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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\textrm{OG6}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>OG6</mtext> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\textrm{Kum}}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>Kum</mtext> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\textrm{Kum}}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>Kum</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.</p>

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

  • 期刊论文

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  • 发表日期：

    2026年2月13日

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  • Ignacio Barros,
  • Pietro Beri,
  • Laure Flapan,
  • Brandon Williams

摘要

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 \({\textrm{OG6}}\) OG6 and \({\textrm{Kum}}_n\) Kum n -type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized \({\textrm{Kum}}_2\) Kum 2 -type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.