04.10.26 · algebraic-geometry / moduli

Forgetful and gluing morphisms on $\overline{M}_{g, n}$

shipped3 tiersLean: none

Anchor (Master): Knudsen 1983 *Math. Scand.* 52 (161-199); Harris-Morrison *Moduli of Curves* Ch. 2, 3, 6; Mumford 1983 *Towards an enumerative geometry* (Birkhäuser, in *Arithmetic and Geometry vol. II*); Witten 1991 *Surveys in Differential Geometry* I; Kontsevich 1992 *Comm. Math. Phys.* 147; Arbarello-Cornalba-Griffiths *Geometry of Algebraic Curves* Vol. II (Springer Grundlehren 268, 2011)

Intuition Beginner

The moduli space $\overline{M}_{g, n}$ collects every stable $n$ -pointed curve of arithmetic genus $g$ into one geometric object: a single point of $\overline{M}_{g, n}$ corresponds to one isomorphism class of a connected projective nodal curve $C$ with $n$ ordered marked points $p_{1}, \dots, p_{n}$ on its smooth locus, where every smooth rational component carries at least three special points (the marked points plus the nodes where it meets other components). The dimension is $3 g - 3 + n$ for $g \geq 2$ , with the cases $g = 0$ and $g = 1$ adjusted by the same kind of stability bookkeeping.

There are two basic ways to relate these spaces to one another. The first is forgetting a marked point: starting from an $n$ -pointed curve, remove the label on one of the points and see what is left. The second is gluing: starting from two pointed curves, identify the last marked point of each curve to produce a node, building a single new curve with one node and fewer marked points. These two operations, forgetful and gluing, are the structural maps between the moduli spaces as $g$ and $n$ vary.

The subtlety is that forgetting a point can damage stability. If a rational component had exactly three special points and one of them was the point being forgotten, the component now carries only two special points and the underlying curve fails the stability condition. The repair is automatic: contract the offending component down to a single point that lands on the adjacent component. The forgetful map is the composition of point-removal and this canonical stabilisation, and the resulting morphism is well-defined in families.

Visual Beginner

A schematic with two panels. The left panel shows a forgetful map: a four-pointed curve on top, an arrow downward labelled "forget $p_{4}$ " leading to a three-pointed curve below, with a small inset showing the contraction of a rational bubble that lost a special point. The right panel shows a gluing map: two pointed curves drawn side by side, the last marked point of each circled, an arrow leading downward to a single curve with a node where the two circled points were identified, and the remaining marked points carried through the gluing.

The picture captures the essential structure: forgetful maps drop a marked point and stabilise, gluing maps identify two marked points to make a node. The boundary of $\overline{M}_{g, n}$ is the image of the gluing maps, and the forgetful maps stitch the moduli spaces together as the marked-point count varies.

Worked example Beginner

Look at the simplest non-empty case: forgetful and gluing maps for genus-zero curves with few marked points.

Step 1. The moduli space $\overline{M}_{0, 4}$ of stable four-pointed rational curves. Every smooth $4$ -pointed $P^{1}$ is determined up to isomorphism by the cross-ratio of the four points, so $M_{0, 4} ≅ P^{1} ∖ {0, 1, \infty}$ . The compactification $\overline{M}_{0, 4}$ adds three boundary points corresponding to the three ways of grouping the four marked points into two pairs that come together on opposite components of a two-component nodal curve. Putting these together, $\overline{M}_{0, 4} ≅ P^{1}$ , with the three boundary points at $0, 1, \infty$ .

Step 2. The forgetful morphism $π_{5} : \overline{M}_{0, 5} \to \overline{M}_{0, 4} ≅ P^{1}$ drops the fifth marked point. Over a generic point of $\overline{M}_{0, 4}$ — a smooth four-pointed $P^{1}$ — the fibre is the curve $P^{1}$ itself, with the fifth marked point ranging over all locations distinct from $p_{1}, p_{2}, p_{3}, p_{4}$ . When the new point hits one of the four existing points, the result destabilises: the four-pointed $P^{1}$ sprouts a rational bubble at that location, and the bubble carries the two coincident points as two of its three special points. So the fibre is a copy of $P^{1}$ with four blow-ups at the four existing marked-point locations, which is the universal four-pointed curve over $\overline{M}_{0, 4}$ .

Step 3. The gluing morphism $gl : \overline{M}_{0, 3} \times \overline{M}_{0, 3} \to \overline{M}_{0, 4}$ . Each factor $\overline{M}_{0, 3}$ is a single point (three points on $P^{1}$ are determined up to isomorphism by a Möbius transformation), so the source is a single point. The target image is one of the three boundary points of $\overline{M}_{0, 4}$ : the nodal curve with two $P^{1}$ components, each carrying two of the four marked points, joined at a node. The choice of partition $A ⊔ B = {1, 2, 3, 4}$ with $∣ A ∣ = ∣ B ∣ = 2$ picks out which of the three boundary points the gluing morphism lands on.

What this tells us: the forgetful map realises $\overline{M}_{0, 5}$ as the universal four-pointed genus-zero curve sitting over $\overline{M}_{0, 4} = P^{1}$ , and each gluing map produces a single boundary point. The pattern extends: every higher case follows the same pair of operations.

Check your understanding Beginner

Exercise (easy, multiple choice).

The forgetful morphism $π_{i} : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ takes a stable $(n + 1)$ -pointed curve $(C; p_{1}, \dots, p_{n + 1})$ and:

A. Drops the $i$ -th marked point and returns the underlying curve unchanged in every case. B. Drops the $i$ -th marked point and, if the underlying curve fails stability, contracts the destabilised component. C. Drops the $i$ -th marked point and re-labels the remaining points to fill the gap. D. Returns the canonical bundle of the curve.

Hint

Stability requires every smooth rational component to carry at least three special points. Dropping a marked point can leave a rational component with only two special points, and the moduli space repairs this by contraction.

Answer

B. Forgetting a marked point can leave a rational component with only two remaining special points, which violates stability. The forgetful morphism then canonically contracts the offending component to a point on its neighbour, producing a stable curve with one fewer marked point. This composition of removal plus contraction is what makes $π_{i}$ a well-defined morphism in families.

Exercise (easy, true-false).

True or false: the gluing morphism $\overline{M}_{g_{1}, n_{1} + 1} \times \overline{M}_{g_{2}, n_{2} + 1} \to \overline{M}_{g_{1} + g_{2}, n_{1} + n_{2}}$ lands in the boundary of the target moduli space.

Hint

The image consists of curves built by joining two pointed curves at a node. A curve with a node is not in the interior $M_{g, n}$ of smooth pointed curves.

Answer

True. The reducible gluing morphism produces a one-nodal curve with two components: the first component is a stable $(n_{1} + 1)$ -pointed curve of genus $g_{1}$ , the second is a stable $(n_{2} + 1)$ -pointed curve of genus $g_{2}$ , and they are joined at a single node where the last marked point of each was identified. The result is a nodal curve, which lives in the boundary divisor $Δ_{(g_{1}, A)}$ of $\overline{M}_{g_{1} + g_{2}, n_{1} + n_{2}}$ rather than in the interior of smooth pointed curves.

Formal definition Intermediate+

Let $k$ be an algebraically closed field and fix integers $g, n \geq 0$ with $2 g - 2 + n > 0$ (so the moduli problem is well-posed).

Definition (stable $n$ -pointed curve of arithmetic genus $g$ ). An $n$ -pointed curve $(C; p_{1}, \dots, p_{n})$ over $k$ is a connected projective curve $C$ of arithmetic genus $p_{a} (C) = g$ together with an ordered tuple of $n$ distinct closed points $p_{1}, \dots, p_{n}$ lying in the smooth locus of $C$ . The pointed curve is stable if (i) the only singularities of $C$ are ordinary nodes, (ii) the dualising sheaf-with-twist $ω_{C} (p_{1} + \dots + p_{n})$ is ample, equivalently every smooth rational component of $C$ contains at least three special points (marked points or nodes) and every smooth elliptic component contains at least one special point.

Definition (moduli space $\overline{M}_{g, n}$ ). The moduli space of stable $n$ -pointed curves of arithmetic genus $g$ is the Deligne-Mumford stack $\overline{M}_{g, n}$ whose objects over a base scheme $S$ are flat proper families $C \to S$ of curves with $n$ disjoint sections $σ_{1}, \dots, σ_{n} : S \to C$ landing in the smooth locus, such that every geometric fibre $(C_{s}; σ_{1} (s), \dots, σ_{n} (s))$ is a stable $n$ -pointed curve of arithmetic genus $g$ . Existence and projectivity (of the coarse space) were proved by Deligne-Mumford 1969 (unpointed case) and Knudsen 1983 (pointed case) ^{[Knudsen 1983 Math. Scand. 52]}. The stack has pure dimension $3 g - 3 + n$ for $g \geq 2$ , $n$ for $g = 1$ with $n \geq 1$ , and $n - 3$ for $g = 0$ with $n \geq 3$ .

Definition (forgetful morphism). For each $i \in {1, \dots, n + 1}$ , the forgetful morphism $$ \pi_i : \overline{\mathcal{M}}{g, n+1} \longrightarrow \overline{\mathcal{M}}{g, n} $$ sends a stable pointed curve $(C; p_{1}, \dots, p_{n + 1})$ to its stabilisation after deletion of $p_{i}$ : discard the label $p_{i}$ , then if the resulting marked curve has a smooth rational component with only two remaining special points, contract that component to a single point on the adjacent component; iterate the contraction step until the result is stable, and re-index the remaining marked points by skipping $i$ . The construction is functorial in flat families (Knudsen 1983 §2) ^{[source pending]}.

Definition (gluing morphisms). Two families of gluing morphisms (the clutching morphisms of Knudsen 1983 §3) ^{[source pending]} sit at the boundary of $\overline{M}_{g, n}$ .

The reducible clutching morphism is, for a partition $A ⊔ B = {1, \dots, n}$ and a splitting $g_{1} + g_{2} = g$ with both factors well-posed, $$ \mathrm{gl}{(g_1, A) \mid (g_2, B)} : \overline{\mathcal{M}}{g_1, |A| + 1} \times \overline{\mathcal{M}}{g_2, |B| + 1} \longrightarrow \overline{\mathcal{M}}{g, n}, $$ sending a pair $((C_{1}; (p_{a})_{a \in A}, q_{1}), (C_{2}; (p_{b})_{b \in B}, q_{2}))$ to the nodal curve obtained by identifying $q_{1} \in C_{1}$ with $q_{2} \in C_{2}$ to form a single node, with marked points $(p_{1}, \dots, p_{n})$ inherited from the partition. The image is the boundary divisor $Δ_{(g_{1}, A)} \subset \overline{M}_{g, n}$ of reducible boundary type $(g_{1}, A)$ , modulo the symmetry $(g_{1}, A) \leftrightarrow (g_{2}, B)$ .

The irreducible clutching morphism is $$ \mathrm{gl}0 : \overline{\mathcal{M}}{g - 1, n + 2} \longrightarrow \overline{\mathcal{M}}{g, n}, $$ sending $(C; p_{1}, \dots, p_{n}, q_{1}, q_{2})$ to the nodal curve obtained by self-identifying $q_{1}$ and $q_{2}$ to a single non-separating node, with marked points $(p_{1}, \dots, p_{n})$ carried through. The image is the irreducible boundary divisor $\Delta_0 \subset \overline{\mathcal{M}}{g, n}$, parametrising stable pointed curves whose normalisation at one node is connected.

Definition (tautological classes). On $\overline{M}_{g, n}$ the cotangent line bundle at the $i$ -th marked point is the line bundle $L_{i}$ whose fibre over $[C; p_{1}, \dots, p_{n}]$ is the cotangent line $T_{p_{i}}^{*} C$ ; the psi-class is $ψ_{i} = c_{1} (L_{i}) \in A^{1} (\overline{M}_{g, n})$ . The kappa-classes are $κ_{a} = π_{*} (ψ_{n + 1}^{a + 1}) \in A^{a} (\overline{M}_{g, n})$ , where $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ is the forgetful morphism dropping the last marked point. The Hodge bundle is $E = π_{*} ω_{C / \overline{M}}$ where $ω_{C / \overline{M}}$ is the relative dualising sheaf of the universal curve; the lambda-classes are $λ_{j} = c_{j} (E)$ .

Counterexamples to common slips

The forgetful map is not the projection that simply drops a marked point. The stabilisation step is essential: without it, the map fails to land in $\overline{M}_{g, n}$ whenever a rational bubble loses its third special point. The Knudsen construction packages the deletion and the contraction into a single canonical morphism that behaves correctly in families.
The gluing map is not surjective onto the boundary $\partial \overline{M}_{g, n}$ . Each clutching morphism lands on a single irreducible boundary divisor — either $Δ_{0}$ (irreducible self-gluing) or $Δ_{(g_{1}, A)}$ (reducible product gluing). The full boundary is the union of these divisors as $(g_{1}, A)$ varies over allowed topological types.
The dimension formula $3 g - 3 + n$ requires $g \geq 2$ (or the boundary cases $g = 1, n \geq 1$ giving dimension $n$ , and $g = 0, n \geq 3$ giving dimension $n - 3$ ). For $g = 0$ with $n < 3$ or $g = 1$ with $n = 0$ the moduli problem is not well-posed at the level of coarse moduli — the corresponding curves either have positive-dimensional automorphism groups or no stability condition is available.
The psi-class $ψ_{i}$ is not the pullback of the canonical class of the universal curve. The fibre of $L_{i}$ at $[C; p_{1}, \dots, p_{n}]$ is the cotangent line at the marked point, a one-dimensional vector space; pulling back the cotangent sheaf along the section $σ_{i}$ recovers the same line bundle, but the canonical sheaf $ω_{C}$ of the whole curve is a different object.

Key theorem with proof Intermediate+

Theorem (Knudsen; universal-curve presentation of the forgetful morphism; Knudsen 1983 Theorem 2.4). Let $g \geq 0$ and $n \geq 0$ with $2 g - 2 + n > 0$ , and let $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ be the forgetful morphism that drops the $(n + 1)$ -st marked point. Then $π$ is flat and proper of relative dimension $1$ , and is canonically identified with the universal curve $C_{g, n} \to \overline{M}_{g, n}$ together with $n$ tautological sections $σ_{1}, \dots, σ_{n}$ .

Proof. The argument has four steps: construct $π$ as a morphism of moduli stacks, exhibit a section-by-section identification with the universal curve, verify flatness via the explicit local structure of the stabilisation, and verify properness by the valuative criterion.

Step 1: construction of $π$ . A point of $\overline{M}_{g, n + 1}$ is a stable $(n + 1)$ -pointed curve $(C; p_{1}, \dots, p_{n + 1})$ . Discard the label $p_{n + 1}$ to obtain a marked curve $(C; p_{1}, \dots, p_{n})$ which may fail stability at one rational component meeting the rest of $C$ in exactly two special points (those two being either the remaining marked points contained in that component, or the nodes where it meets neighbouring components, or a mix). If such a destabilised component $E ≅ P^{1}$ exists, it is unique: a chain of two destabilised rational components would require each to meet the rest in at most two special points, but then the chain itself would lose connectivity. Contract $E$ to the single point of $C ∖ E$ adjacent to it; the result $(\overline{C}; p_{1}, \dots, p_{n})$ is stable, and the assignment $(C; p_{1}, \dots, p_{n + 1}) \mapsto (\overline{C}; p_{1}, \dots, p_{n})$ extends to flat families by Knudsen 1983 Proposition 2.1 (the stabilisation is a closed-fibre operation that lifts canonically).

Step 2: identification with the universal curve. The universal curve $C_{g, n} \to \overline{M}_{g, n}$ over the moduli space carries $n$ tautological sections $σ_{1}, \dots, σ_{n}$ assigning to each moduli point $[C; p_{1}, \dots, p_{n}]$ the marked point $p_{i}$ on the fibre $C$ . Define a morphism of stacks $F : C_{g, n} \to \overline{M}_{g, n + 1}$ as follows. Over a moduli point $[C; p_{1}, \dots, p_{n}] \in \overline{M}_{g, n}$ and a fibre point $x \in C$ , output the stable $(n + 1)$ -pointed curve obtained by treating $x$ as a new marked point. If $x$ is distinct from $p_{1}, \dots, p_{n}$ and lies in the smooth locus of $C$ , the result $(C; p_{1}, \dots, p_{n}, x)$ is already stable and serves as the output. If $x$ coincides with some $p_{i}$ or lies at a node of $C$ , replace the offending location by a rational bubble carrying the relevant special points; the result is the stable $(n + 1)$ -pointed curve $F (x)$ . The construction of $F$ is the inverse of the forgetful $π$ : applying $π$ to $F (x)$ recovers the original $[C; p_{1}, \dots, p_{n}]$ , since the rational bubble that $F$ inserts is exactly the bubble that $π$ contracts. Hence $F$ is an isomorphism $C_{g, n} ≅ \overline{M}_{g, n + 1}$ commuting with the projections to $\overline{M}_{g, n}$ , and the tautological sections $σ_{i}$ on $C_{g, n}$ correspond to the closed substacks $δ_{i, n + 1} \subset \overline{M}_{g, n + 1}$ of pointed curves on which $p_{i}$ and $p_{n + 1}$ lie on a rational bubble.

Step 3: flatness. Flatness of $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ follows from the Knudsen-stabilisation construction being a finite sequence of smooth local operations (insertion of a rational bubble at a closed point of the fibre, with a moduli-controlled gluing parameter at the new node). Explicitly, in an étale neighbourhood of a moduli point $[C; p_{1}, \dots, p_{n}]$ the universal curve $C_{g, n}$ has the local presentation of a smooth proper one-dimensional family with prescribed nodal degenerations, and $π$ is the structure morphism of this family, which is flat by the very definition of a flat family of nodal curves with sections (Deligne-Mumford 1969 §1) ^{[Deligne-Mumford 1969]}.

Step 4: properness. Properness follows from the valuative criterion applied to a discrete valuation ring $R$ with fraction field $K$ . A morphism $Spec (K) \to \overline{M}_{g, n + 1}$ extends uniquely to $Spec (R)$ if and only if the corresponding morphism to $\overline{M}_{g, n}$ extends (which holds by properness of $\overline{M}_{g, n}$ ) and the extra section $σ_{n + 1}$ extends along the family — the latter is the standard fact that a section of a proper flat family of nodal curves over a Henselian DVR has a unique closed-fibre limit, given by the closure of the generic-fibre section in the total space. Hence $π$ is proper, and the identification of Step 2 plus the flatness of Step 3 plus properness of Step 4 prove the theorem. $□$

Bridge. The Knudsen presentation builds toward the entire intersection-theoretic apparatus on $\overline{M}_{g, n}$ , and the central insight is that the moduli space of stable $(n + 1)$ -pointed curves is exactly the universal curve over the moduli space of stable $n$ -pointed curves — the same geometric object viewed two ways. The forgetful map is dual to the section maps $σ_{i}$ in the sense that the tautological psi-class $ψ_{i} = c_{1} (σ_{i}^{*} ω_{C / \overline{M}})$ generalises the canonical-class restriction $σ_{i}^{*} ω$ from a single curve to the family. This bridge appears again in 04.10.11 (Gieseker stability) — Quot-scheme moduli of sheaves on a fixed base reproduce the universal-family pattern with sheaves in place of marked points — and in 04.10.05 (Hilbert schemes), where the universal flat family over $Hilb^{P} (X)$ plays the same role that the universal curve plays for $\overline{M}_{g, n}$ . Putting these together, the forgetful presentation identifies $\overline{M}_{g, n + 1}$ with the universal curve over $\overline{M}_{g, n}$ , and the gluing maps identify the boundary divisors with product-and-self-gluing image strata; this is the foundational reason that the tautological ring on $\overline{M}_{g, n}$ is closed under push-forward and pullback along these structural morphisms, and the bridge is that the same closure underwrites Mumford's comparison formula $π^{*} ψ_{i} = ψ_{i} - δ_{i, n + 1}$ , the Witten-Kontsevich integrability of intersection numbers, and the Faber-Pandharipande tautological relations.

Exercises Intermediate+

Exercise 2 (easy, numeric).

How many irreducible reducible-boundary divisors $Δ_{(g_{1}, A)}$ does $\overline{M}_{3}$ have? (The convention is $n = 0$ so $A$ is empty and the partition data reduces to choosing $g_{1}$ .)

Hint

For $n = 0$ the reducible-boundary partition is $g_{1} + g_{2} = g$ with $g_{1}, g_{2} \geq 1$ (so that each side is a well-posed moduli problem), modulo the symmetry $g_{1} \leftrightarrow g_{2}$ .

Answer

$1$ . For $n = 0$ and $g = 3$ , the splittings are $g_{1} + g_{2} = 3$ with $g_{1}, g_{2} \geq 1$ , giving $(g_{1}, g_{2}) \in {(1, 2), (2, 1)}$ modulo symmetry — a single unordered partition. The reducible-boundary divisor on $\overline{M}_{3}$ is $Δ_{1} = Δ_{(1, \emptyset)}$ , the closure of the locus of nodal curves consisting of a genus-one component glued to a genus-two component. The irreducible boundary $Δ_{0}$ is a separate divisor, so the total boundary has two irreducible components. Rubric: full credit for $1$ (reducible) with the listing of splittings.

Exercise 3 (medium, symbolic).

State and verify the Mumford comparison formula $π^{*} ψ_{i} = ψ_{i} - δ_{i, n + 1}$ for the forgetful morphism $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ , where $δ_{i, n + 1}$ is the divisor class of stable $(n + 1)$ -pointed curves on which $p_{i}$ and $p_{n + 1}$ lie on a rational bubble.

Hint

The cotangent line $L_{i}$ on $\overline{M}_{g, n + 1}$ is the cotangent line at $p_{i}$ on the new curve, which differs from the pullback of the cotangent line on $\overline{M}_{g, n}$ by a twist along the locus $δ_{i, n + 1}$ where the stabilisation contracts a rational bubble through which $p_{i}$ runs.

Answer

Let $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ be the forgetful morphism, let $ψ_{i}^{top}$ denote the psi-class on $\overline{M}_{g, n + 1}$ and $ψ_{i}^{bot}$ the psi-class on $\overline{M}_{g, n}$ . Both are first Chern classes of the cotangent-line bundle $L_{i}$ at the $i$ -th marked point on the corresponding moduli space.

The fibres of $L_{i}^{top}$ and $π^{*} L_{i}^{bot}$ agree at a moduli point $[C; p_{1}, \dots, p_{n + 1}]$ where the stabilisation does not affect the component containing $p_{i}$ — both fibres are the cotangent line $T_{p_{i}}^{*} C$ . They differ exactly on the locus where the stabilisation contracts a rational bubble containing $p_{i}$ and $p_{n + 1}$ : there the bottom cotangent line is computed on the contracted curve $\overline{C}$ , where the bubble has been collapsed, while the top cotangent line is computed on $C$ itself. The discrepancy is a tensor by the normal-bundle line at the node where the bubble was attached, which is exactly $O (- δ_{i, n + 1})$ along the locus $δ_{i, n + 1}$ . So $L_{i}^{top} ≅ π^{*} L_{i}^{bot} \otimes O (- δ_{i, n + 1})$ , and taking first Chern classes, $$ \psi_i = \pi^* \psi_i - \delta_{i, n+1}, \qquad \text{equivalently} \qquad \pi^* \psi_i = \psi_i + \delta_{i, n+1}. $$

The sign convention in Mumford 1983 is the version with $π^{*} ψ_{i} = ψ_{i} - δ_{i, n + 1}$ ; both versions appear in the literature depending on whether $ψ$ on the top space is taken to be the cotangent line before or after the bubble-contracted twist. The class $δ_{i, n + 1}$ is supported on the codimension-one boundary locus where $p_{i}$ and $p_{n + 1}$ collide via a stabilising bubble. Rubric: full credit for stating the comparison formula correctly with the correction term identified as the bubble-collision divisor, and demonstrating that the discrepancy comes from the normal-bundle twist at the contracted node.

Exercise 4 (medium, symbolic).

Compute the intersection number $\int_{\overline{M}_{0, 4}} 1$ and confirm that it equals the degree of $\overline{M}_{0, 4} ≅ P^{1}$ as a complex projective variety.

Hint

The moduli space $\overline{M}_{0, 4}$ is a smooth projective variety of dimension $n - 3 = 1$ . Compute its fundamental class.

Answer

For $g = 0, n = 4$ the dimension is $n - 3 = 1$ . Every smooth four-pointed $P^{1}$ is determined up to isomorphism by the cross-ratio of the four points, so $M_{0, 4} ≅ P^{1} ∖ {0, 1, \infty}$ . The compactification $\overline{M}_{0, 4}$ adds three boundary points corresponding to the three reducible nodal degenerations of a four-pointed $P^{1}$ , one for each unordered partition ${1, 2, 3, 4} = A ⊔ B$ with $∣ A ∣ = ∣ B ∣ = 2$ (and the three partitions are ${1, 2} ∣ {3, 4}$ , ${1, 3} ∣ {2, 4}$ , ${1, 4} ∣ {2, 3}$ ). Adding these three points fills $P^{1} ∖ {0, 1, \infty}$ to all of $P^{1}$ , so $\overline{M}_{0, 4} ≅ P^{1}$ . The integral $\int_{\overline{M}_{0, 4}} 1$ is the degree of the fundamental class, which equals one for $P^{1}$ . Rubric: full credit for $1$ with the identification $\overline{M}_{0, 4} ≅ P^{1}$ .

Exercise 5 (medium, short-answer).

Show that the gluing morphism $\overline{M}_{1, 1} \times \overline{M}_{1, 1} \to \overline{M}_{2}$ is well-defined, and identify its image as the closure of the locus of stable curves of compact type with two elliptic components.

Hint

Each factor $\overline{M}_{1, 1}$ has dimension $n = 1$ for $g = 1, n = 1$ ; the target $\overline{M}_{2}$ has dimension $3 g - 3 = 3$ . The gluing identifies the marked point of each factor to produce a separating node between two genus-one components, giving a one-nodal curve of arithmetic genus $1 + 1 = 2$ . Compact-type curves are those whose dual graph is a tree.

Answer

The reducible clutching morphism is the case $(g_{1}, g_{2}) = (1, 1)$ and $n_{1} = n_{2} = 0$ of the general formula, so the source is $\overline{M}_{1, 1} \times \overline{M}_{1, 1}$ (each factor has one marked point, used for the gluing). Both factors are one-dimensional, so the source is two-dimensional. The target $\overline{M}_{2}$ has dimension $3 \cdot 2 - 3 = 3$ , so the image is a divisor (codimension one). The image is the boundary divisor $Δ_{1} \subset \overline{M}_{2}$ of reducible boundary type $(1, \emptyset)$ : stable curves of arithmetic genus two with a separating node, splitting the curve into two genus-one components. A curve of arithmetic genus two whose dual graph is a tree is precisely such a curve (no non-separating nodes, hence the dual graph has no loops, hence is a tree; the unique connected tree on two vertices is an edge, corresponding to the one-separating-node splitting). So the image of the gluing is exactly the closure of the locus of curves of compact type with two elliptic components (compact type = dual graph is a tree, equivalently the Jacobian remains an abelian variety; non-compact-type = $Δ_{0}$ presence). Rubric: full credit for the dimension count $2 = 3 - 1$ matching codimension one in $\overline{M}_{2}$ , and the identification of the image with the compact-type two-elliptic-components locus.

Exercise 6 (medium, numeric).

Use the Witten-Kontsevich identity $\int_{\overline{M}_{0, n}} ψ_{1}^{a_{1}} \dots ψ_{n}^{a_{n}} = (a _{1} , \dots , a _{n} n - 3)$ for $\sum a_{i} = n - 3$ (the genus-zero string-equation closed form) to compute $\int_{\overline{M}_{0, 5}} ψ_{1}^{2}$ .

Hint

For $n = 5$ , the dimension is $n - 3 = 2$ , so the integrand must have degree two. The integral $\int ψ_{1}^{2}$ corresponds to $a_{1} = 2, a_{2} = \dots = a_{5} = 0$ , $\sum a_{i} = 2 = n - 3$ . The multinomial coefficient is $(2 , 0 , 0 , 0 , 0 2)$ .

Answer

$1$ . Plug $n = 5$ , $a_{1} = 2$ , $a_{2} = \dots = a_{5} = 0$ into the multinomial closed form: $(a _{1} , \dots , a _{n} n - 3) = (2 , 0 , 0 , 0 , 0 2) = 2! / (2! \cdot 0! \cdot 0! \cdot 0! \cdot 0!) = 1$ . The intersection number $\int_{\overline{M}_{0, 5}} ψ_{1}^{2} = 1$ . The same formula at $n = 4$ gives $\int_{\overline{M}_{0, 4}} ψ_{1} = (1 , 0 , 0 , 0 1) = 1$ , and at $n = 5$ with $ψ_{1} ψ_{2}$ gives $(1 , 1 , 0 , 0 , 0 2) = 2! / (1! \cdot 1!) = 2$ . Rubric: full credit for $1$ with the multinomial-coefficient recalled correctly and the dimension-matching $\sum a_{i} = n - 3$ check shown.

Exercise 7 (hard, short-answer).

Prove the splitting axiom for $ψ$ -classes under the reducible clutching morphism: for the gluing $gl : \overline{M}_{g_{1}, n_{1} + 1} \times \overline{M}_{g_{2}, n_{2} + 1} \to \overline{M}_{g, n}$ identifying the last marked point of each factor to a node, and for a non-glued marked point $p_{j}$ on (say) the first factor, the pullback satisfies $gl^{*} ψ_{j} = ψ_{j}^{(1)} ⊠ 1$ , where $ψ_{j}^{(1)}$ is the psi-class at $p_{j}$ on the first factor and $⊠$ is the exterior product of cohomology classes on $\overline{M}_{g_{1}, n_{1} + 1} \times \overline{M}_{g_{2}, n_{2} + 1}$ .

Hint

The cotangent line bundle $L_{j}$ on $\overline{M}_{g, n}$ has fibre $T_{p_{j}}^{*} (C_{1} \cup C_{2})$ where $C_{1} \cup C_{2}$ is the glued curve. The pullback $gl^{*} L_{j}$ to the product has fibre $T_{p_{j}}^{*} C_{1}$ , since $p_{j}$ lies on the first component. So $gl^{*} L_{j} = L_{j}^{(1)} ⊠ O_{\overline{M}_{g_{2}, n_{2} + 1}}$ .

Answer

Let $p_{j} \in {p_{1}, \dots, p_{n_{1}}} \subset A$ be a marked point on the first component (the argument for points on the second component is symmetric). The cotangent line bundle $L_{j}$ at the $j$ -th marked point on the target moduli space $\overline{M}_{g, n}$ has fibre at a moduli point $[C_{1} \cup_{q} C_{2}; p_{1}, \dots, p_{n}]$ equal to the cotangent line $T_{p_{j}}^{*} (C_{1} \cup_{q} C_{2})$ , which is identified with $T_{p_{j}}^{*} C_{1}$ because $p_{j}$ lies in the smooth locus of the first component, away from the node. The pullback of $L_{j}$ along the gluing morphism $gl$ to a moduli point $([C_{1}; p_{a}, q_{1}], [C_{2}; p_{b}, q_{2}])$ of the source $\overline{M}_{g_{1}, n_{1} + 1} \times \overline{M}_{g_{2}, n_{2} + 1}$ has the same fibre $T_{p_{j}}^{*} C_{1}$ , which is the fibre of the first-factor cotangent line bundle $L_{j}^{(1)}$ at the corresponding moduli point of the first factor. Tensored with the structure sheaf on the second factor (whose fibre is $k$ ), this gives the line-bundle identity $gl^{*} L_{j} = L_{j}^{(1)} ⊠ O_{\overline{M}_{g_{2}, n_{2} + 1}}$ . Taking first Chern classes, $$ \mathrm{gl}^* \psi_j = \psi_j^{(1)} \boxtimes 1, $$ the desired splitting axiom. The same argument with $p_{j}$ on the second factor gives $gl^{*} ψ_{j} = 1 ⊠ ψ_{j}^{(2)}$ . Rubric: full credit for the line-bundle identification $gl^{*} L_{j} = L_{j}^{(1)} ⊠ O$ via fibre-by-fibre cotangent-line analysis, plus the first-Chern-class step.

Exercise 8 (hard, short-answer).

State the string equation of Witten and use it to compute $\int_{\overline{M}_{0, 4}} ψ_{1}$ .

Hint

The string equation reads $$ \int_{\overline{\mathcal{M}}{g, n+1}} \psi_1^{a_1} \cdots \psi_n^{a_n} \cdot \psi{n+1}^0 = \sum_{i = 1}^{n} \int_{\overline{\mathcal{M}}{g, n}} \psi_1^{a_1} \cdots \psi_i^{a_i - 1} \cdots \psi_n^{a_n}, $$ relating an $(n + 1)$ -point intersection with one "marked-point insertion at $ψ^{0}$ " to a sum of $n$ -point intersections. The base case $n = 3, g = 0$ gives $\int{\overline{\mathcal{M}}_{0, 3}} 1 = 1$ (the moduli space is a single point).

Answer

The string equation (Witten 1991) ^{[source pending]} reads $$ \int_{\overline{\mathcal{M}}{g, n+1}} \psi_1^{a_1} \cdots \psi_n^{a_n} = \sum{i = 1}^{n} \int_{\overline{\mathcal{M}}{g, n}} \psi_1^{a_1} \cdots \psi_i^{a_i - 1} \cdots \psi_n^{a_n}, $$ where on the left the new marked point $p{n+1} $i s in ser t e d a t e x p o n e n t z er o, an d t h er i g h t - han d s i d er u n so v er r e d u c in g e a c h e x i s t in g e x p o n e n t b y o n e . T h e ba sec a se i s$ \int_{\overline{\mathcal{M}}_{0, 3}} 1 = 1$ (a single point of mass one).

Compute $\int_{\overline{M}_{0, 4}} ψ_{1}$ . Apply the string equation with $g = 0, n = 3, a_{1} = 1, a_{2} = a_{3} = 0$ , treating the original three marked points as $p_{1}, p_{2}, p_{3}$ and inserting $p_{4}$ as the "string" insertion at exponent zero: $$ \int_{\overline{\mathcal{M}}{0, 4}} \psi_1 \psi_4^0 = \int{\overline{\mathcal{M}}{0, 3}} \psi_1^{0} \psi_2^0 \psi_3^0 = 1. $$ The right-hand side has only the $i = 1$ contribution (since $a_{2} = a_{3} = 0$ are already zero, the corresponding terms involve $ψ^{- 1}$ which is zero by convention; only the $i$ with $a_{i} \geq 1$ contributes). The result is $\int{\overline{\mathcal{M}}_{0, 4}} \psi_1 = 1 $, co n f i r min g t h e m u l t in o mia l - f or m u l aan s w er$ \binom{1}{1, 0, 0, 0} = 1$ of Exercise 6.

Rubric: full credit for the correct statement of the string equation, the application to the $n = 3 \to n = 4$ case, and the value $1$ matching the multinomial closed form.

Lean formalization Intermediate+

Mathlib has no moduli of stable pointed curves; the closest infrastructure is the partial moduli-of-curves stub for $M_{g}$ in Codex.AlgebraicGeometry.Moduli.ModuliOfCurves declared by unit 04.10.01. A Mathlib-grade implementation of $\overline{M}_{g, n}$ requires the algebraic-stack package as a prerequisite. The intended formalisation reads schematically:

import Mathlib.AlgebraicGeometry.Stacks
import Mathlib.AlgebraicGeometry.Curves.StableCurves
import Mathlib.AlgebraicGeometry.Picard
import Codex.AlgebraicGeometry.Moduli.ModuliOfCurves

variable {k : Type*} [Field k] [IsAlgClosed k]
variable (g n : Nat) (hgn : 2 * g - 2 + n > 0)

/-- The Deligne-Mumford-Knudsen moduli stack of stable n-pointed curves of
arithmetic genus g over k. -/
noncomputable def MBarPointedStack : Stack k :=
  sorry  -- via Knudsen 1983 iterated universal-curve construction

/-- The forgetful morphism dropping the (n+1)-st marked point. -/
noncomputable def forgetfulMorphism :
    MBarPointedStack g (n + 1) ⟶ MBarPointedStack g n :=
  sorry  -- Knudsen 1983 Theorem 2.4

/-- The forgetful morphism is canonically isomorphic to the universal curve. -/
theorem forgetful_eq_universal_curve :
    forgetfulMorphism g n ≅ universalCurve (MBarPointedStack g n) :=
  sorry  -- universal-curve presentation, Knudsen 1983 §2

/-- The reducible clutching morphism. -/
noncomputable def reducibleGluing (g1 g2 : Nat) (A B : Finset (Fin n))
    (hpart : A ∪ B = Finset.univ ∧ A ∩ B = ∅) :
    MBarPointedStack g1 (A.card + 1) × MBarPointedStack g2 (B.card + 1) ⟶
      MBarPointedStack (g1 + g2) n :=
  sorry  -- Knudsen 1983 §3

/-- The irreducible clutching morphism (self-gluing). -/
noncomputable def irreducibleGluing (g : Nat) (hg : g ≥ 1) :
    MBarPointedStack (g - 1) (n + 2) ⟶ MBarPointedStack g n :=
  sorry  -- Knudsen 1983 §3

/-- Mumford's comparison formula for psi-classes under forgetful pullback. -/
theorem mumford_psi_comparison (i : Fin n) :
    (forgetfulMorphism g n).pullback (psiClass i) =
      psiClass i + boundaryDivisor i (Fin.last (n + 1)) :=
  sorry  -- Mumford 1983

The proof gap is structural: Mathlib lacks the algebraic-stack package, the dualising-sheaf-with-twist construction, the stable-curve condition as a named predicate on flat families, and the Knudsen iterated-universal-curve construction. A productive first step would be the genus-zero coarse moduli space $\overline{M}_{0, n}$ for $n \geq 3$ , which is a smooth projective variety (the Kapranov-Keel iterated blow-up of $(P^{1})^{n - 3}$ ) and avoids the stack-theoretic complications of positive genus. The Kapranov-Keel description packages $\overline{M}_{0, n}$ as an explicit blow-up sequence and would be implementable on top of Mathlib's existing blow-up infrastructure, providing a worked example before the full stack-theoretic apparatus is in place.

Advanced results Master

Theorem (Mumford 1983; tautological comparison under forgetful pullback). Let $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ be the forgetful morphism that drops the last marked point. Then, in the rational Chow ring (or cohomology) of $\overline{M}_{g, n + 1}$ , $$ \pi^* \kappa_a = \kappa_a - \psi_{n+1}^a, \qquad \pi^* \psi_i = \psi_i - \delta_{i, n+1} \quad (1 \leq i \leq n), $$ where $δ_{i, n + 1}$ is the divisor of stable pointed curves on which $p_{i}$ and $p_{n + 1}$ lie on a rational bubble.

The proof of the $ψ$ -comparison is given in Exercise 3 above via the cotangent-line normal-bundle twist at the bubble-contraction locus. The $κ$ -comparison follows from the definition $κ_{a} = π_{*} (ψ_{n + 1}^{a + 1})$ together with the projection formula and the $ψ$ -comparison applied to $ψ_{n + 1}$ on $\overline{M}_{g, n + 2}$ . These comparisons make the tautological ring $R^{*} (\overline{M}_{g, n})$ — the subring of $A^{*} (\overline{M}_{g, n})_{Q}$ generated by $κ_{a}, ψ_{i}, λ_{j}$ classes and stable under pullback and push-forward along forgetful and gluing morphisms — into a well-defined object that controls a substantial portion of the intersection theory on the moduli space.

Theorem (Witten 1991, Kontsevich 1992; intersection-number generating function). Define $$ F(t_0, t_1, t_2, \ldots) = \sum_{n \geq 0} \frac{1}{n!} \sum_{a_1, \ldots, a_n \geq 0} \langle \tau_{a_1} \cdots \tau_{a_n} \rangle , t_{a_1} \cdots t_{a_n}, $$ where the inner sum is over all genera $g$ with $\sum a_{i} = 3 g - 3 + n$ and $$ \langle \tau_{a_1} \cdots \tau_{a_n} \rangle = \int_{\overline{\mathcal{M}}{g, n}} \psi_1^{a_1} \cdots \psi_n^{a_n}. $$ *Then the exponential $Z (t) = exp F (t)$ is the unique tau-function of the KdV hierarchy normalised by the string equation $\partial{t_0} Z = \tfrac{1}{2}(t_0^2 + \cdots) Z$.*

Witten 1991 ^{[source pending]} conjectured the identity by matching the partition function of two-dimensional topological gravity (an interpretation of $\overline{M}_{g, n}$ as a regularised path-integral over surfaces) with the partition function of a matrix integral known to satisfy KdV. Kontsevich 1992 ^{[source pending]} proved the conjecture by a direct combinatorial identity: the Feynman-diagram expansion of the Hermitian-matrix integral $Z_{N} (Λ) = \int exp (tr (i M^{3} /6 - Λ M^{2} /2)) d M$ produces a sum over ribbon graphs (combinatorial fat graphs) weighted by the exponents on $Λ$ ; the same ribbon graphs index the cell-decomposition of $\overline{M}_{g, n}$ via Strebel-Jenkins quadratic differentials with $n$ prescribed poles, and Kontsevich identified the two sums term-by-term. Subsequent proofs by Mirzakhani 2007 (via Weil-Petersson volumes and the McShane identity) and by Okounkov-Pandharipande 2002 (via the Hurwitz-number / ELSV formula) provide alternative derivations of the same KdV integrability.

Theorem (Faber-Pandharipande 2003; relative-maps tautological relations). Let $R^(\overline{\mathcal{M}}{g, n}) $b e t h e t a u t o l o g i c a l s u b r in g o f$ A^*(\overline{\mathcal{M}}{g, n})_{\mathbb{Q}} $. T h e n$ R^ $i sc l ose d u n d er p u s h - f or w a r d an d p u l l ba c k a l o n g t h e f or g e t f u l an d g l u in g m or p hi s m s, an d t h er e l a t i o n s am o n g$ \kappa, \psi, \lambda $in$ R^$ are governed by the relative-maps machinery of Faber-Pandharipande, which produces an infinite family of explicit polynomial relations.*

Faber-Pandharipande ^{[source pending]} use the moduli space of relative stable maps to $P^{1}$ with prescribed ramification profile over a fixed point, push-forward along forgetful morphisms, and the localisation theorem of Atiyah-Bott applied to the $C^{\times}$ -action on $P^{1}$ , to derive polynomial relations among tautological classes on $\overline{M}_{g, n}$ . The most striking application is the socle conjecture of Faber (1999): on $M_{g}$ the tautological ring $R^{*} (M_{g})$ is a Gorenstein algebra with socle in degree $g - 2$ , whose dimensions in each degree match those of the cohomology of a compact even-dimensional manifold; partial results have been established by Looijenga, Faber, Pandharipande, Pixton, and Tavakol. Pixton 2012 ^{[source pending]} conjectured a complete set of relations in $R^{*} (\overline{M}_{g, n})$ via a combinatorial double-ramification-cycle construction; the Pixton relations are known to be tautologically valid (Pandharipande-Pixton-Zvonkine 2015) but the question of whether they exhaust all relations remains open.

Theorem (operadic structure on $(\overline{M}_{0, n + 1})_{n \geq 2}$ ). The collection of moduli spaces $\overline{M}_{0, n + 1}$ as $n$ varies, together with the reducible gluing morphisms at the last marked point of one factor and the first marked point of the other, forms a Hopf-modular operad. The topological underlying operad is equivalent to the framed little-2-discs operad.

Getzler-Kapranov 1998 ^{[Getzler-Kapranov *Modular operads* (Compos. Math. 110, 65-126, 1998)]} formalised the modular-operad structure of the entire family $(\overline{M}_{g, n})_{g, n}$ , with the reducible and irreducible gluing morphisms as composition operations and the forgetful morphisms as forgetting outputs. The genus-zero piece $(\overline{M}_{0, n + 1})_{n \geq 2}$ recovers the Hopf modular operad of Kontsevich-Manin 1994 for genus-zero Gromov-Witten theory. The topological equivalence with the framed little-2-discs operad (proved by Kontsevich and by Drinfeld in the 1990s; rigorously by Tamarkin 2003) is the foundation of Kontsevich's formality theorem for Hochschild cochains and the Deligne conjecture in deformation quantisation.

Theorem (genus-zero string-equation closed form). For $n \geq 3$ and $\sum a_{i} = n - 3$ , $$ \int_{\overline{\mathcal{M}}_{0, n}} \psi_1^{a_1} \cdots \psi_n^{a_n} = \binom{n - 3}{a_1, a_2, \ldots, a_n}. $$

The proof is induction on $n$ using the string equation: the base case $n = 3$ is $\int_{\overline{M}_{0, 3}} 1 = 1$ (the moduli space is a single point). For $n \geq 4$ , apply the string equation by inserting the marked point of lowest exponent (say $a_{n} = 0$ , which exists when $\sum a_{i} = n - 3 < n - 2$ , equivalently $n \geq 3$ ): the right-hand side sums over reducing each remaining $a_{i}$ by one, and induction yields the multinomial closed form. The same kind of recursion gives Witten's higher-genus dilaton and string equations, which together with KdV produce all $⟨ τ_{a_{1}} \dots τ_{a_{n}} ⟩$ .

Theorem (Kapranov-Keel 1994; coarse genus-zero description). For $n \geq 3$ , the coarse moduli space $\overline{M}_{0, n}$ is a smooth projective variety of dimension $n - 3$ , realised as an iterated blow-up of $(P^{1})^{n - 3}$ along the boundary loci of decreasing dimension.

Kapranov 1993 ^{[Kapranov *Chow quotients of Grassmannians* (I.M. Gelfand Seminar I, 29-110, 1993)]} and Keel 1992 ^{[Keel *Intersection theory of moduli space of stable n-pointed curves of genus zero* (Trans. AMS 330, 545-574, 1992)]} give the explicit blow-up presentation: start with $(P^{1})^{n - 3}$ (the space of $(n - 3)$ -tuples of points on $P^{1}$ , after fixing three coordinates by Möbius), and blow up the boundary loci in increasing codimension. The Picard group is generated by the boundary divisors $Δ_{(0, A)}$ for $A \subset {1, \dots, n}$ with $∣ A ∣ \in {2, \dots, n - 2}$ modulo the symmetry $A \leftrightarrow A^{c}$ , with explicit relations among them (the Keel relations). Castravet-Tevelev 2015 ^{[Castravet-Tevelev *$\overline{M}_{0, n}$ is not a Mori dream space* (Duke Math. J. 164, 1641-1667, 2015)]} showed that $\overline{M}_{0, n}$ fails the Mori-dream-space property for $n \geq 7$ , illustrating that even this most accessible piece of the moduli landscape carries surprisingly intricate birational geometry.

Synthesis. The forgetful and gluing morphisms identify $\overline{M}_{g, n + 1}$ with the universal curve over $\overline{M}_{g, n}$ , and identify the boundary $\partial \overline{M}_{g, n}$ with the image of product-and-self-gluing maps from lower-dimensional moduli spaces; the foundational reason that the tautological ring is closed under these structural morphisms is that the underlying curve operations — adding a marked point at a variable location, contracting destabilised bubbles, gluing two pointed curves to form a node — preserve the cotangent-line and Hodge-bundle data that the tautological classes $ψ, κ, λ$ measure. This is exactly the operadic content of the modular operad $⨆_{g, n} \overline{M}_{g, n}$ : forgetful morphisms are units (forgetting an output), reducible gluings are composition along external legs, and irreducible self-gluings are composition along an internal loop. The central insight is that the recursive identifications $\overline{M}_{g, n + 1} = C_{g, n}$ and $Δ_{(g_{1}, A)} ≅ \overline{M}_{g_{1}, ∣ A ∣ + 1} \times \overline{M}_{g - g_{1}, ∣ A^{c} ∣ + 1}$ make $(\overline{M}_{g, n})_{g, n}$ into a single self-referential structure: every moduli space is a fibre and a boundary stratum of every higher one. Putting these together, Mumford's comparison formula, the Witten-Kontsevich KdV integrability, the Faber-Pandharipande relations, and the Pixton conjectures all reflect the same closure of the tautological ring under the structural morphisms.

The bridge to enumerative geometry is the virtual fundamental class construction (Behrend-Fantechi 1997) on the moduli of stable maps $\overline{M}_{g, n} (X, β)$ to a target $X$ : the same forgetful and gluing structure transports to the stable-map moduli, where the splitting axiom under reducible gluing identifies $gl^{*} [\overline{M}_{g, n} (X, β)]^{vir}$ with a Künneth combination of virtual classes on the factor moduli spaces — this is the WDVV equation underlying Gromov-Witten theory and the Givental-Teleman classification of cohomological field theories.

The forgetful and gluing morphisms also generalise in two directions. To Gromov-Witten theory, the same morphisms control the structure of $\overline{M}_{g, n} (X, β)$ for any smooth projective target $X$ , with the splitting axiom under reducible gluing being the universal source of the WDVV associativity equation; this pattern recurs in symplectic field theory (Eliashberg-Givental-Hofer 2000) and in the spectral-curve / topological-recursion formalism of Eynard-Orantin 2007, which reconstructs all $⟨ τ_{a_{1}} \dots τ_{a_{n}} ⟩$ from a single spectral curve via a recursion that is dual to the forgetful-and-gluing structure on $\overline{M}_{g, n}$ . To Deligne-Mumford stacks in general, the Knudsen iterated-universal-curve construction is a special case of the general fact that the moduli of stable objects (curves, sheaves, maps) of one type extends to a moduli of stable objects of a richer type by an iterated universal-family construction, with the forgetful morphism being a flat-proper structure morphism and the boundary being identified by gluing morphisms parametrised by the combinatorial types of degeneration. The synthesis is structural: $\overline{M}_{g, n}$ is the universal moduli of stable pointed curves and the universal curve over $\overline{M}_{g, n - 1}$ at the same time, with the boundary identifies with products of lower moduli; the entire intersection theory on $\overline{M}_{g, n}$ — the tautological ring, Witten-Kontsevich, Faber-Pandharipande — is a corollary of this self-referential structure.

Full proof set Master

Proposition (universal-curve presentation; proof sketch in detail). The forgetful morphism $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ is canonically isomorphic to the universal-curve morphism $C_{g, n} \to \overline{M}_{g, n}$ .

Proof. The morphism in one direction is $π$ itself: take a stable $(n + 1)$ -pointed curve, stabilise after deleting the last marked point. The morphism in the other direction is the tautological-section completion $F : C_{g, n} \to \overline{M}_{g, n + 1}$ assigning to a moduli point $[C; p_{1}, \dots, p_{n}]$ and a fibre point $x \in C$ the stable $(n + 1)$ -pointed curve $F (x)$ defined as follows.

Case 1: $x$ is in the smooth locus of $C$ and distinct from $p_{1}, \dots, p_{n}$ . Then $F (x) = (C; p_{1}, \dots, p_{n}, x)$ , already stable.

Case 2: $x$ coincides with some $p_{i}$ . Then $F (x)$ is constructed by inserting a $P^{1}$ -bubble at $p_{i}$ : replace a small neighbourhood of $p_{i}$ in $C$ by the union of $C$ (with $p_{i}$ replaced by a node) and a new $P^{1}$ -component glued at that node, with $p_{i}$ and the new marked point $p_{n + 1}$ both placed on the new $P^{1}$ (at two distinct smooth points). The resulting curve is stable because the new bubble carries three special points (the node where it meets $C$ , and the two marked points $p_{i}, p_{n + 1}$ ).

Case 3: $x$ is a node of $C$ , say $x = q \in C_{1} \cap C_{2}$ where $C_{1}, C_{2}$ are adjacent components. Then $F (x)$ is constructed by inserting a $P^{1}$ -bubble at the node: replace the node $q$ by a new $P^{1}$ -component carrying two nodes (one connecting to $C_{1}$ , one to $C_{2}$ ) and one marked point $p_{n + 1}$ in its smooth locus, again giving a stable curve with three special points on the new bubble.

The construction extends to flat families. To verify $F$ is the inverse of $π$ : applying $π$ to $F (x)$ in Case 1 simply removes the label and recovers $(C; p_{1}, \dots, p_{n})$ (no stabilisation needed); in Case 2 removes the label on $p_{n + 1}$ , leaving the bubble with only two special points ( $p_{i}$ and the node), which the stabilisation then contracts to recover $(C; p_{1}, \dots, p_{n})$ ; in Case 3 removes the label on $p_{n + 1}$ , leaving the bubble with only two special points (the two nodes), contracted to recover the original node. So $π \circ F = id_{C_{g, n}}$ and similarly $F \circ π = id_{\overline{M}_{g, n + 1}}$ , proving $F$ is an isomorphism. The tautological sections $σ_{i} : \overline{M}_{g, n} \to C_{g, n}$ correspond under $F$ to the closed substacks $δ_{i, n + 1} \subset \overline{M}_{g, n + 1}$ of pointed curves on which $p_{i}$ and $p_{n + 1}$ lie on a rational bubble — exactly the locus where Case 2 produces a bubble of positive length. $□$

Proposition (gluing morphisms are closed immersions). The reducible gluing $gl_{(g_{1}, A) ∣ (g_{2}, B)} : \overline{M}_{g_{1}, ∣ A ∣ + 1} \times \overline{M}_{g_{2}, ∣ B ∣ + 1} \to \overline{M}_{g_{1} + g_{2}, n}$ and the irreducible gluing $gl_{0} : \overline{M}_{g - 1, n + 2} \to \overline{M}_{g, n}$ are closed immersions onto their image divisors $Δ_{(g_{1}, A)}$ and $Δ_{0}$ respectively, up to the symmetry swap $(g_{1}, A) \leftrightarrow (g - g_{1}, A^{c})$ in the reducible case and the $Z /2$ involution swapping $q_{1} \leftrightarrow q_{2}$ in the irreducible case.

Proof. The gluing morphism is constructed by Knudsen 1983 §3 as the morphism of stacks classifying nodal degenerations: in a smooth family of $(n + 1)$ -pointed curves over a base $S$ , the locus where the last two marked points collide and the curve develops a node is a closed substack, and Knudsen's clutching morphism factors through this closed substack with image the relevant boundary divisor.

Specifically, choose a moduli point $[C; p_{1}, \dots, p_{n}] \in Δ_{(g_{1}, A)}$ — a one-nodal curve with two components of genera $g_{1}, g_{2}$ and marked points partitioned by $A ⊔ B$ . The preimage of $[C; p_{1}, \dots, p_{n}]$ under $gl_{(g_{1}, A) ∣ (g_{2}, B)}$ is the set of pairs $([C_{1}; (p_{a})_{a \in A}, q_{1}], [C_{2}; (p_{b})_{b \in B}, q_{2}])$ glued by identifying $q_{1}$ and $q_{2}$ . The two components $C_{1}, C_{2}$ and the gluing points $q_{1}, q_{2}$ are recovered uniquely from $[C; p_{1}, \dots, p_{n}]$ — $C_{1}$ is the connected component of the partial normalisation $C$ containing the marked points $A$ , $C_{2}$ the component containing $B$ , and $q_{1} = q_{2} =$ the preimage of the node — so the preimage is a single point, and the gluing morphism is set-theoretically injective.

The infinitesimal injectivity follows from the same identification on the level of tangent spaces: the normal-bundle exact sequence at the node identifies the tangent space of the boundary divisor at $[C]$ with the tangent space of the source at $([C_{1}; q_{1}], [C_{2}; q_{2}])$ . The image is closed by properness of the gluing morphism, and the image is exactly the boundary divisor $Δ_{(g_{1}, A)}$ (or $Δ_{0}$ in the irreducible case) by Knudsen's construction. Modding out by the symmetry swap (in the reducible case) or the $Z /2$ involution (in the irreducible case) gives the closed-immersion factorisation. $□$

Proposition (dimension of $\overline{M}_{g, n}$ ). For $(g, n)$ with $2 g - 2 + n > 0$ , $$ \dim \overline{\mathcal{M}}_{g, n} = \begin{cases} 3g - 3 + n & \text{if } g \geq 2, \ n & \text{if } g = 1 \text{ and } n \geq 1, \ n - 3 & \text{if } g = 0 \text{ and } n \geq 3. \end{cases} $$

Proof. For $g \geq 2$ : the moduli space $M_{g}$ of smooth genus- $g$ curves has dimension $3 g - 3$ (Riemann 1857, recovered rigorously by Mumford 1965). Adding $n$ marked points to a fixed smooth curve adds $n$ dimensions, one for each free location on the curve. Compactifying to $\overline{M}_{g, n}$ adds boundary loci of strictly smaller dimension, so the dimension is unchanged: $dim \overline{M}_{g, n} = 3 g - 3 + n$ . For $g = 1, n \geq 1$ : the moduli space $M_{1, 1}$ of one-pointed elliptic curves has dimension one (the $j$ -invariant determines the curve, and the marked point can be normalised to the identity of the group law). Adding $n - 1$ more marked points adds $n - 1$ dimensions, giving total $1 + (n - 1) = n$ . For $g = 0, n \geq 3$ : the moduli space $M_{0, n}$ of $n$ -pointed smooth $P^{1}$ 's has $n - 3$ dimensions because three marked points can be normalised to $(0, 1, \infty)$ by a Möbius transformation, fixing the curve uniquely. The remaining $n - 3$ points are free, giving $dim M_{0, n} = n - 3$ ; compactification by stable nodal curves does not change this. $□$

Proposition (string equation, Witten 1991). For $g, n$ with $2 g - 2 + n > 0$ and exponents $a_{1}, \dots, a_{n} \geq 0$ with $\sum a_{i} + 0 = 3 g - 3 + (n + 1)$ (equivalently $\sum a_{i} = 3 g - 2 + n$ ), $$ \int_{\overline{\mathcal{M}}{g, n+1}} \psi_1^{a_1} \cdots \psi_n^{a_n} \cdot \psi{n+1}^0 = \sum_{i = 1}^{n} \int_{\overline{\mathcal{M}}_{g, n}} \psi_1^{a_1} \cdots \psi_i^{a_i - 1} \cdots \psi_n^{a_n}, $$ where on the right the convention is that $ψ^{- 1}$ vanishes (so only $i$ with $a_{i} \geq 1$ contributes).

Proof. Apply the Mumford comparison formula $π^{*} ψ_{i} = ψ_{i} - δ_{i, n + 1}$ from Exercise 3, and use the projection formula for $π : \overline{M}_{g, n + 1} \to \overline{M}_{g, n}$ . On the left, integrate the product $ψ_{1}^{a_{1}} \dots ψ_{n}^{a_{n}}$ against the fundamental class of $\overline{M}_{g, n + 1}$ . Substitute $ψ_{i} = π^{*} ψ_{i} + δ_{i, n + 1}$ (rearranged Mumford comparison) and expand the product. The cross-terms involving $δ_{i, n + 1}$ are supported on the locus where $p_{i}$ and $p_{n + 1}$ collide via a bubble; on such a bubble the cotangent line $L_{i}$ is isomorphic to the cotangent line of the bubble at $p_{i}$ , which is the structure-sheaf line bundle on the bubble component, so $ψ_{i} \cdot δ_{i, n + 1} = - ψ_{n + 1} \cdot δ_{i, n + 1}$ (the psi-class restriction identity). Pushing the resulting cross-terms forward along $π$ gives the right-hand side of the string equation, with each $a_{i}$ reduced by one. The terms with $δ_{i, n + 1}^{k}$ for $k \geq 2$ vanish on dimensional grounds (the divisor $δ_{i, n + 1}$ has codimension one, so higher powers exceed the dimension of the fibre of $π$ for the relevant degrees). The detailed computation is given in Witten 1991 §2 ^{[Witten 1991]} and in Arbarello-Cornalba-Griffiths Vol. II §XVII.5 ^{[Arbarello-Cornalba-Griffiths]}. $□$

Theorem (Witten-Kontsevich; stated without full proof — full proof in Kontsevich 1992 ^{[source pending]} and in Mirzakhani 2007). The generating function $Z = exp F$ is the unique tau-function of the KdV hierarchy normalised by the string equation.

Kontsevich's proof uses the Strebel-Jenkins decomposition of $\overline{M}_{g, n}$ by ribbon graphs: every stable $n$ -pointed curve with prescribed positive real numbers (perimeters) at the marked points carries a unique Strebel quadratic differential, and the corresponding critical-trajectory graph is a ribbon graph with $n$ external legs. Integrating $ψ_{1}^{a_{1}} \dots ψ_{n}^{a_{n}}$ over the moduli space is then reduced to a combinatorial sum over ribbon graphs, which Kontsevich identifies term-by-term with the Feynman expansion of the Hermitian-matrix integral $Z_{N} (Λ)$ . The matrix integral is shown to satisfy the KdV hierarchy by standard matrix-model methods (the integral is a tau-function of the Virasoro constraints), completing the proof.

Connections Master

Moduli of curves 04.10.01. The moduli space $\overline{M}_{g, n}$ generalises the Deligne-Mumford compactification $\overline{M}_{g}$ by adding $n$ marked points; for $n = 0$ the two notions coincide. The forgetful and gluing morphisms developed here are the structural morphisms that organise the family of moduli spaces ${\overline{M}_{g, n}}_{g, n}$ into a coherent geometric structure (a modular operad), and every property of $\overline{M}_{g}$ stated in 04.10.01 (Mumford 1983 tautological classes, Witten-Kontsevich intersection numbers, Faber socle conjecture) extends to $\overline{M}_{g, n}$ via these morphisms.
Hilbert scheme 04.10.05. The construction of $\overline{M}_{g, n}$ via tri-canonical embedding (Mumford 1965) realises stable curves as points of a Hilbert scheme of subschemes of a fixed projective space, then takes the GIT quotient by the action of the projective linear group of changes of basis. The Knudsen 1983 construction of the pointed-curve moduli iterates the universal-curve passage, with each step a Hilbert-scheme construction at the next level of marked points. Both moduli spaces fit into the larger framework of moduli of geometric objects parametrised by Hilbert polynomials.
Riemann-Roch for curves 04.04.01. The dimension formula $dim \overline{M}_{g, n} = 3 g - 3 + n$ comes from a deformation-theoretic count $dim H^{1} (C, T_{C} (- p_{1} - \dots - p_{n})) = 3 g - 3 + n$ for a stable $n$ -pointed curve, which is a consequence of Riemann-Roch applied to the twisted tangent sheaf. The proof that $\overline{M}_{g, n}$ is a Deligne-Mumford stack with the expected dimension reduces to a Riemann-Roch computation on the underlying curves.
Canonical sheaf 04.08.02. The cotangent line bundle $L_{i}$ at the $i$ -th marked point is the line bundle whose fibre is the cotangent line $T_{p_{i}}^{*} C$ on the underlying stable pointed curve, and the psi-class $ψ_{i} = c_{1} (L_{i})$ is the first Chern class. The Hodge bundle $E = π_{*} ω_{C / \overline{M}}$ is the push-forward of the relative dualising sheaf along the universal-curve morphism, with the relative dualising sheaf being the relative version of the canonical sheaf studied in 04.08.02. The lambda-classes $λ_{j} = c_{j} (E)$ are the Chern classes of this Hodge bundle.
Adjunction formula 04.05.07. The relative dualising sheaf $ω_{C / \overline{M}}$ on the universal curve satisfies a relative adjunction formula: for the embedding of the universal curve as a closed substack of a smooth ambient variety (Hilbert-scheme presentation), the canonical bundle of the curve is the restriction-with-twist of the canonical bundle of the ambient variety by the normal-bundle determinant, paralleling the adjunction formula on a surface developed in 04.05.07. The psi-class identity $ψ_{i} = c_{1} (σ_{i}^{*} ω_{C / \overline{M}})$ uses this relative dualising sheaf as the universal canonical input.
Gieseker stability 04.10.11. The same Quot-scheme + GIT quotient pattern that constructs the Gieseker moduli of sheaves in 04.10.11 underlies Mumford's 1965 construction of $\overline{M}_{g}$ via tri-canonically embedded curves; the Knudsen pointed-curve construction iterates this pattern at every marked-point level. The universal family over $M_{X}^{ss} (P)$ in the Gieseker moduli plays the same structural role as the universal curve over $\overline{M}_{g, n}$ .
Hilbert-Mumford criterion 04.10.03. The construction of $\overline{M}_{g}$ and $\overline{M}_{g, n}$ as GIT quotients of Hilbert schemes uses the Hilbert-Mumford numerical criterion to identify the stable-curve locus inside the Hilbert scheme of tri-canonically embedded curves. The stable-curve condition of Deligne-Mumford (every smooth rational component has at least three special points; equivalently $ω_{C} (\sum p_{i})$ is ample) corresponds exactly to GIT stability under the natural projective-linear action.
GIT 04.10.02. Mumford's 1965 construction of $M_{g}$ as a GIT quotient extends to $\overline{M}_{g, n}$ via Knudsen's iterated universal-curve argument. The forgetful morphisms inherit their flatness and properness from the GIT-quotient structure of the underlying Hilbert-scheme construction, and the gluing morphisms identify boundary strata as products of GIT quotients of smaller Hilbert schemes.

Historical & philosophical context Master

Riemann's 1857 count of moduli for genus- $g$ curves, $dim M_{g} = 3 g - 3$ for $g \geq 2$ ^{[— Riemann 1857 *Theorie der Abelschen Functionen*]}, anticipated by sixty years the rigorous construction by David Mumford in 1965 via Geometric Invariant Theory. Pierre Deligne and David Mumford 1969 ^{[Deligne-Mumford 1969 *Publ. Math. IHES* 36]} introduced the compactification $\overline{M}_{g}$ by stable curves with at-worst-nodal singularities; the same paper proved $\overline{M}_{g}$ is irreducible, settling a classical question and inaugurating the modern moduli-space program. Finn Knudsen 1983 ^{[Knudsen 1983 *Math. Scand.* 52]} extended the construction to the pointed case $\overline{M}_{g, n}$ , with explicit forgetful and gluing morphisms making the family of moduli spaces into a coherent geometric structure. The pointed-curve moduli was implicit in Deligne-Mumford 1969 but Knudsen's paper systematised the iterated universal-curve construction and proved projectivity of the coarse moduli space via the ample-divisor formula $ω^{\otimes a} (\sum_{i} a_{i} p_{i})$ for sufficiently positive coefficients.

David Mumford 1983 ^{[Mumford 1983 — *Towards an enumerative geometry of the moduli space of curves*]} defined the tautological classes $κ_{a}, ψ_{i}, λ_{j}$ on $\overline{M}_{g, n}$ and proved the comparison formula $π^{*} ψ_{i} = ψ_{i} - δ_{i, n + 1}$ for the forgetful pullback, opening the way to a systematic intersection theory. Edward Witten 1991 ^{[Witten 1991 *Surveys in Differential Geometry* I]} conjectured that the generating function of $ψ$ -class intersection numbers satisfies the KdV hierarchy, motivated by the matching between two-dimensional topological gravity and matrix models. Maxim Kontsevich 1992 ^{[Kontsevich 1992 *Comm. Math. Phys.* 147]} proved Witten's conjecture via Hermitian-matrix integrals and Strebel-Jenkins ribbon-graph decompositions, providing the first complete computation of intersection numbers on $\overline{M}_{g, n}$ in closed form. Alternative proofs by Maryam Mirzakhani 2007 (via Weil-Petersson volumes and the McShane identity) and by Andrei Okounkov and Rahul Pandharipande 2002 (via Hurwitz numbers and the ELSV formula) provide independent derivations of the same KdV integrability, illustrating the convergence of geometry, topology, and integrable-system theory at the moduli of pointed curves.

Carel Faber 1999 conjectured the socle structure of the tautological ring on $M_{g}$ ; Faber-Pandharipande 2003 ^{[Faber-Pandharipande 2003 — *Inventiones* 139]} derived explicit polynomial relations among tautological classes from the moduli of relative stable maps to $P^{1}$ with the localisation theorem. Aaron Pixton 2012 ^{[Pixton 2012 *arXiv:1207.1918*]} conjectured a combinatorially-described complete set of tautological relations via double-ramification cycles; Pandharipande-Pixton-Zvonkine 2015 proved the Pixton relations are tautologically valid, but completeness remains open. Ezra Getzler and Mikhail Kapranov 1998 ^{[Getzler-Kapranov 1998 *Compos. Math.* 110]} formalised the modular-operad structure of $⨆_{g, n} \overline{M}_{g, n}$ under forgetful and gluing morphisms, providing the operadic framework that Maxim Kontsevich used in his 1999 formality theorem and that became the foundation of the Givental-Teleman classification of cohomological field theories.

Bibliography Master

@article{Knudsen1983,
  author  = {Knudsen, Finn F.},
  title   = {The projectivity of the moduli space of stable curves II: the stacks $M_{g,n}$},
  journal = {Mathematica Scandinavica},
  volume  = {52},
  year    = {1983},
  pages   = {161--199}
}

@book{HarrisMorrison,
  author    = {Harris, Joe and Morrison, Ian},
  title     = {Moduli of Curves},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {187},
  year      = {1998}
}

@incollection{Mumford1983,
  author    = {Mumford, David},
  title     = {Towards an enumerative geometry of the moduli space of curves},
  booktitle = {Arithmetic and Geometry, Vol. II},
  editor    = {Artin, Michael and Tate, John},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {36},
  year      = {1983},
  pages     = {271--328}
}

@article{DeligneMumford1969,
  author  = {Deligne, Pierre and Mumford, David},
  title   = {The irreducibility of the space of curves of given genus},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {36},
  year    = {1969},
  pages   = {75--109}
}

@article{Witten1991,
  author  = {Witten, Edward},
  title   = {Two-dimensional gravity and intersection theory on moduli space},
  journal = {Surveys in Differential Geometry},
  volume  = {1},
  year    = {1991},
  pages   = {243--310}
}

@article{Kontsevich1992,
  author  = {Kontsevich, Maxim},
  title   = {Intersection theory on the moduli space of curves and the matrix Airy function},
  journal = {Communications in Mathematical Physics},
  volume  = {147},
  year    = {1992},
  pages   = {1--23}
}

@article{FaberPandharipande2003,
  author  = {Faber, Carel and Pandharipande, Rahul},
  title   = {Relative maps and tautological classes},
  journal = {Inventiones Mathematicae},
  volume  = {139},
  year    = {2003},
  pages   = {173--199}
}

@article{Pixton2012,
  author  = {Pixton, Aaron},
  title   = {Conjectural relations in the tautological ring of $\overline{M}_{g,n}$},
  journal = {arXiv preprint},
  volume  = {arXiv:1207.1918},
  year    = {2012}
}

@book{ArbarelloCornalbaGriffithsII,
  author    = {Arbarello, Enrico and Cornalba, Maurizio and Griffiths, Phillip A.},
  title     = {Geometry of Algebraic Curves, Volume II (with a contribution by J. D. Harris)},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {268},
  year      = {2011}
}

@article{GetzlerKapranov1998,
  author  = {Getzler, Ezra and Kapranov, Mikhail M.},
  title   = {Modular operads},
  journal = {Compositio Mathematica},
  volume  = {110},
  year    = {1998},
  pages   = {65--126}
}

@article{Keel1992,
  author  = {Keel, Sean},
  title   = {Intersection theory of moduli space of stable $n$-pointed curves of genus zero},
  journal = {Transactions of the American Mathematical Society},
  volume  = {330},
  year    = {1992},
  pages   = {545--574}
}

@incollection{Kapranov1993,
  author    = {Kapranov, Mikhail M.},
  title     = {Chow quotients of Grassmannians I},
  booktitle = {I. M. Gelfand Seminar},
  series    = {Advances in Soviet Mathematics},
  volume    = {16},
  publisher = {American Mathematical Society},
  year      = {1993},
  pages     = {29--110}
}

@article{Mirzakhani2007,
  author  = {Mirzakhani, Maryam},
  title   = {Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces},
  journal = {Inventiones Mathematicae},
  volume  = {167},
  year    = {2007},
  pages   = {179--222}
}

@article{OkounkovPandharipande2002,
  author  = {Okounkov, Andrei and Pandharipande, Rahul},
  title   = {Gromov-Witten theory, Hurwitz numbers, and matrix models I},
  journal = {arXiv preprint},
  volume  = {arXiv:math/0101147},
  year    = {2002}
}

@article{CastravetTevelev2015,
  author  = {Castravet, Ana-Maria and Tevelev, Jenia},
  title   = {$\overline{M}_{0,n}$ is not a Mori dream space},
  journal = {Duke Mathematical Journal},
  volume  = {164},
  year    = {2015},
  pages   = {1641--1667}
}

Prerequisites

04.10.01
04.10.05
04.04.01
04.08.02

Tier anchors

beginner: Harris-Morrison *Moduli of Curves* (Springer GTM 187, 1998) Ch. 2 §2.B informal opening; Mumford 1983 *Towards an enumerative geometry of the moduli space of curves* introductory section
intermediate: Harris-Morrison *Moduli of Curves* Ch. 2 §C, Ch. 3 §D; Knudsen 1983 *Math. Scand.* 52 §1-§3; Vakil *Moduli space of curves and Gromov-Witten theory* lecture notes (2006) §2
master: Knudsen 1983 *Math. Scand.* 52 (161-199); Harris-Morrison *Moduli of Curves* Ch. 2, 3, 6; Mumford 1983 *Towards an enumerative geometry* (Birkhäuser, in *Arithmetic and Geometry vol. II*); Witten 1991 *Surveys in Differential Geometry* I; Kontsevich 1992 *Comm. Math. Phys.* 147; Arbarello-Cornalba-Griffiths *Geometry of Algebraic Curves* Vol. II (Springer Grundlehren 268, 2011)

References

Knudsen — The projectivity of the moduli space of stable curves II: the stacks M_{g,n} · Math. Scand. 52 (1983), 161-199; Part I (Mumford-Knudsen) is the unpointed projectivity argument, Part II constructs the pointed moduli via forgetful and gluing morphisms
Harris-Morrison — Moduli of Curves · Springer Graduate Texts in Mathematics 187, 1998; Ch. 2 §C (forgetful), Ch. 3 §D (gluing / boundary), Ch. 6 (intersection theory and tautological classes)
Mumford — Towards an enumerative geometry of the moduli space of curves · in *Arithmetic and Geometry vol. II* (Birkhäuser, 1983), 271-328; defines $\kappa, \psi, \lambda$ tautological classes and proves the forgetful-pullback comparison formula
Deligne-Mumford — The irreducibility of the space of curves of given genus · Publ. Math. IHES 36 (1969), 75-109; unpointed $\overline{\mathcal{M}}_g$ as a stack, with the stable-curve compactification later extended to the pointed case by Knudsen
Witten — Two-dimensional gravity and intersection theory on moduli space · Surveys in Differential Geometry I (1991), 243-310; conjectures that the generating function of $\psi$-class intersection numbers satisfies the KdV hierarchy
Kontsevich — Intersection theory on the moduli space of curves and the matrix Airy function · Commun. Math. Phys. 147 (1992), 1-23; proof of Witten's conjecture via Hermitian-matrix integrals and Strebel-Jenkins triangulations
Faber-Pandharipande — Relative maps and tautological classes · Inventiones Math. 139 (2003); J. Eur. Math. Soc. 7 (2005), 13-49; explicit relations among $\kappa$, $\psi$, $\lambda$ under forgetful pullback
Arbarello-Cornalba-Griffiths — Geometry of Algebraic Curves Vol. II · Springer Grundlehren der mathematischen Wissenschaften 268, 2011; Ch. XII (boundary stratification and tautological ring), Ch. XVII (Witten-Kontsevich)
Pixton — Conjectural relations in the tautological ring · arXiv:1207.1918 (2012); conjecturally-complete set of relations in $R^*(\overline{\mathcal{M}}_{g,n})$

Estimated time

beginner: 18m
intermediate: 45m
master: 90m