Cones of Noether–Lefschetz divisors and moduli spaces of hyperkähler manifolds

We give a general formula for generators of the NL cone on an orthogonal modular variety. This is the cone of effective divisors linearly equivalent to an effective linear combination of irreducible components of Noether-Lefschetz divisors. We apply this to describe, in terms of minimal generators, the NL cone of various moduli spaces of geometric origin such as those of polarized K3 surfaces, cubic fourfolds, and hyperkähler manifolds. Additionally, we establish uniruledness for many moduli spaces of primitively polarized hyperkähler manifolds of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>OG6</mml:mtext></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>Kum</mml:mtext> <mml:mi>n</mml:mi></mml:msub> </mml:math> -type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>Kum</mml:mtext> <mml:mn>2</mml:mn></mml:msub> </mml:math> -type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.