For d ≥ 4, the Noether–Lefschetz locus NLd parametrizes smooth, degree d surfaces in ℙ3 with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the Noether–Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large d, but is true for d ≤ 5. She also showed that for d = 6, 7, there are finitely many reduced, irreducible components of NLd of non-maximal codimension. In this paper, we prove that for any d ≥ 6, there are infinitely many non-reduced irreducible components of NLd of non-maximal codimension.

中文翻译：

---

哈里斯猜想

为了d ≥ 4, Noether-Lefschetz 轨迹荷兰d参数化平滑度d表面在ℙ3Picard 数至少为 2。Harris 的一个猜想指出，非最大余维数的 Noether-Lefschetz 轨迹只有有限多个不可约分量。Voisin 表明该猜想在足够大的情况下是错误的d, 但对于d ≤ 5. 她还表明，对于d = 6, 7, 有有限多个减少, 的不可约成分荷兰d的非最大余维数。在本文中，我们证明对于任何d ≥ 6, 有无穷多个非还原的不可约成分荷兰d的非最大余维数。