Algebraic geometry
Projective bundles and blowing ups
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1129-1133.

We study the blowing up X ˜ of a smooth projective variety X along a smooth center B that is equipped with a projective bundle structure over a variety Z. If B is a point, then X is a projective space. If the Picard number ρ(X) is 1, then dimZ has a lower bound dimX-dimB-1. Moreover, when dimZ is dimX-dimB-1, X is a projective space and B is a linear subspace in X. If X is a projective space n and B is a curve, then either n is 3 and B is a twisted cubic curve or n is an arbitrary integer and B is a line in n . If X is a quadric Q n and B is a curve, then n is 3 and B is a line in Q 3 .

Nous étudions l’éclatement X ˜ d’une variété projective lisse X le long d’un centre lisse B, munie d’une structure de fbré projectif. Si B est un point, X est un espace projectif. Si le nombre de Picard ρ(X) est 1, alors dimZ a une borne inférieure dimX-dimB-1. De plus, lorsque dimZ est dimX-dimB-1, X est un espace projectif et B est un sous-espace linéaire dans X. Si X est l’espace projectif n et B est une courbe, ou n est égale à 3 et B est une courbe cubique tordue, ou n est un entier arbitraire et B est une ligne droite dans n . Si X est une quadrique et B est une courbe, alors n est égale à 3 et B est une ligne droite dans Q 3 .

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DOI: 10.5802/crmath.249
Classification: 14J45, 14E30
Li, Duo 1

1 School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, China.
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Li, Duo. Projective bundles and blowing ups. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1129-1133. doi : 10.5802/crmath.249. http://www.numdam.org/articles/10.5802/crmath.249/

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