Let X⊂PN be a smooth complex connected variety. The tangent and normal bundles produce a natural exact sequence: 0→TX→TPN|X→NX/PN→0. The sequence is known to split iff X is a linear subspace. The authors strengthen this. Let Y⊂X be an irreducible (possibly singular) curve, such that the sequence above splits, when it is restricted to Y. Then X is a linear subspace of PN. The proof is quite easy. First, it is shown that if the restriction splits then (KX+nH,Y)X≤0. Here n=dim(X) and H is the hyperplane section. Thus KX+nH is not very ample. Then an old result is used: If (under the conditions above) KX+nH is not very ample, then (X,H) is either (Pn,O(1)) or (Qn,O(1)) (a hyperquadric) or X is a projective bundle with the fibre being linear subspaces of PN. Finally, the last two cases are ruled out.

On a theorem of van de Ven

IONESCU, Paltin;
2008

Abstract

Let X⊂PN be a smooth complex connected variety. The tangent and normal bundles produce a natural exact sequence: 0→TX→TPN|X→NX/PN→0. The sequence is known to split iff X is a linear subspace. The authors strengthen this. Let Y⊂X be an irreducible (possibly singular) curve, such that the sequence above splits, when it is restricted to Y. Then X is a linear subspace of PN. The proof is quite easy. First, it is shown that if the restriction splits then (KX+nH,Y)X≤0. Here n=dim(X) and H is the hyperplane section. Thus KX+nH is not very ample. Then an old result is used: If (under the conditions above) KX+nH is not very ample, then (X,H) is either (Pn,O(1)) or (Qn,O(1)) (a hyperquadric) or X is a projective bundle with the fibre being linear subspaces of PN. Finally, the last two cases are ruled out.
2008
Ionescu, Paltin; Repetto, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1509313
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