Complete Surfaces inside the Genus-Four Moduli Space

Definitions and mathematical statement

Let $\mathcal M_4$ denote the moduli space of smooth genus-$4$ complex curves, understood in the usual sense of smooth, connected, projective curves over ${\mathbb{C}}$. In the standard coarse algebraic moduli space interpretation, a closed complex algebraic subvariety $S\subseteq\mathcal M_4$ is called complete if its structure morphism $S\to\operatorname{Spec}{\mathbb{C}}$ is proper.

Does there exist a closed complex algebraic subvariety $S\subseteq\mathcal M_4$ such that \[ \dim_{{\mathbb{C}}}S=2 \qquad\text{and}\qquad S\longrightarrow\operatorname{Spec}{\mathbb{C}}\ \text{is proper}? \] The source imposes no smoothness condition on $S$. Completeness is required inside the moduli space of smooth curves, so a surface lying only in a compactification does not satisfy the question. The supplied record does not explicitly distinguish the coarse moduli space from the moduli stack or specify whether "complex subvariety" is intended algebraically or analytically; the formulation above makes the standard algebraic interpretation explicit.

Short English statement

Does the moduli space of smooth genus-$4$ complex curves contain a complete complex surface? In the algebraic interpretation stated above, the surface must be proper over ${\mathbb{C}}$, must have complex dimension $2$, and need not be smooth.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003296$, title "Complete Surfaces inside the Genus-Four Moduli Space," Oberwolfach Reports Open Problems collection, source problem code catalogue entry. The record gives proposed year $2016$ and does not name a proposer.
• Supplied report citation: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. https://doi.org/10.4171/owr/2016/56. The literature-source list also labels this reference "OWR 56 (2017)."
• Supplied literature source: Grushevsky--Mondello--Salvati Manni--Tsimerman, "compact subvarieties." https://arxiv.org/abs/2404.06009.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports that the Diaz and Looijenga bounds allow dimension $2$ in $\mathcal M_4$, while the checked literature supplies no existence or nonexistence theorem for a complete complex surface. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.