Unexpected loss of maximality: the case of Hilbert square of real surfaces

TL;DR

This work characterizes when the Hilbert square $X^{[2]}$ of a real surface $X$ is maximal in the Smith sense. It combines Smith theory, equivariant cohomology, and a cut-and-paste construction of $X^{[2]}$ to show that, for maximal $X$ with $H_1(X(\mathbb{C}),\mathbb{F}_2)=0$, maximality of $X^{[2]}$ is equivalent to $X(\mathbb{R})$ being connected; in particular, K3-surfaces (with $h^{2,0}>0$) cannot yield maximal $X^{[2]}$. The paper then derives precise Betti-number formulas, reduces the problem to computing $\beta_1(X^{[2]}(\mathbb{R}))$, and applies Mayer–Vietoris and boundary-operator arguments to obtain a complete picture in many cases, including elliptic, ruled, abelian, and rational surfaces, with broad implications for cubic 4-folds. Collectively, the results illuminate how real topology interacts with Hilbert schemes of points and provide a versatile toolkit for constructing (non-)maximal examples across a wide range of surfaces.

Abstract

We explore the maximality of the Hilbert square of maximal real surfaces, and find that in many cases the Hilbert square is maximal if and only if the surface has connected real locus. In particular, the Hilbert square of no maximal K3-surface is maximal. Nevertheless, we exhibit maximal surfaces with disconnected real locus whose Hilbert square is maximal.

Figures (1)

• Figure 1: The decomposition of $X^{[2]}(\mathbb{R})$

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• Corollary 2.2
• Definition 2.3