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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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