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arXiv · 2312.11246

Non-density of the exceptional components of the Noether-Lefschetz locus

Gregorio Baldi·Bruno Klingler·Emmanuel Ullmo
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Abstract

We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^3$ acquires extra classes. In particular we show that the so called exceptional components of the Noether-Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether-Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective threefold and $L$ a very ample line bundle. Both results are applications of the Zilber-Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

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Gregorio Baldi, Bruno Klingler, Emmanuel Ullmo. 2023-12-18. Non-density of the exceptional components of the Noether-Lefschetz locus. https://arxiv.org/abs/2312.11246

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