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On zero cycles of some hypersurfaces with automorphism. (Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme.) (French) Zbl 0786.14006

This paper gives a simple proof of Bloch’s conjecture \[ (p_ g(S)=0\Longrightarrow\text{CH}^ 0_ 0(S)\simeq\text{Alb}(S)) \] for any Godeaux surface, quotient of a complete intersection by a finite group. The proof is also applied in higher dimension and gives the following result: Let \(X\subset\mathbb{P}^ 4\) be a quintic hypersurface invariant under an involution acting trivially on \(H^{3,0}(X)\). Then \(\text{CH}_ 0(X)^ -=\{0\}\). To get this result one needs to show that \((JX)^ -\) is parametrized by algebraic cycles, which is done using the Noether-Lefschetz locus for invariant quintic fourfolds containing \(X\).
Reviewer: C.Voisin (Orsay)

MSC:

14C25 Algebraic cycles
14C05 Parametrization (Chow and Hilbert schemes)
14J70 Hypersurfaces and algebraic geometry
14J50 Automorphisms of surfaces and higher-dimensional varieties
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References:

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