Segre’s reflexivity and an inductive characterization of hyperquadrics

Kachi, Yasuyuki; Satô, Eiichi

doi:10.1090/memo/0763

Cited by 21 publications

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“…Theorem 2) that if m > [n/2]+1, then N = n + 1 and X is itself a hyperquadric. This gives a substantial improvement to the Main Theorem 0.2 of [7], where the same claim is proved under the assumption that a general hyperquadric in the family is smooth and that m ≥ 3n/5 + 1. Our proof here, based on ideas contained in [5] and [9], is much simpler and is completely different from that in [7].…”

Section: Introductionmentioning

confidence: 70%

“…This gives a substantial improvement to the Main Theorem 0.2 of [7], where the same claim is proved under the assumption that a general hyperquadric in the family is smooth and that m ≥ 3n/5 + 1. Our proof here, based on ideas contained in [5] and [9], is much simpler and is completely different from that in [7]. However, we should point out that a more general result, without assuming the quadric subspaces pass all through a fixed point, is proven in [7].…”

Section: Introductionmentioning

confidence: 70%

“…Our proof here, based on ideas contained in [5] and [9], is much simpler and is completely different from that in [7]. However, we should point out that a more general result, without assuming the quadric subspaces pass all through a fixed point, is proven in [7].…”

Section: Introductionmentioning

confidence: 92%

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Inductive characterizations of hyperquadrics

2007

Math. Ann.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 70%

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Inductive characterizations of hyperquadrics

2007

Math. Ann.

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“…Definition 1.1 (cf. also [26,31,52]) An irreducible, non-degenerate projective variety X ⊂ P N is said to be a local quadratic entry locus variety of type δ ≥ 0, briefly an LQEL-variety of type δ, if, for general x, y ∈ X distinct points and for general p ∈ x, y ⊆ S X, the union of the irreducible components of the entry locus of p passing through x and through y is a quadric hypersurface of dimension δ = δ(X ) in the given embedding X ⊂ P N . Equivalently an LQEL-variety of type δ ≥ 0 is an irreducible projective variety X ⊂ P N if through two general points there passes a quadric hypersuface of dimension δ = δ(X ) contained in X .…”

Section: Definitions Preliminary Results and Examplesmentioning

confidence: 99%

Varieties with quadratic entry locus, I

Russo

2008

Math. Ann.

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Quadratic entry locus manifold of type $\delta$ $X\subset\mathbb P^N$ of dimension $n\geq 1$ are smooth projective varieties such that the locus described on $X$ by the points spanning secant lines passing through a general point of the secant variety $SX\subseteq\mathbb P^N$ is a smooth quadric hypersurface of dimension $\delta=2n+1-\dim(SX)$ equal to the secant defect of $X$. These manifolds appear widely and naturally among projective varieties having special geometric properties and/or extremal tangential behaviour. We prove that, letting $\delta=2r_X +1\geq 3$ or $\delta=2r_X+2$, then $2^{r_X}$ divides $n-\delta$. This is obtained by the study of the projective geometry of the Hilbert scheme $Y_x\subset \mathbb(T_x^*)$ of lines passing through a general point $x$ of $X$, allowing an inductive procedure. The Divisibility Property described above allows unitary and simple proofs of many results on $QEL$-manifolds such as the complete classification of those of type $\delta\geq n/2$, of Cremona transformation of type $(2,3)$, $(2,5)$. In particular we propose a new and very short proof of the fact that Severi varieties have dimension 2,4, 8 or 16 and also an almost self contained half page proof of their classification due to Zak.Comment: 16 pages; some misprints and imprecisions corrected; some references added; final version as appeared in Math. An

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“…On the other hand, even if X is smooth, a geometric quotient for V does not necessarily exist (see Example 2). We refer to [KS02] for a general introduction to this question and related ones.…”

Section: Introductionmentioning

confidence: 99%

On covering and quasi-unsplit families of curves

Bonavero¹,

Casagrande²,

Druel³

2007

J. Eur. Math. Soc.

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Abstract. Given a covering family V of effective 1-cycles on a complex projective variety X, we find conditions allowing one to construct a geometric quotient q : X → Y , with q regular on the whole of X, such that every fiber of q is an equivalence class for the equivalence relation naturally defined by V . Among other results, we show that on a normal and Q-factorial projective variety X with canonical singularities and dim X ≤ 4, every covering and quasi-unsplit family V of rational curves generates a geometric extremal ray of the Mori cone NE(X) of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for V .

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Segre’s reflexivity and an inductive characterization of hyperquadrics

Cited by 21 publications

References 0 publications

Inductive characterizations of hyperquadrics

Inductive characterizations of hyperquadrics

Varieties with quadratic entry locus, I

On covering and quasi-unsplit families of curves

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