2000 character limit reached
Non-density of the exceptional components of the Noether-Lefschetz locus
Published 18 Dec 2023 in math.AG | (2312.11246v2)
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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