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Picard groups on moduli of \(K3\) surfaces with Mukai models. (English) Zbl 1333.14033

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 \(g\neq 11\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14D22 Fine and coarse moduli spaces

Citations:

Zbl 0701.14044
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