Universal Curve with Unordered Marked Points

• Universal curve with unordered marked points is a fundamental moduli object encoding smooth curves of fixed genus with an arbitrary ordering of marked points.
• The nonsplitting of the homotopy exact sequence of profinite fundamental groups reveals key obstructions linked to nontrivial monodromy and algebraic structure.
• Advanced techniques like continuous relative completion and graded Lie algebra sections extend classical topological nonsplitting results to the algebraic context.

A universal curve with unordered marked points is a fundamental object in the study of the algebraic geometry of moduli spaces, encoding families of smooth, proper algebraic curves of fixed genus and an unordered collection of distinct marked points. Over a field of characteristic zero, the geometry of such universal curves is closely linked to the structure of their algebraic (profinite) fundamental groups, and profound obstructions arise when considering the exact sequences associated with their universal families. Recent results rigorously confirm that these obstructions manifest as a nonsplitting of homotopy exact sequences for the algebraic fundamental groups of both ordinary and hyperelliptic universal curves with unordered markings, generalizing and extending earlier topological results.

1. Moduli Stacks and Universal Curves with Unordered Marked Points

Let $\mathcal{M}_{g,n/k}$ denote the Deligne–Mumford stack over a field $k$ of characteristic zero classifying smooth, proper, genus-$g$ curves equipped with $n$ ordered distinct marked points. The stack $\mathcal{M}_{g,[n]/k} := [\mathcal{M}_{g,n/k}/S_n]$ encodes the moduli problem where the $n$ marked points are unordered, that is, up to the action of the symmetric group $S_n$ by permutation.

The universal curve $$\pi_{g,n/k} \colon \mathcal{M}_{g,n+1/k} \to \mathcal{M}_{g,n/k}$$ forgets the last marking; its fiber over $(C; x_1, \dots, x_n)$ is the smooth curve $C$ with an additional marked point. Descending this to the quotient yields the universal curve with unordered markings: $$\pi_{g,[n]/k} \colon \mathcal{M}_{g,[n]+1/k} = [\mathcal{M}_{g,n+1/k}/S_n] \longrightarrow \mathcal{M}_{g,[n]/k}.$$ Thus, $\pi_{g,[n]/k}$ is the universal curve of genus $g$ with $n$ unordered marked points.

2. Profinite Fundamental Groups and Homotopy Exact Sequence

Fixing a geometric point $$\bar y \colon \mathrm{Spec}(\bar k) \to \mathcal{M}_{g,[n]/k}$$, its fiber under $\pi_{g,[n]/k}$ is the curve $C$ over $\bar k$, and selecting $\bar x \in C(\bar k)$ yields a basepoint. The algebraic (profinite) fundamental groups assemble into the homotopy exact sequence: $$1 \longrightarrow \pi_1(C, \bar x) \longrightarrow \pi_1(\mathcal{M}_{g,[n]+1/k}, \bar x) \longrightarrow \pi_1(\mathcal{M}_{g,[n]/k}, \bar y) \longrightarrow 1.$$ Denote $X = \mathcal{M}_{g,[n]/k}$ and $X' = \mathcal{M}_{g,[n]+1/k}$; the above sequence describes how the fundamental group of the universal curve fiber sits inside the total space and base.

3. Nonsplitting Theorem for the Homotopy Exact Sequence

For any field $k$ of characteristic zero and genus $g \ge 3$, the homotopy exact sequence above does not admit a continuous algebraic splitting; explicitly, there is no section $\pi_1(X,\bar y) \to \pi_1(X',\bar x)$ compatible with the sequence. This nonexistence persists after passing through the quotient stack construction from the ordered to the unordered case, as any splitting in the unordered setting induces a splitting in the ordered case.

The proof hinges on several key ingredients:

• Any putative section on the unordered moduli yields one on the ordered moduli space.
• Profinite group sections induce sections after continuous relative completion.
• Such a section yields an $\mathrm{Sp}$-equivariant graded Lie algebra section of the associated graded Lie algebra of the unipotent radical.
• Hain’s classification shows that only the $n$ tautological sections exist for weight $-1$; these do not globally extend due to incompatibility with fiber homology monodromy.

The upshot is that any section would have to satisfy strong algebraic compatibilities that cannot be met; the nontrivial monodromy on the fibers precludes the construction of such a splitting for $g \ge 3$ (Watanabe et al., 15 Nov 2025).

4. Relative Completion and Obstruction Theory

Relative (continuous) completion provides the essential machinery for detecting the nonexistence of sections. For a profinite group $\Gamma$ and a Zariski-dense representation $\rho \colon \Gamma \to R$ with $R$ reductive (here $R = \mathrm{Sp}(H)$), the continuous relative completion $\mathcal{G}$ is a proalgebraic group fitting into

$$1 \longrightarrow U \longrightarrow \mathcal{G} \longrightarrow R \longrightarrow 1,$$

with $U$ prounipotent and universal for lifting $\rho$.

In the case of moduli, $$\Gamma = \pi_1(\mathcal{M}_{g,n}) \simeq \mathcal{G}_{g,n}$$ (mapping class group). Relative completion translates the existence of a splitting of the original profinite exact sequence into the existence of an $\mathrm{Sp}(H)$-equivariant graded Lie-algebra section, but classification results confirm that only "tautological" weight-$-1$ sections exist, and they do not assemble to a genuine group-theoretic section due to incompatibility at the level of homology. This provides a conceptual and technical obstruction in the profinite and algebraic framework, extending methods from topological settings.

5. Extension to Universal Hyperelliptic Curves

Analogous questions arise for the universal hyperelliptic curve. The hyperelliptic locus $\mathrm{Hyp}_{g,n} \subset \mathcal{M}_{g,n}$ parametrizes hyperelliptic curves with ordered marked points, and $\mathrm{Hyp}_{g,[n]} = [\mathrm{Hyp}_{g,n}/S_n]$ handles the unordered case. The universal hyperelliptic curve

$$\pi^{\mathrm{hyp}}_{g,[n]} \colon \mathrm{Hyp}_{g,[n]+1} \longrightarrow \mathrm{Hyp}_{g,[n]}$$

admits its own homotopy exact sequence of fundamental groups.

Employing the same relative completion framework, but for the hyperelliptic mapping class group $\Delta_{g,n}$, leads to a corresponding nonsplitting result: for $g \ge 3$, the homotopy exact sequence

$$1 \longrightarrow \pi_1(C,\bar x) \longrightarrow \pi_1(\mathrm{Hyp}_{g,[n]+1/k},\bar x) \longrightarrow \pi_1(\mathrm{Hyp}_{g,[n]/k},\bar y) \longrightarrow 1$$

does not split (Watanabe et al., 15 Nov 2025). This conclusion follows by classifying the possible weight-$-1$ sections (the "signed tautological" sections $\zeta_i^\pm$), none of which extend to genuine group-theoretic sections, as confirmed via their incompatibility on the fiber homology.

6. Comparison to Topological Results and Context

In the analytic and topological regime, Chen previously established that the universal unordered surface bundle over $\mathcal{M}_{g,[n]}^{an}$ does not admit a section, i.e., the associated exact sequence of topological orbifold fundamental groups does not split. The recent algebraic results not only generalize this, showing that the nonsplitting persists at the level of profinite (algebraic) fundamental groups in characteristic zero, but also introduce the machinery of continuous relative completion and associated graded Lie algebras to detect and explain the obstruction. This development integrates mixed Hodge theory and $\ell$-adic weights directly into the structure of the algebraic fundamental group sequence, showing the interplay between algebraic geometry, arithmetic, and topology (Watanabe et al., 15 Nov 2025).

7. Summary and Significance

The main theorems assert that for every characteristic-0 field, the exact sequences of algebraic fundamental groups associated to both ordinary and hyperelliptic universal curves with unordered marked points never split when $g \ge 3$. The obstruction is rooted in the geometry and representation theory of the moduli stacks, manifesting in the impossibility of constructing an $\mathrm{Sp}$-equivariant section at the level of graded Lie algebras after relative completion. This result clarifies the fundamental group-theoretic structure of these moduli problems and situates the algebraic theory as a natural extension of previously known topological nonsplitting phenomena (Watanabe et al., 15 Nov 2025).