We discuss the Picard group of the moduli space K g of quasi-polarized K3 surfaces of genus g ≤ 12 and g = 11. In this range, K g is unirational, and a general element in K g is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators for the Picard group Pic Q (K g ) using the Noether-Lefschetz (NL) theory. This verifies the NL conjecture on the moduli of K3 surfaces in these cases.

The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be solved in terms of the Noether-Lefschetz intersection numbers of the fibration and the reduced invariants of a K3 surface. Toward a generalization of this result to families with singular fibers, we introduce completed Noether-Lefschetz numbers using toroidal compactifications of the period space of elliptic K3 surfaces. As an application, we prove quasi-modularity for some genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and show that they satisfy a holomorphic anomaly equation. Sketch of a programAn old approach to computing Gromov-Witten invariants on X is to embed ι : X ֒→ P as a complete intersection in a more positive variety P , often with a torus action. Composing with ι induces an embedding of Kontsevich spaces M g (X, β) ֒→ M g (P, ι * β).

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