Greer, Francois; Li, Zhiyuan; Tian, Zhiyu Picard groups on moduli of \(K3\) surfaces with Mukai models. (English) Zbl 1333.14033 Int. Math. Res. Not. 2015, No. 16, 7238-7257 (2015). The authors study the moduli space \(\mathcal{K}_g\) of primitively quasi-polarized \(K3\) surfaces of genus \(g\) for \(g\leq 12\) and \(g\neq 11\). In this range S. Mukai [in: Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. I, 357–377 (1988; Zbl 0701.14044)] proved that \(\mathcal{K}_g\) is unirational and the general element in \(\mathcal{K}_g\) is a complete intersection with respect to a vector bundle on a homogeneous space. The main result in this paper is to prove that the Picard group \(\text{Pic}_\mathbb{Q}(\mathcal{K}_g)\) is spanned by Noether-Lefschetz divisors. Moreover, the authors find the generators for \(\text{Pic}_\mathbb{Q}(\mathcal{K}_g)\) using Noether-Lefschetz theory. This verifies Noether-Lefschetz’s conjecture for \(g\) in the range \(g\leq 12\) and \(g\neq 11\). Reviewer: Ljudmila Kamenova (New York) Cited in 8 Documents MSC: 14J28 \(K3\) surfaces and Enriques surfaces 14D22 Fine and coarse moduli spaces Keywords:\(K3\) surfaces; Mukai models; Noether-Lefschetz theory Citations:Zbl 0701.14044 PDFBibTeX XMLCite \textit{F. Greer} et al., Int. Math. Res. Not. 2015, No. 16, 7238--7257 (2015; Zbl 1333.14033) Full Text: DOI arXiv