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On the Noether-Lefschetz theorem and some remarks on codimension-two cycles. (English) Zbl 0552.14011
The theorem referred to in the title states that a surface \(S\subset {\mathbb{P}}^ 3\) of degree 4 or more, having general moduli, contains no curves other than intersections \(S\cap T\) with other surfaces. It was originally stated by (and is usually attributed to) Max Noether; it was first proved by Lefschetz.
The purpose of the present paper is to consider possible extensions of this theorem to higher-dimensional settings, e.g. to curves on 3-folds in \({\mathbb{P}}^ 4\). One difficulty is that all proofs to date of the Noether- Lefschetz theorem make essential use of Hodge theory, which is not directly applicable in such a setting. Accordingly, the first part of the paper gives an algebraic proof of the theorem, via a degeneration argument, and concludes with a discussion of what would be needed to make such an argument work, for example, in \({\mathbb{P}}^ 4\). - In the second part of the paper, possible ways of applying Hodge theory to the problem via the construction of normal functions are explored.

MSC:
14M07 Low codimension problems in algebraic geometry
14M10 Complete intersections
14J30 \(3\)-folds
14H45 Special algebraic curves and curves of low genus
14H10 Families, moduli of curves (algebraic)
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References:
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