On the Noether-Lefschetz theorem and some remarks on codimension-two cycles.

*(English)*Zbl 0552.14011The 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.

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) |

##### Keywords:

codimension-two cycles; curves on 3-folds as complete intersections; Noether-Lefschetz theorem; Hodge theory##### References:

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