Joins and intersections

Flenner, Hubert; Gastel, L.J. van; Vogel, Wolfgang

doi:10.1007/bf01445234

Cited by 67 publications

(99 citation statements)

References 8 publications

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“…We have then

α_{K} = 0

K \supseteq E (x) \supseteq E

, and so

0 = α_{K *} false(β_{K} false) = {((), α_{L *}, false(β_{L} false))}_{K} .

Since

L \supseteq K

is Galois it follows from the theorem above

0 = α_{L *}^{\circ 2} (α_{L *} (β_{L})) = {((), α_{k (x) *}^{\circ 3}, false(β false))}_{L} .

Now

L \subseteq k (x)

is purely transcendental and so the restriction map

{CH}_{1} false(X_{k false(x false)} false) ⟶ {CH}_{1} false(X_{L} false)

is an isomorphism by [, Proposition 2.1.8]. Hence we have

α_{k (x) *}^{\circ 3} false(β false) = 0

as claimed.…”

Section: Rost Nilpotence For Threefolds and Zero Cyclesmentioning

confidence: 91%

See 1 more Smart Citation

On the Rost nilpotence theorem for threefolds

Gille

2017

Bull. London Math. Soc.

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We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if α •N k(X) * CH0(X k(X) ) = 0 for some integer N > 0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three-dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model.

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“…We have then

α_{K} = 0

K \supseteq E (x) \supseteq E

, and so

0 = α_{K *} false(β_{K} false) = {((), α_{L *}, false(β_{L} false))}_{K} .

Since

L \supseteq K

is Galois it follows from the theorem above

0 = α_{L *}^{\circ 2} (α_{L *} (β_{L})) = {((), α_{k (x) *}^{\circ 3}, false(β false))}_{L} .

Now

L \subseteq k (x)

is purely transcendental and so the restriction map

{CH}_{1} false(X_{k false(x false)} false) ⟶ {CH}_{1} false(X_{L} false)

is an isomorphism by [, Proposition 2.1.8]. Hence we have

α_{k (x) *}^{\circ 3} false(β false) = 0

as claimed.…”

Section: Rost Nilpotence For Threefolds and Zero Cyclesmentioning

confidence: 91%

“…Proof By homotopy invariance and the localization sequence for Chow groups this is clear if

E

is a purely transcendental field extension, see, for example, [, Proposition 2.1.8], and so we can assume that

E \supseteq k

is algebraic. Let

α \in {CH}_{n - 1} false(X false)

be such that

α_{E} = 0

.…”

Section: Chow Motives Rost Nilpotence and Zero Cyclesmentioning

confidence: 99%

On the Rost nilpotence theorem for threefolds

Gille

2017

Bull. London Math. Soc.

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“…(To be precise, apply Bertini's theorem for the normalizationX of X and the pull-backH of H , and take the push-forward ofX ∩H .) Moreover, since X is integral, (X \ L) ∩ H satisfies Serre's condition S 1 by [7], (3.4.6), and hence (X \ L) ∩ H is reduced (see [1], (VII.2.2)). On the other hand, since X ⊆ P N is nondegenerate, so is π L,X (X \ L) ⊆ P N −2 , and also so is its hyperplane section (see [7], p.116).…”

Section: Proof Of Corollarymentioning

confidence: 99%

“…z,X (π z,X (x))) + 1 by a local computation (see for example, [7], p.269, (8.4.3)). In particular, if l(X ∩ z, x ) = 2 for a point x ( = z) ∈ X, then π z,X is an embedding at x by (1.1.2).…”

Section: Atsushi Nomamentioning

confidence: 99%

Untitled

2010

Trans. Amer. Math. Soc.

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Let X be a nondegenerate projective variety of degree d and codimension e in a projective space P N defined over an algebraically closed field. We study the following two problems: Is the length of the intersection of X and a line L in P N at most d − e + 1 if L ⊆ X? Is the scheme-theoretic intersection of all hypersurfaces of degree at most d − e + 1 containing X equal to X? To study the second problem, we look at the locus of points from which X is projected nonbirationally.

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“…, x k are general, because in that case, Terracini's Lemma (cf. [4], Proposition 4.3.2) says that the general point on the secant k-plane spanned by these points is a regular point of S k (X ). And upper semi-continuity of the local dimension follows for example from the fact that every closed semi-algebraic set can be locally triangulated (cf.…”

Section: Theorem 36 Let X ⊂ a 2r Be An Irreducible Curve And Assume mentioning

confidence: 99%

Algebraic Boundaries of $$\mathrm{SO}(2)$$ SO ( 2 ) -Orbitopes

Sinn

2013

Discrete Comput Geom

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Let X ⊂ A 2r be a real curve embedded into an even-dimensional affine space. We characterise when the r th secant variety to X is an irreducible component of the algebraic boundary of the convex hull of the real points X (R) of X. This fact is then applied to 4-dimensional SO(2)-orbitopes and to the so called Barvinok-Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of 4-dimensional SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

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Joins and intersections

Cited by 67 publications

References 8 publications

On the Rost nilpotence theorem for threefolds

On the Rost nilpotence theorem for threefolds

Untitled

Algebraic Boundaries of $$\mathrm{SO}(2)$$ SO ( 2 ) -Orbitopes

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