## VARIETIES WITH QUADRATIC ENTRY LOCUS, II (2008)

### BibTeX

@MISC{Ionescu08varietieswith,

author = {Paltin Ionescu and Francesco Russo},

title = {VARIETIES WITH QUADRATIC ENTRY LOCUS, II},

year = {2008}

}

### OpenURL

### Abstract

We continue the study, begun in [Ru07], of secant defective manifolds having “simple entry loci”. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also [IR07], their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of rational homogeneous manifolds. We also provide evidence that the classification of linearly normal dual defective manifolds with Picard group generated by the hyperplane section should follow along the same lines.

### Citations

150 | Threefolds whose canonical bundles are not numerically effective - Mori - 1982 |

132 |
The Adjunction Theory of Complex Projective Varieties
- Beltrametti, Sommese
- 1995
(Show Context)
Citation Context ...anifold if moreover assumed to be linearly normal. Combined with Remark 3.8, this would imply that maximal dual defective manifolds with Pic(X) ∼ = Z〈OX(1)〉 are homogeneous, as already conjectured in =-=[BS95]-=-. Our last application is the following: Proposition 4.6. Let X ⊂ P N be as in (1), with Pic(X) ∼ = Z〈OX(1)〉 and def(X) > 0. Assume moreover that def(X) > n−6 3 and N � 5n+2 3 . Then SX ̸= P N . Proof... |

118 | Higher-dimensional Algebraic Geometry - Debarre |

107 | Rationally connected varieties - Kollár, Miyaoka, et al. - 1992 |

101 | Projective manifolds with ample tangent bundles - Mori - 1979 |

68 | L.: Tangents and secants of algebraic varieties - Zak - 1993 |

48 |
Varieties with small dual varieties
- Ein
- 1986
(Show Context)
Citation Context ... improvement of one of the main results from [KS02] and is based on ideas developped in [Ru07], [IR07] and the present paper. The other application is a new proof of the classification, due to L. Ein =-=[Ei86]-=-, of manifolds with small dual. To the best of our knowledge, it is for the first time that classification of dual defective manifolds is connected to (and, conjecturally, even reduced to, cf. Remark ... |

23 | Rational curves on algebraic varieties, Ergeb - Kollár - 1996 |

22 | Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective - Hwang - 2001 |

16 | Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a suo punti tripli apparenti - Severi - 1901 |

11 | Varieties with one apparent double point
- Ciliberto, Mella, et al.
(Show Context)
Citation Context ...of the projective tangent space at x ∈ X. In Theorem 2.1 we show that the generic δ-partial tangential projection of a QELM is birational. The proof is based on a degeneration technique introduced in =-=[CMR04]-=- (see also [CR06]). The key point is that the general fibre of the (full) tangential projection is a degeneration of the general entry locus. Conversely, if the generic δpartial tangential projection ... |

11 |
Varieties with minimal secant degree and linear systems of maximal dimension on surfaces
- Ciliberto, Russo
(Show Context)
Citation Context ...tangent space at x ∈ X. In Theorem 2.1 we show that the generic δ-partial tangential projection of a QELM is birational. The proof is based on a degeneration technique introduced in [CMR04] (see also =-=[CR06]-=-). The key point is that the general fibre of the (full) tangential projection is a degeneration of the general entry locus. Conversely, if the generic δpartial tangential projection of X is birationa... |

11 | Determinazione delle varietà a tre dimensioni di Sr, (r ≥ 7), i cui S3 tangenti si tagliano a due a - Scorza - 1907 |

10 | Varieties with small secant varieties: The extremal - Fujita, Roberts - 1981 |

9 | Geometry of chains of minimal rational curves
- Hwang, Kebekus
(Show Context)
Citation Context ... In Section 3 we compare various properties of LQEL and CC-manifolds, Proposition 3.2. Examples show that the former class is much larger than the first. In particular, using results by Hwang-Kebekus =-=[HK05]-=- (see also [IR07]), we see that complete intersections (of high dimension with respect to the multidegree) are CCM, while the only LQELM which are complete intersections are the hyperquadrics, Proposi... |

9 | Sulle varietà a quattro dimensioni di Sr (r ≥ 9) i cui S4 tangenti si tagliano a due a - Scorza - 1909 |

8 |
On the homotopy groups of complex projective algebraic manifolds
- BARTH, LARSEN
- 1972
(Show Context)
Citation Context ...nto two categories. Manifolds admitting “non-trivial projections” are those for which SX ̸= P N , see [Se01, Za93], while manifolds “of small codimension” correspond to the other case, SX = P N , see =-=[BL72]-=-. Let x, y ∈ X be two general points and let p be a general point on the line 〈x, y〉. Consider the closure of the locus of secants to X passing through p. Its trace on X, denoted Σp, is called the ent... |

8 |
Almost-lines and quasi-lines on projective manifolds
- Bădescu, Beltrametti, et al.
- 2000
(Show Context)
Citation Context ...at the other extreme we found the stereographic projection of quadric hypersurfaces, the only QEL-manifolds of type equal to their dimension. 3. LQEL versus CC We recall the following definition from =-=[BBI00]-=-. Definition 3.1 (([BBI00])). A smooth rational curve C ⊂ X, where X is a projective manifold of dimension n, is a quasi-line if NC|X ≃ n−1 ⊕ OP1(1). The relation between the notions LQEL-manifold and... |

7 | Rationality properties of manifolds containing quasi-lines - Ionescu, Naie - 2003 |

6 | Inductive characterizations of hyperquadrics
- Fu
- 2008
(Show Context)
Citation Context ...th Pic(X) ∼ = Z〈OX(1)〉 are linear sections of rational homogeneous manifolds (which are completely classified), see Remark 3.8. The last section contains two applications. The first one, due to B. Fu =-=[Fu08]-=-, gives a substantial improvement of one of the main results from [KS02] and is based on ideas developped in [Ru07], [IR07] and the present paper. The other application is a new proof of the classific... |

6 |
Segre’s Reflexivity and an Inductive Characterization of Hyperquadrics
- Kachi, Sato
(Show Context)
Citation Context ...P N as above, such that through two general points x, y ∈ X there passes an r-dimensional quadric hypersurface, say Q r , contained in X. Observe that Q r ⊆ Σp if p ∈ 〈x, y〉; in particular r � δ, see =-=[KS02]-=-. When r = 1, we call such manifolds conic-connected (CCM for short). The extremal case r = δ was called “manifolds with local quadratic entry locus” (abbreviated as LQELM), while the special case whe... |

4 |
The sum of powers as canonical expression
- Bronowski
- 1933
(Show Context)
Citation Context ... is a rational variety of dimension n − δ. Therefore, Fx is rational, as the family Fx → Qx has a section. Being birational to Fx by Lemma 1.2, X is rational too. □ Remark 2.4. J. Bronowski claims in =-=[Br32]-=- that X ⊂ P 2n+1 is a variety with one apparent double point if and only if the projection of X to P n from a general tangent space to X is a birational map; he also formulates a generalization to arb... |

2 |
Homogeneous projective varieties with degenerate secants
- Kaji
(Show Context)
Citation Context ...ld with Picard group Z is homogeneous. Note that homogeneous manifolds arising from irreducible representations of (semi)simple complex algebraic groups are QEL-manifolds, see [Za93, Chapter III] and =-=[Ka99]-=-, and the secant defective ones are classified completely, see loc. cit. Moreover, the results in [Ru07, Section 3] confirm the conjecture for δ > n 2 . The next two finiteness results for QEL-manifol... |

2 |
Varieties with quadratic entry locus, I, preprint 2006, also math.AG/0701889
- Russo
(Show Context)
Citation Context ...ifolds may be studied in the context of Mori Theory, see [Mo79, Mo82, KMM92] and also [De01, Ko96, Hw01]. The interested reader can find further motivation and various examples in the introduction to =-=[Ru07]-=-. The present paper continues the line of investigation started in [Ru07], see also [IR07]. We acknowledge once more our intellectual debt to the classical work by G. Scorza, see [Sc08, Sc09]. The con... |

1 |
Conic-connected manifolds, preprint 2006, also math. AG/0701885
- Ionescu, Russo
(Show Context)
Citation Context ...ntinue the study, begun in [Ru07], of secant defective manifolds having “simple entry loci”. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also =-=[IR07]-=-, their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of r... |

1 |
Inductive characterizations of hyperquadrics, preprint
- FU
- 2006
(Show Context)
Citation Context ...th Pic(X) ∼ = Z〈OX(1)〉 are linear sections of rational homogeneous manifolds (which are completely classified), see Remark 3.8. The last section contains two applications. The first one, due to B. Fu =-=[Fu]-=-, gives a substantial improvement of one of the main results from [KS] and is based on ideas developped in [Ru], [IR] and the present paper. The other application is a new proof of the classification,... |