance, [B], [ChC1], =-=[ChC2]-=-, [CMR], [Ch], [CR]). Definition 2.4. Consider the notations of Section 2.1. Given k ≤ k0 and (x1, . . .,xk) ∈ U general, πk : X → Xk stands for the linear projection of X onto its image Xk from the l... |

23 | Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127, AMS - Zak - 1993 |

19 |
A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings
- Fulton, Hansen
- 1979
(Show Context)
Citation Context ...smooth part of X then the relative tangent variety of X with respect to Y is defined as T(Y, X) = ⋃ TyX. Let us recall the following useful consequence [Z2, Ch. I, Thm. 1.4] of FultonHansen’s Theorem =-=[FH]-=-. Lemma 2.2. Let X ⊂ P N be a projective variety and let Y ⊂ X be an irreducible closed subset contained in the smooth part of X. Then either: y∈Y 4(a) dim(T(Y, X)) = dim(Y ) + dim(X) = dim(S(Y, X)) ... |

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 ... low dimension. A smooth non-degenerate curve C ⊂ P N (N ≥ 3) cannot be projected isomorphically onto a plane curve. For n = 2 a complete list of surfaces with this property was achieved by Severi in =-=[S]-=-: Theorem 1.1. Let X ⊂ P 5 be a smooth irreducible complex and non-degenerate projective surface. If X can be projected isomorphically into P 4 , then X is the Veronese surface v2(P 2 ) ⊂ P 5 . The ca... |

11 | Varieties with one apparent double point
- Ciliberto, Mella, et al.
(Show Context)
Citation Context ...ccount. Let us remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance, [B], [ChC1], [ChC2], =-=[CMR]-=-, [Ch], [CR]). Definition 2.4. Consider the notations of Section 2.1. Given k ≤ k0 and (x1, . . .,xk) ∈ U general, πk : X → Xk stands for the linear projection of X onto its image Xk from the linear s... |

11 |
Varieties with minimal secant degree and linear systems of maximal dimensions on surfaces
- Ciliberto, Russo
- 2004
(Show Context)
Citation Context ...s remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance, [B], [ChC1], [ChC2], [CMR], [Ch], =-=[CR]-=-). Definition 2.4. Consider the notations of Section 2.1. Given k ≤ k0 and (x1, . . .,xk) ∈ U general, πk : X → Xk stands for the linear projection of X onto its image Xk from the linear space 〈Tx1X, ... |

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 ...ducible complex and non-degenerate projective surface. If X can be projected isomorphically into P 4 , then X is the Veronese surface v2(P 2 ) ⊂ P 5 . The case n = 3 was first considered by Scorza in =-=[Sc1]-=- and completed by Fujita in [Fu] (see Theorem 2.1). When n = 4 only some partial results are 1known, see [Sc2] and [FuR], where an infinite list of examples is shown. Hence the problem of getting a c... |

9 |
The possible dimensions of the higher secant varieties
- Catalano-Johnson
- 1996
(Show Context)
Citation Context ...1). See Example 3.9 for further examples. Remark 3.4. For any sequence of non-negative integers z = (z1, . . . , zr) there exists a projective variety X ⊂ P N such that ζ = z and k0 = r, as proved in =-=[CJ]-=-. In the following subsection we recall some arithmetic properties of the defective sequence of a smooth projective variety. 3.1 Additivity and superadditivity of the defective sequence of smooth proj... |

8 |
Threefolds with degenerate secant variety: on a theorem of
- Chiantini, Ciliberto
(Show Context)
Citation Context ...(X) when there is no ambiguity. As said in the introduction, the study of 1-defective, not necessarily smooth, varieties of small dimension goes back to Severi [S] and Scorza [Sc1] (see also [Fu] and =-=[ChC1]-=-), who completed the classification for dimension two and three, respectively. Theorem 2.1. Let X ⊂ P N be a non-degenerate 1-defective projective variety. (a) If dim(X) = 2 and N ≥ 5, then X is eithe... |

8 |
Varieties with small secant varieties: the extremal
- Fujita, Roberts
- 1981
(Show Context)
Citation Context ...nese surface v2(P 2 ) ⊂ P 5 . The case n = 3 was first considered by Scorza in [Sc1] and completed by Fujita in [Fu] (see Theorem 2.1). When n = 4 only some partial results are 1known, see [Sc2] and =-=[FuR]-=-, where an infinite list of examples is shown. Hence the problem of getting a complete classification for arbitrary dimension seems far from being reached. However, if N is big enough, Zak’s Theorem o... |

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 ...s the Veronese surface v2(P 2 ) ⊂ P 5 . The case n = 3 was first considered by Scorza in [Sc1] and completed by Fujita in [Fu] (see Theorem 2.1). When n = 4 only some partial results are 1known, see =-=[Sc2]-=- and [FuR], where an infinite list of examples is shown. Hence the problem of getting a complete classification for arbitrary dimension seems far from being reached. However, if N is big enough, Zak’s... |

4 |
Projective threefolds with small secant varieties
- Fujita
- 1982
(Show Context)
Citation Context ...e projective surface. If X can be projected isomorphically into P 4 , then X is the Veronese surface v2(P 2 ) ⊂ P 5 . The case n = 3 was first considered by Scorza in [Sc1] and completed by Fujita in =-=[Fu]-=- (see Theorem 2.1). When n = 4 only some partial results are 1known, see [Sc2] and [FuR], where an infinite list of examples is shown. Hence the problem of getting a complete classification for arbit... |

4 |
Tangents and secants of algebraic varieties: notes of a course
- RUSSO
- 2003
(Show Context)
Citation Context ...y π(C) and by π −1 (D) the strict transforms of C in Y and of D in X, respectively. 2.2 Tangential projections Let us recall the definition of tangential projection. We refer the interested reader to =-=[Ru]-=- for a more detailed account. Let us remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance,... |

4 |
Le superficie degli iperspazi con una doppia infinità di curve piane o spaziali, Atti della R. Accademia delle Scienze di Torino 56
- Segre
- 1921
(Show Context)
Citation Context ...CGH, p. 110]). If Dx is a plane conic then X is a projective surface with a two dimensional family of plane conics so that either X ⊂ P3 , or X = v2(P2 ) ⊂ P5 , or one of its projections into P4 (cf. =-=[Se2]-=-). But these cases can be excluded because Dx is not a conic for general x ∈ X. 6��� 3 The drop sequence and the defective sequence of a projective variety Let k ≤ k0 be a positive integer. The gener... |

3 |
The sum of powers as canonical expressions
- Bronowski
- 1932
(Show Context)
Citation Context ...for a more detailed account. Let us remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance, =-=[B]-=-, [ChC1], [ChC2], [CMR], [Ch], [CR]). Definition 2.4. Consider the notations of Section 2.1. Given k ≤ k0 and (x1, . . .,xk) ∈ U general, πk : X → Xk stands for the linear projection of X onto its ima... |

3 |
On the superadditivity of secant defects
- Fantechi
- 1990
(Show Context)
Citation Context ...0. An immediate corollary of this result is what we call superadditivity of the defective sequence (cf. [Z2, Ch. V, Thm. 1.8], having in mind that definitions of δk do not coincide; see also [Z1] and =-=[Fa]-=- for a more general statement): Corollary 3.7. Let X ⊂ P N be a smooth projective variety with drop sequence (ζ1, . . .,ζk0). Then ζi ≥ δ1 for all i, and the defective sequence of X verifies the super... |

3 |
Linear systems of hyperplane sections on varieties of small codimension, Funktsional. Anal. i Prilozhen 19
- Zak
- 1985
(Show Context)
Citation Context ... δ1(Z) = 0. An immediate corollary of this result is what we call superadditivity of the defective sequence (cf. [Z2, Ch. V, Thm. 1.8], having in mind that definitions of δk do not coincide; see also =-=[Z1]-=- and [Fa] for a more general statement): Corollary 3.7. Let X ⊂ P N be a smooth projective variety with drop sequence (ζ1, . . .,ζk0). Then ζi ≥ δ1 for all i, and the defective sequence of X verifies ... |

2 |
Lectures on the structure of projective embeddings
- Chiantini
- 2004
(Show Context)
Citation Context ... Let us remark that tangential projections have been used in other problems regarding projective varieties with special properties on their projections (see, for instance, [B], [ChC1], [ChC2], [CMR], =-=[Ch]-=-, [CR]). Definition 2.4. Consider the notations of Section 2.1. Given k ≤ k0 and (x1, . . .,xk) ∈ U general, πk : X → Xk stands for the linear projection of X onto its image Xk from the linear space 〈... |

1 |
Classification of n-dimensional subvarieties of G(1, 2n) that can be projected to G(1
- Arrondo, Ugaglia
- 2005
(Show Context)
Citation Context ... isomorphically into P2n , then N ≤ N(n) with equality if and only if X is the second Veronese embedding v2(Pn ) ⊂ PN(n) . The main result in the paper is an extension of this theorem, conjectured in =-=[ASU]-=-, where a similar statement was proved for subvarieties of grassmannians of lines: Theorem 1.3. Let X ⊂ P N be a smooth irreducible complex and non-degenerate projective variety of dimension n and let... |

1 |
Sulle Vn contenenti più di ∞ n−k Sk I
- Segre
- 1948
(Show Context)
Citation Context ...f F by the (n − 2)-tangential projection of X is a conic for a general choice of points x1, . . . , xn−2 ∈ F. Hence dim(L) = (n − 2)n + 2 − (n − 2)(n − 1) = n. Now we prove that L ⊂ L ′ is linear. By =-=[Se1]-=- it suffices to show that L contains a 3(n − 2)-dimensional family G parameterizing the planes L(T). A dimension count shows that dim(G) = dim(V ) − (n − 2)dim(XT), where XT = {x ∈ X | TxX ⊂ T }. We c... |