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<p>We explore maximality with respect to the classical Smith bound on the total Betti number of the real locus. For a large class of surfaces, we prove that the Hilbert square of a real surface is maximal if and only if the surface is maximal and has connected real locus. In particular, the Hilbert square of no K3 or abelian surface is maximal. We also exhibit various types of maximal surfaces, including ones with disconnected real locus, whose Hilbert square is maximal.</p>

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On the maximality problem for the Hilbert square of real surfaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月1日

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  • Viatcheslav Kharlamov,
  • Rareş Răsdeaconu

摘要

We explore maximality with respect to the classical Smith bound on the total Betti number of the real locus. For a large class of surfaces, we prove that the Hilbert square of a real surface is maximal if and only if the surface is maximal and has connected real locus. In particular, the Hilbert square of no K3 or abelian surface is maximal. We also exhibit various types of maximal surfaces, including ones with disconnected real locus, whose Hilbert square is maximal.