Hyperbolicity and specialness of symmetric powers

Cadorel, Benoit; Campana, Frédéric; Rousseau, Erwan

doi:10.5802/jep.185

Cited by 1 publication

(3 citation statements)

References 48 publications

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“…Along the way, we obtain restrictions on varieties which satsify relations that share some properties with the Y − F (Y ) relation (Proposition 1.20, Corollary 1.21) and answer a question of Cadorel-Campana-Rousseau[6] related to a connection between uniruledness and symmetric products of varieties (Example 1 19…”

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confidence: 78%

“…In fact, the following result implies that Y itself is uniruled if d ≥ 2: Corollary 1.13. (Cadorel-Campana-Rousseau, Corollary 4.2 on p. 9 -10 of [6]) Suppose that X is compact and Kähler.…”

Section: Proof the Blowup At The Diagonal Blmentioning

confidence: 99%

“…In Question 1 on p. 10 of [6], it is asked whether X (m) being unirational (resp. rational, stably rational) for some m ≥ 2 implies that X is unirational (resp.…”

Section: Example 119 (Symmetric Products and Stable Birationality)mentioning

confidence: 99%

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Characterizing cubic hypersurfaces via projective geometry

Park¹

2022

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We use the cut and paste relation [Y [2] ] = [Y ][P m ] + L 2 [F (Y )] in K 0 (Var k ) of Galkin-Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. The same method also gives a family of other cut and paste relations that can only possibly be satisfied by cubic hypersurfaces.Suppose that Y is not a 1-dimensional family of quadrics or a 2-dimensional family of (projective) (m − 2)-planes (each isomorphic to P m−2 ). If m ≥ 7, then the following are equivalent:2. Y is a cubic hypersurface or the intersection of two quadric hypersurfaces. This statement can be replaced with Y being a cubic hypersurface if we assume that F (Y ) is connected. 3 (p. 12 of [16]).Remark 0.3. Here are some comments on the assumptions of Theorem 0.2. 1. If Hartshorne's conjecture (part 1 of Remark 1.7) holds in codimension 2, then the conditions on Y can be weakened to 2d − 4 ≤ n and n ≥ 7. Note that the uniruledness property already implies d ≤ n if Y is a hypersurface and d 1 + d 2 ≤ n if Y has codimension 2 and d 1 , d 2 are the degrees of the hypersurfaces whose intersection is equal to Y . Another possible replacement of the conditions is to take Y contained in some hypersurface of degree d such that 2d − 4 ≤ n.

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mentioning

confidence: 78%

Section: Proof the Blowup At The Diagonal Blmentioning

confidence: 99%