Irregular Stable Nodal Curve: Theory & Moduli

• Irregular stable nodal curves are genus-zero, nodal projective curves with marked points carrying higher-order jet data, generalizing classical stable curves.
• The stability condition adapts Deligne–Mumford criteria by counting higher pole orders as weighted points, ensuring a finite automorphism group.
• This framework compactifies moduli spaces of meromorphic connections with irregular singularities, crucial for analyzing Painlevé-type equations and non-abelian Hodge theory.

An irregular stable nodal curve is a generalization of the Deligne–Mumford notion of stable pointed curves, incorporating higher-order pole behavior by equipping marked points with additional jet data, and extending the combinatorial, stability, and boundary structures fundamental to the theory of stable curves. This construction is central in the compactification of moduli spaces of meromorphic connections and differential equations with irregular singularities, especially in genus zero. The framework is motivated by the need to systematically handle moduli of connections (or bundles with connections) that feature poles of order exceeding one, such as those arising in Painlevé-type equations and broader non-abelian Hodge theory.

1. Definition and Data of Irregular Stable Nodal Curves

An irregular stable nodal curve of arithmetic genus zero with polar divisor of total order $d$ is specified by the data

$$(X,\, p_1^{(a_1-1)},\, p_2^{(a_2-1)},\,\dots,\,p_n^{(a_n-1)})$$

with the following components:

• $X$ is a connected projective curve of arithmetic genus zero with only nodal singularities;
• For each marked smooth point $p_i\in X$, there is an assigned pole order $a_i\ge 1$, and an associated $(a_i-1)$-jet

$$j^{\,a_i-1}p_i \in \operatorname{Jet}^{a_i-1}_{p_i}(X) \cong \{\text{Taylor expansions of order } a_i-1 \text{ at } p_i\};$$

• The sum of all pole orders satisfies $\sum_{i=1}^n a_i = d$.

Combinatorially, each curve is encoded by a dual graph $\Gamma = (V \cup \{\ell_1, \dots, \ell_n\}, E)$, where vertices represent irreducible components $X_v \cong \mathbb{P}^1$, edges represent nodes, and legs (half-edges) correspond to marked points, each decorated by their pole order and jet data. The notation $(X, \{p_i^{(a_i-1)}\})$ suppresses further specifics but encodes all essential structure.

2. Stability Condition and its Generalization

The stability of such curves generalizes the genus zero Deligne–Mumford condition. Traditionally, every component must have at least three special points (marked points or nodes) to ensure a finite automorphism group. In the irregular setting, a point of pole order $a_i>1$ is counted as $a_i$ ordinary points, reflecting the richer infinitesimal data encoded by the jets. The precise stability inequality for each irreducible component $X_v$ (genus zero) is

$$\#\{\text{nodes on }X_v\} + \sum_{p_i\in X_v} a_i \geq 3.$$

This formula ensures stability in the sense that $\operatorname{Aut}(X,\{p_i^{(a_i-1)}\})$ is finite, as in the pure Deligne–Mumford case. This generalization is essential for moduli problems involving irregular singularities, as it dictates the admissible degenerations and boundary points in the compactified moduli space.

3. Local Model at Nodes and Gluing of Irregular Jets

Near a node $n \in X$, the standard local description involves coordinates $x$ (on the first branch) and $y$ (on the second), with the node defined by $xy = 0$. If a marked point with pole order $a$ is near the node on one branch, the jet is described by

$x \mapsto x + \sum_{k=2}^a c_k x^k \mod x^a.$

When two poles coalesce into a node, the jets on both branches must glue to define a well-posed local expansion across the node. The gluing condition is that the jets are reciprocal up to order $a-1$, that is,

$$xy = 0, \quad (x + \sum_{k=2}^a \alpha_k x^k)(y + \sum_{k=2}^a \beta_k y^k) = 0 \mod (xy)^a,$$

matching the corresponding coefficients to guarantee single-valuedness of the formal connection data. This condition is entirely absent in the ordinary stable pointed curve theory and is necessary to control the degenerations involving higher-order poles.

4. Moduli Spaces, Boundary Strata, and Compactification

Denote by

$\Mbar_{0,\, (a_1-1),\, \dots,\, (a_n-1)}^{\mathrm{irr}}$

the compactified moduli space of genus zero curves with $n$ irregular marked points (with orders $a_i$ and jet data), up to isomorphism.

• For all $a_i = 1$, one recovers $\Mbar_{0,n}$, the classical Deligne–Mumford moduli space.
• Boundary strata correspond to stable dual graphs: e.g., when poles of order $a$ and $b$ coalesce, the limiting curve comprises two components $X_v$ and $X_w$ joined at a node, with the jets now associated to the two branches and glued as above.
• Higher codimension strata arise from simultaneous collisions, resulting in trees of $\mathbb{P}^1$’s with more complex jet decorations.

For the specific case of rank-2 connections of total pole order 4 (a “Painlevé V” type example), the moduli space is described as an open subset of $\mathbb{P}^1_q \times \mathbb{P}^1_t$ with certain divisors removed, further compactified by a sequence of blowups to a weak Del Pezzo surface of degree 5, followed by contraction of a $(-2)$-curve to yield a mildly singular surface $\overline M$. The boundary divisors explicitly correspond to the various irreducible degenerations and collisions of the poles.

5. Key Local and Global Formulas

Several crucial formulas organize the geometry and moduli theory in this context:

Formula Type	Expression	Interpretation
Local normal form at irregular pole	$$\Omega = (A_a\,dz/z^a + A_{a-1}\,dz/z^{a-1} + \ldots + A_1\,dz/z) + (\text{holomorphic})$$	Principal part of connection at a pole of order $a$
Stability condition	$$\#\{\text{nodes on }X_v\} + \sum_{p_i\in X_v} a_i \ge 3$$	Generalized stability per component
Gluing of jets at node	$$(x+\alpha_2x^2+\dots+\alpha_ax^a)(y+\beta_2y^2+\dots+\beta_ay^a) = 0 \pmod{(xy)^a}$$	Gluing condition for jets at nodal branch
Fuchs relation for connections	$$\sum_{i=1}^n(\kappa_i^+ + \kappa_i^-) = -\deg(E) - \sum_{\text{nodes}}(\text{node‐contribution})$$	Compatibility for residues/formal types in global connections

This analytic and combinatorial machinery ensures that both local and global aspects of the moduli theory are compatible with higher-order singularity data.

6. Explicit Example: PV-Type Irregular Curve (Total Degree Four)

The simplest nontrivial example features a rank-2 connection with one double pole, two simple poles, and an apparent simple pole, modeled as

$$(\mathbb{P}^1;\, 0^{(0)},\, 1^{(1)},\, \infty^{(0)},\, q^{(0)}),$$

with corresponding jet data at the marked points. Degenerations where the simple moving pole $q$ approaches another fixed pole yield limiting curves composed of two $\mathbb{P}^1$'s meeting at a node, with jet data distributed and glued according to the described rules.

The moduli space is initially an open subset of $\mathbb{P}^1_q \times \mathbb{P}^1_t$, then compactified via three blow-ups to a weak Del Pezzo surface, and finally contracting a $(-2)$-curve to obtain the singular surface $\overline M$. The six boundary divisors correspond precisely to the six non-equivalent pairwise collisions of poles, each parametrizing a degeneration to a curve with two components and jet decorations summing to the total polar order.

7. Comparison with Classical Compactification

The construction of irregular stable nodal curves parallels the Deligne–Mumford approach by allowing nodal degenerations to compactify the moduli space. The key distinctions are:

• The moduli spaces are generally singular (e.g., possess canonical Du Val singularities in the degree 4 case) due to the contraction of exceptional curves, while classical $\Mbar_{0,n}$ is smooth.
• Jet data $j^{a_i-1}$ at each marked point must be tracked, dramatically enriching the combinatorial data of the dual graph versus the ordinary point-marked case.
• Stability involves both nodes and weighted contributions from pole orders: $\#(\text{nodes})+\sum_i a_i \ge 3$, rather than the classical $\#(\text{marked points})\ge 3$.
• At each node, a nontrivial matching condition for the associated jets is enforced to guarantee the descended formal type of the connection, a phenomenon without parallel in the pure pointed setting.

This suggests that the compactified space of irregular stable nodal curves serves as a natural generalization of the Deligne–Mumford space, capturing the necessary data for compactifying moduli of meromorphic connections with irregular singularities.

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Summary:

The theory of irregular stable nodal curves expands the moduli-theoretic toolkit for connections on rational curves with higher-order poles. Stability conditions, combinatorial stratification via dual graphs and jet data, and sophisticated boundary gluing prescribe a modular compactification analogous in philosophy—but considerably richer in structure—to the classical case. The result is a singular but natural compact moduli space whose boundary stratification reflects the intricate collision theory of higher-order poles and jets (Morbello, 6 Nov 2025).