04.10.35 · algebraic-geometry / moduli

Moduli of stable maps and Gromov-Witten invariants

shipped3 tiersLean: none

Anchor (Master): Behrend-Fantechi 1997 *Invent. Math.* 128 (the intrinsic normal cone and the virtual fundamental class); Li-Tian 1998 *J. AMS* 11 (virtual classes for general targets); Kontsevich 1995 *Enumeration of rational curves via torus actions* (Birkhäuser, in *The Moduli Space of Curves*); Fulton-Pandharipande 1997; Cox-Katz 1999 Ch. 7-8; McDuff-Salamon *J-holomorphic Curves and Symplectic Topology* (AMS Colloq. 52, 2nd ed. 2012)

Intuition Beginner

Classical enumerative geometry asks counting questions: how many lines in the plane pass through two given points, how many conics pass through five points in general position, how many rational cubics pass through eight points. Each answer is a finite number, and each question fixes a target space (the plane, or some other variety) together with a shape of curve to draw inside it and a list of geometric constraints the curve must satisfy. The difficulty is the same one that haunted the moduli of curves: the space of all curves of a given shape inside a target is not compact, so a naive count can miss limits or run to infinity.

The fix mirrors the Deligne-Mumford story. Instead of recording only smooth curves mapped into the target, we record maps from possibly-degenerate marked curves into the target, and we admit exactly the degenerate maps needed to fill in missing limits. A map of this admitted kind is a stable map. The rule for which maps to keep is the same in spirit as before: the map must have only finitely many symmetries. Collecting all stable maps of a fixed genus, with a fixed number of marked points, landing in a fixed homology class of the target, gives a compact moduli space.

Once that space exists and is compact, the enumerative count becomes an integral over it. We use the marked points: each marked point gives an evaluation that records where it lands in the target, and pulling back a geometric constraint (a point, a line, a hyperplane) along that evaluation cuts the moduli space down. The number we want is the size of the leftover finite set, packaged as an integral. This integral is a Gromov-Witten invariant: a rigorous virtual count of curves drawn inside the target meeting the constraints.

Visual Beginner

A schematic in three panels. The left panel shows a smooth marked genus-zero curve mapped into a target surface, the image a smooth rational curve passing through three marked dots. The middle panel shows a degeneration: the source curve has broken into two genus-zero pieces joined at a node, and the map sends each piece to a curve in the target, the two image pieces meeting where the node maps. The right panel shows a contracted component: a small genus-zero bubble in the source that the map crushes to a single point of the target, drawn with the rule that the bubble carries at least three special points (marked points plus nodes) so that it cannot wobble.

The third panel carries the whole stability idea. A genus-zero component crushed to a point by the map keeps no information from the target, so its symmetry must be pinned down by special points on the source side. With three or more special points, the symmetry is finite. With fewer, the bubble would slide freely and the count would break. Components on which the map is not constant need no such condition: the image already pins them down. This is the single rule that makes the moduli space compact and well-behaved.

Worked example Beginner

Count the lines in the plane through two distinct points, using the language of stable maps, and see the answer come out as one.

Step 1. Fix the target as the plane $P^{2}$ and the shape as the class of a line. A line is the image of a map from a genus-zero curve whose image has degree one. So we look at genus-zero stable maps to $P^{2}$ of degree one, with two marked points to carry the two constraints.

Step 2. Impose the constraints. We want the first marked point to land on a fixed point $a$ and the second to land on a fixed point $b$ . Each constraint is a condition on where a marked point evaluates in the target.

Step 3. Solve. A degree-one map from a smooth genus-zero curve to the plane has image a line, and the map is an isomorphism onto that line. Asking the image to pass through $a$ and $b$ pins the line down: two distinct points determine one line. The marked points then sit at the preimages of $a$ and $b$ , with no remaining freedom.

Step 4. Read off the count. There is one stable map, so the count is one. No degenerate maps enter, because a single line through two points needs no breaking into pieces.

What this tells us. The bookkeeping reproduces the schoolbook fact that two points determine a line, but it does so in a frame that scales. Replacing "line through two points" by "conic through five points" or "rational cubic through eight points" keeps the same three steps: fix the target and the degree, add marked points for the constraints, and integrate. For conics through five points the same machine returns the number one, and for rational cubics through eight points it returns twelve, both as virtual counts over a compact moduli space of stable maps.

Check your understanding Beginner

Exercise (easy, multiple choice).

In a stable map, a genus-zero component on which the map is constant (it is crushed to a single point of the target) must carry at least how many special points (marked points together with nodes) to be admitted?

A. 0 B. 1 C. 2 D. 3

Hint

A crushed genus-zero component carries no information from the target, so its symmetries can only be pinned down by special points on the source. A genus-zero curve with fewer than three marked-or-node points has an infinite symmetry group.

Answer

D. A contracted genus-zero component must carry at least three special points. With fewer, the symmetry group of the genus-zero source fixing those points would be infinite (a one-parameter family fixing two points, a larger group fixing fewer), and the map would have infinite automorphisms. The rule for genus-zero contracted components matches the Deligne-Mumford rule that a smooth rational component of a stable curve meets the rest in at least three points; components where the map is non-constant need no condition, because the image already pins them down.

Formal definition Intermediate+

Throughout, $X$ is a smooth projective variety over an algebraically closed field $k$ of characteristic zero, and $β \in H_{2} (X, Z)$ (equivalently a curve class in the group of one-cycles modulo numerical equivalence) is a fixed class.

Definition (prestable curve). A prestable $n$ -pointed curve of arithmetic genus $g$ over $k$ is a connected projective nodal curve $C$ of arithmetic genus $g$ together with $n$ distinct marked points $p_{1}, \dots, p_{n}$ in the smooth locus of $C$ . No stability condition is imposed on $C$ itself; the marked points and nodes are the special points.

Definition (stable map). A stable map of genus $g$ , with $n$ marked points, to $X$ in class $β$ is a datum $(C, p_{1}, \dots, p_{n}, f)$ where $(C, p_{1}, \dots, p_{n})$ is a prestable $n$ -pointed curve of arithmetic genus $g$ and $f : C \to X$ is a morphism with $f_{*} [C] = β$ , satisfying the stability condition: every irreducible component of $C$ on which $f$ is constant and which has geometric genus zero carries at least three special points, and every component on which $f$ is constant and which has geometric genus one carries at least one special point. Equivalently, the datum $(C, p_{1}, \dots, p_{n}, f)$ has finite automorphism group, where an automorphism is an automorphism $φ$ of $C$ fixing each marked point with $f \circ φ = f$ .

Definition (moduli stack of stable maps). The moduli stack of stable maps $\overline{M}_{g, n} (X, β)$ is the stack whose objects over a base scheme $S$ are families of stable maps: a flat proper family $C \to S$ of prestable $n$ -pointed genus- $g$ curves with $n$ disjoint sections, together with a morphism $f : C \to X$ whose every geometric fibre is a stable map to $X$ in class $β$ . By Behrend-Manin 1996 and Fulton-Pandharipande 1997, $\overline{M}_{g, n} (X, β)$ is a proper Deligne-Mumford stack of finite type over $k$ .

Definition (evaluation morphisms). For each $i \in {1, \dots, n}$ the evaluation morphism $ev_{i} : \overline{M}_{g, n} (X, β) \to X$ sends a stable map $(C, p_{1}, \dots, p_{n}, f)$ to the image point $f (p_{i}) \in X$ . The evaluation morphisms are the source of all constraints: pulling back a cohomology class $γ_{i} \in H^{*} (X)$ along $ev_{i}$ records the requirement that the $i$ -th marked point lie on a cycle Poincaré-dual to $γ_{i}$ .

Definition (expected dimension). The expected (virtual) dimension of $\overline{M}_{g, n} (X, β)$ is $$ \mathrm{vdim} = (\dim X - 3)(1 - g) + \int_\beta c_1(T_X) + n. $$ This is the Euler characteristic of the deformation-obstruction complex of a stable map: deformations of the map come from $H^{0} (C, f^{*} T_{X})$ and deformations of the source from the curve moduli, while obstructions live in $H^{1} (C, f^{*} T_{X})$ , and the alternating sum is computed by Hirzebruch-Riemann-Roch 04.05.10 on the source curve.

The non-equidimensionality issue. Unlike $\overline{M}_{g, n}$ , the space $\overline{M}_{g, n} (X, β)$ is in general not equidimensional and not smooth: different boundary strata can have different dimensions, and the actual dimension at a stable map exceeds the expected dimension by $dim H^{1} (C, f^{*} T_{X})$ , the obstruction space. When $H^{1} (C, f^{*} T_{X}) = 0$ at every point (the convex or unobstructed case, which holds for $g = 0$ and $X = P^{r}$ since $T_{P^{r}}$ is generated by global sections), the moduli space is smooth of the expected dimension. In general it is not, and the naive fundamental class has the wrong dimension to integrate constraints against. The repair is the virtual fundamental class.

Counterexamples to common slips

Stability is a condition on contracted components only. A component on which $f$ is non-constant may be a genus-zero curve with one or even zero marked points and still be admitted; its image in $X$ pins down its symmetries. The slip is to impose the three-special-point rule on every genus-zero component.
The class $β$ is an effective curve class. For $β = 0$ the only stable maps are constant, and $\overline{M}_{g, n} (X, 0) = \overline{M}_{g, n} \times X$ when $2 g - 2 + n > 0$ — the map collapses to a single point of $X$ and the source must be Deligne-Mumford stable on its own.
The expected dimension is not the actual dimension in general. Reading off the count from $dim \overline{M}_{g, n} (X, β)$ rather than from $vdim$ gives wrong numbers whenever the obstruction space is non-zero, which is the rule rather than the exception once $g \geq 1$ or $X$ fails to be convex.

Key theorem with proof Intermediate+

Theorem (virtual fundamental class; Behrend-Fantechi 1997, Li-Tian 1998). Let $X$ be a smooth projective variety and $β \in H_{2} (X, Z)$ . The moduli stack $\overline{M}_{g, n} (X, β)$ carries a perfect obstruction theory, and hence a virtual fundamental class $$ [\overline{\mathcal{M}}{g,n}(X, \beta)]^{vir} \in A{\mathrm{vdim}}\big(\overline{\mathcal{M}}{g,n}(X, \beta)\big) $$ *in the Chow group of the expected dimension $\mathrm{vdim} = (\dim X - 3)(1 - g) + \int\beta c_1(T_X) + n $. T h ec l a ss i s in d e p e n d e n t o f c h o i ces, i s d e f or ma t i o n - in v a r ian t in$ X $an d$ \beta $, an d in t h e u n o b s t r u c t e d c a se (w h er e t h e m o d u l i s p a ce i ss m oo t h o f d im e n s i o n$ \mathrm{vdim}$) equals the ordinary fundamental class.*

Proof (outline of the Behrend-Fantechi construction). The construction packages the deformation theory of stable maps into a two-term complex and intersects an intrinsic cone against it.

Step 1: the relative obstruction theory. Let $M_{g, n}$ be the smooth Artin stack of prestable $n$ -pointed genus- $g$ curves, and let $ν : \overline{M}_{g, n} (X, β) \to M_{g, n}$ be the forgetful morphism remembering only the source. Over a stable map $(C, f)$ , the deformations and obstructions of $f$ relative to fixed source are governed by $H^{0} (C, f^{*} T_{X})$ and $H^{1} (C, f^{*} T_{X})$ . These assemble into the relative cotangent complex of $ν$ and its dual: writing $π : C \to \overline{M}_{g, n} (X, β)$ for the universal curve and $f : C \to X$ for the universal map, the complex $$ E^\bullet = \big(R\pi_* \mathfrak{f}^* T_X\big)^\vee $$ is a perfect complex of amplitude $[- 1, 0]$ and admits a morphism $E^{∙} \to L_{ν}$ to the relative cotangent complex of $ν$ .

Step 2: perfectness and the obstruction theory. The morphism $ϕ : E^{∙} \to L_{ν}$ is a perfect relative obstruction theory in the sense of Behrend-Fantechi 1997: it is an isomorphism on $h^{0}$ and a surjection on $h^{- 1}$ . The verification is a base-change-and-cohomology argument on the universal curve, using that $R π_{*} f^{*} T_{X}$ has cohomology only in degrees $0$ and $1$ because $π$ has relative dimension one. The rank of $E^{∙}$ is $vdim - dim M_{g, n}$ pointwise, by Hirzebruch-Riemann-Roch 04.05.10 applied to $f^{*} T_{X}$ on the source curve.

Step 3: the intrinsic normal cone. Behrend-Fantechi attach to any Deligne-Mumford stack an intrinsic normal cone $C$ , a stack-theoretic cone of pure dimension zero embedded in the stack $h^{1} / h^{0} (L_{ν}^{\lor})$ . The obstruction theory $ϕ$ provides a closed embedding of $C$ into the vector-bundle stack $E_{∙} = h^{1} / h^{0} (E^{∙}^{\lor})$ .

Step 4: Gysin pullback. The virtual class is the refined Gysin pullback of the intrinsic normal cone along the zero section of $E_{∙}$ : $$ [\overline{\mathcal{M}}{g,n}(X, \beta)]^{vir} = 0^!{E_\bullet}, [\mathfrak{C}] \in A_{\mathrm{vdim}}\big(\overline{\mathcal{M}}{g,n}(X, \beta)\big). $$ Intersecting the cone with the zero section cuts the dimension down to the rank-corrected value $vdim$ . Deformation invariance follows because the obstruction theory is functorial in $X$ and $β$ , and the unobstructed case gives the ordinary class because then $E\bullet $i s a v ec t or b u n d l e,$ \mathfrak{C} $i s i t sz er osec t i o n, an d t h e G y s in p u l l ba c k i s t h e i d e n t i t y .$ \square$

Definition (Gromov-Witten invariant). For cohomology classes $γ_{1}, \dots, γ_{n} \in H^{*} (X, Q)$ , the Gromov-Witten invariant is $$ \langle \gamma_1, \ldots, \gamma_n \rangle_{g, \beta} = \int_{[\overline{\mathcal{M}}_{g,n}(X, \beta)]^{vir}} \mathrm{ev}_1^* \gamma_1 \cup \cdots \cup \mathrm{ev}_n^* \gamma_n, $$ a rational number, non-zero only when $\sum_{i} de g γ_{i} = 2 vdim$ (the dimension constraint). When the $γ_{i}$ are Poincaré-dual to subvarieties in general position, the invariant is a virtual count of genus- $g$ curves in class $β$ meeting all the subvarieties.

Bridge. The virtual fundamental class builds toward every modern enumerative count, and the foundational reason it is needed is exactly the non-equidimensionality flagged above: the naive fundamental class of $\overline{M}_{g, n} (X, β)$ has the wrong dimension to integrate constraints against, and the obstruction theory is the data that corrects it. This is exactly the same move as in 04.10.22, where ampleness of the dualising sheaf supplied the polarisation making the moduli of stable curves a projective scheme; here the perfect obstruction theory supplies the virtual class making the moduli of stable maps a space one can integrate over. The construction generalises the Deligne-Mumford compactification: setting $β = 0$ and $dim X = 0$ recovers $\overline{M}_{g, n}$ with its ordinary fundamental class, so the stable-curve theory is the constant-target special case. The central insight is that the evaluation morphisms $ev_{i}$ of this unit are the maps-to- $X$ analogue of the marked-point structure that the forgetful and gluing morphisms of 04.10.26 organise, and putting these together, the splitting axiom for Gromov-Witten invariants is dual to the gluing morphisms of 04.10.26: a boundary divisor where the source breaks into two pieces pulls the virtual class back to a product of virtual classes, and this is the bridge from the geometry of the moduli space to the algebra of quantum cohomology. The pattern appears again in 04.12.15, where the same obstruction-theory virtual class is built for log-smooth maps to organise tropical and degeneration counts.

Exercises Intermediate+

Exercise 1 (easy, numeric).

Compute the expected dimension of $\overline{M}_{0, n} (P^{r}, d)$ , the moduli of genus-zero $n$ -pointed stable maps to $P^{r}$ of degree $d$ , using $\int_{β} c_{1} (T_{P^{r}}) = d (r + 1)$ .

Hint

Use $vdim = (dim X - 3) (1 - g) + \int_{β} c_{1} (T_{X}) + n$ with $dim X = r$ , $g = 0$ , and $\int_{β} c_{1} (T_{P^{r}}) = d (r + 1)$ .

Answer

$vdim = (r - 3) (1 - 0) + d (r + 1) + n = r - 3 + d (r + 1) + n$ . For $P^{2}$ and degree $d$ this is $2 - 3 + 3 d + n = 3 d - 1 + n$ . For $P^{2}$ , $d = 1$ (lines), $n = 2$ the value is $3 - 1 + 2 = 4$ , and imposing two point-constraints (each of complex codimension two, real degree four) on the four-dimensional space leaves a finite count — the single line through two points. Since $P^{r}$ is convex in genus zero, the moduli space is smooth of this dimension and the virtual class is the ordinary one. Rubric: full credit for the dimension formula with the correct $c_{1}$ value substituted.

Exercise 2 (easy, multiple choice).

For which of the following is the moduli space $\overline{M}_{g, n} (X, β)$ guaranteed smooth of the expected dimension, so that the virtual class equals the ordinary fundamental class?

A. $g = 0$ , $X = P^{r}$ , any $β$ B. $g = 2$ , $X$ a quintic threefold, any $β$ C. $g = 1$ , $X$ an abelian surface, any $β$ D. Every case, by the Behrend-Fantechi theorem

Hint

Smoothness needs the obstruction space $H^{1} (C, f^{*} T_{X})$ to vanish at every point. This holds when $T_{X}$ is generated by global sections and $g = 0$ (the convex case).

Answer

A. For $g = 0$ and $X = P^{r}$ , the tangent bundle $T_{P^{r}}$ is generated by global sections, so $H^{1} (C, f^{*} T_{P^{r}}) = 0$ on every genus-zero source (this is convexity), the moduli space is smooth of the expected dimension, and the virtual class is the ordinary fundamental class. The higher-genus and Calabi-Yau cases (B, C) have non-zero obstruction spaces and genuinely need the virtual class; D is false in general. Rubric: full credit for A with the convexity justification.

Exercise 3 (medium, symbolic).

Verify by Hirzebruch-Riemann-Roch 04.05.10 that $χ (C, f^{*} T_{X}) = \int_{β} c_{1} (T_{X}) + dim X \cdot (1 - g)$ for a genus- $g$ source curve $C$ and a map $f$ with $f_{*} [C] = β$ , and explain how this produces the $\int_{β} c_{1} (T_{X}) + (dim X) (1 - g)$ part of the expected dimension.

Hint

Riemann-Roch for a vector bundle $V$ of rank $r$ on a curve $C$ of arithmetic genus $g$ reads $χ (C, V) = de g V + r (1 - g)$ . Take $V = f^{*} T_{X}$ , so $r = dim X$ and $de g f^{*} T_{X} = \int_{C} f^{*} c_{1} (T_{X}) = \int_{β} c_{1} (T_{X})$ .

Answer

For a vector bundle $V$ of rank $r$ on a connected projective curve $C$ of arithmetic genus $g$ , Riemann-Roch gives $χ (C, V) = de g V + r (1 - g)$ . With $V = f^{*} T_{X}$ we have $r = dim X$ and, by the projection formula, $de g f^{*} T_{X} = \int_{C} f^{*} c_{1} (T_{X}) = \int_{f_{*} [C]} c_{1} (T_{X}) = \int_{β} c_{1} (T_{X})$ . Hence $$ \chi(C, f^* T_X) = \int_\beta c_1(T_X) + (\dim X)(1 - g). $$ The deformations of the map live in $H^{0} (C, f^{*} T_{X})$ and the obstructions in $H^{1} (C, f^{*} T_{X})$ , so their difference is $χ (C, f^{*} T_{X})$ ; adding the contribution of deformations of the $n$ -pointed source curve, which is $dim M_{g, n} = 3 g - 3 + n - (dim H^{0} - dim H^{1}$ of the source automorphisms $)$ , and reorganising gives $vdim = (dim X - 3) (1 - g) + \int_{β} c_{1} (T_{X}) + n$ . The $- 3 (1 - g)$ piece is exactly the $3 g - 3$ contribution of the source-curve moduli written for the map problem. Rubric: full credit for the Riemann-Roch computation of $χ (f^{*} T_{X})$ and the identification of $H^{0} / H^{1}$ as deformations/obstructions of the map.

Exercise 4 (medium, short-answer).

The genus-zero three-point invariant $⟨ γ_{1}, γ_{2}, γ_{3} ⟩_{0, 0}$ in class $β = 0$ on a smooth projective $X$ equals the classical triple intersection $\int_{X} γ_{1} \cup γ_{2} \cup γ_{3}$ . Explain why, identifying the relevant moduli space and the virtual class.

Hint

For $β = 0$ every stable map is constant. With $g = 0$ and $n = 3$ the source must be a single $P^{1}$ with three marked points, which has no moduli, so $\overline{M}_{0, 3} (X, 0) = X$ and all three evaluation maps are the identity.

Answer

In class $β = 0$ every stable map is constant, sending the whole source to a single point of $X$ . For $(g, n) = (0, 3)$ the stability condition forces the source to be one $P^{1}$ with three marked points; this pointed curve is rigid ( $\overline{M}_{0, 3}$ is a point), so the only datum is the image point in $X$ . Hence $\overline{M}_{0, 3} (X, 0) ≅ X$ , and each evaluation morphism $ev_{i} : X \to X$ is the identity. The expected dimension is $(dim X - 3) + 0 + 3 = dim X$ , the moduli space is smooth of this dimension, and the virtual class is the ordinary fundamental class $[X]$ . Therefore $$ \langle \gamma_1, \gamma_2, \gamma_3 \rangle_{0, 0} = \int_{[X]} \mathrm{ev}_1^* \gamma_1 \cup \mathrm{ev}_2^* \gamma_2 \cup \mathrm{ev}3^* \gamma_3 = \int_X \gamma_1 \cup \gamma_2 \cup \gamma_3, $$ the classical triple intersection. This is the constant term of the quantum cohomology product: the quantum product reduces to the cup product when all positive-degree classes $β$ are dropped. Rubric: full credit for identifying $\overline{\mathcal{M}}{0,3}(X,0) \cong X$, the identity evaluation maps, and the resulting classical intersection.

Exercise 5 (hard, short-answer).

State the splitting (WDVV) axiom for genus-zero Gromov-Witten invariants and explain how it expresses associativity of the quantum cohomology product, via the boundary structure of $\overline{M}_{0, 4} (X, β)$ .

Hint

$\overline{M}_{0, 4} ≅ P^{1}$ has three boundary points, the three ways of splitting four marked points into two pairs. Pulling back along the forgetful map $\overline{M}_{0, 4} (X, β) \to \overline{M}_{0, 4}$ and using that the three boundary points are linearly equivalent gives the WDVV relation. Insert a sum over a basis ${T_{e}}$ of $H^{*} (X)$ with dual basis ${T^{e}}$ at the node.

Answer

The moduli $\overline{M}_{0, 4} ≅ P^{1}$ has three boundary points $D_{(12∣34)}, D_{(13∣24)}, D_{(14∣23)}$ , one for each partition of ${1, 2, 3, 4}$ into two pairs, and as divisors on $P^{1}$ they are linearly equivalent. Pull back this equivalence along the forgetful morphism $\overline{M}_{0, 4} (X, β) \to \overline{M}_{0, 4}$ , cap against $ev_{i}^{*} γ_{i}$ and the virtual class, and apply the splitting axiom: at a node where the source breaks into two genus-zero pieces, the virtual class factors as a product, and summing over a basis ${T_{e}}$ of $H^{*} (X)$ with Poincaré-dual basis ${T^{e}}$ inserted at the two branches gives the WDVV equation $$ \sum_{\beta_1 + \beta_2 = \beta} \sum_e \langle \gamma_1, \gamma_2, T_e \rangle_{0, \beta_1} \langle T^e, \gamma_3, \gamma_4 \rangle_{0, \beta_2} = \sum_{\beta_1 + \beta_2 = \beta} \sum_e \langle \gamma_1, \gamma_3, T_e \rangle_{0, \beta_1} \langle T^e, \gamma_2, \gamma_4 \rangle_{0, \beta_2}. $$ Encoding the genus-zero invariants in the quantum product $T_{a} * T_{b} = \sum_{β, e} ⟨ T_{a}, T_{b}, T_{e} ⟩_{0, β} q^{β} T^{e}$ , the WDVV equation is exactly the statement $(T_{a} * T_{b}) * T_{c} = T_{a} * (T_{b} * T_{c})$ — associativity of the quantum cohomology ring. The three boundary points correspond to the two sides of associativity plus the symmetry, and the linear equivalence of the three points on $P^{1}$ is the geometric origin of the algebraic identity. Rubric: full credit for the WDVV equation with the basis-sum at the node, the linear equivalence of the three boundary divisors of $\overline{M}_{0, 4}$ , and the identification with associativity of the quantum product.

Exercise 6 (hard, numeric).

Let $N_{d}$ be the number of degree- $d$ rational plane curves through $3 d - 1$ general points of $P^{2}$ . Kontsevich's recursion reads $$ N_d = \sum_{\substack{d_1 + d_2 = d \ d_1, d_2 \geq 1}} N_{d_1} N_{d_2}\left[ d_1^2 d_2^2 \binom{3d - 4}{3d_1 - 2} - d_1^3 d_2 \binom{3d - 4}{3d_1 - 1} \right]. $$ Given $N_{1} = 1$ , compute $N_{2}$ and $N_{3}$ .

Hint

For $N_{2}$ the only splitting is $d_{1} = d_{2} = 1$ (the sum is over ordered pairs). For $N_{3}$ sum over $(d_{1}, d_{2}) \in {(1, 2), (2, 1)}$ . Use $(1 2) = 2$ , $(2 2) = 1$ , $(1 5) = 5$ , $(4 5) = 5$ , $(2 5) = 10$ , $(3 5) = 10$ .

Answer

$N_{2}$ . Here $d = 2$ , $3 d - 4 = 2$ , and the only ordered splitting is $(d_{1}, d_{2}) = (1, 1)$ with $N_{1} = 1$ : $$ N_2 = 1 \cdot 1 \cdot \left[ 1^2 \cdot 1^2 \binom{2}{1} - 1^3 \cdot 1 \binom{2}{2} \right] = 1 \cdot (1 \cdot 2 - 1 \cdot 1) = 2 - 1 = 1. $$ So $N_{2} = 1$ : one conic through five general points.

$N_{3}$ . Here $d = 3$ , $3 d - 4 = 5$ , and the ordered splittings are $(1, 2)$ and $(2, 1)$ with $N_{1} = 1$ , $N_{2} = 1$ . For $(d_{1}, d_{2}) = (1, 2)$ : $$ 1 \cdot 1 \cdot \left[ 1^2 \cdot 2^2 \binom{5}{1} - 1^3 \cdot 2 \binom{5}{2} \right] = 4 \cdot 5 - 2 \cdot 10 = 20 - 20 = 0. $$ For $(d_{1}, d_{2}) = (2, 1)$ : $$ 1 \cdot 1 \cdot \left[ 2^2 \cdot 1^2 \binom{5}{4} - 2^3 \cdot 1 \binom{5}{5} \right] = 4 \cdot 5 - 8 \cdot 1 = 20 - 8 = 12. $$ Summing, $N_{3} = 0 + 12 = 12$ : twelve rational cubics through eight general points, the classical answer. Rubric: full credit for $N_{2} = 1$ and $N_{3} = 12$ with both ordered splittings evaluated.

Advanced results Master

Theorem (Kontsevich-Manin axioms; Kontsevich-Manin 1994, Behrend 1997). The system of Gromov-Witten classes $I_{g,n,\beta} : H^(X)^{\otimes n} \to H^(\overline{\mathcal{M}}{g,n}) $, d e f in e d b y$ I{g,n,\beta}(\gamma_1, \ldots, \gamma_n) = \mathrm{PD}, p_\big( \prod_i \mathrm{ev}i^* \gamma_i \cap [\overline{\mathcal{M}}{g,n}(X, \beta)]^{vir} \big) $w h er e$ p $i s t h e f or g e t f u l ma pt o$ \overline{\mathcal{M}}_{g,n}$, satisfies a list of axioms: the fundamental-class, divisor, point-mapping, splitting, genus-reduction, and deformation axioms. These axioms determine the invariants from genus-zero three-point data together with the classical cohomology.*

The splitting axiom factors the Gromov-Witten class through the gluing morphisms of 04.10.26: the pullback of $I_{g, n, β}$ along a boundary divisor where the source splits into two pieces equals the product of the Gromov-Witten classes of the pieces, summed over the curve classes and a basis of $H^{*} (X)$ inserted at the node. The divisor axiom states that $⟨ D, γ_{1}, \dots, γ_{n} ⟩_{g, β} = (\int_{β} D) ⟨ γ_{1}, \dots, γ_{n} ⟩_{g, β}$ for a divisor class $D$ , reducing insertions of divisors to a multiplicative factor.

Theorem (quantum cohomology ring; Kontsevich-Manin 1994, Ruan-Tian 1995). Define the small quantum product on $H^(X) \otimes \mathbb{Q}[[q]]$ by* $$ T_a * T_b = \sum_{\beta} \sum_e \langle T_a, T_b, T_e \rangle_{0, \beta}, q^\beta, T^e, $$ where ${T_{e}}$ is a homogeneous basis of $H^(X) $an d$ {T^e} $i t s P o in c a r \overset{e}{ˊ} - d u a l ba s i s . T h e p r o d u c t i s a ssoc ia t i v e (a co n se q u e n ceo f t h e W D V V e q u a t i o n), g r a d e d - co mm u t a t i v e, an d ha s u ni tt h e f u n d am e n t a l c l a ss; m o d u l o t h e ma x ima l i d e a l$ (q) $i t r e d u ces t o t h ec l a ss i c a l c u pp r o d u c t . T h er es u l t in g r in g i s t h e * * q u an t u m co h o m o l o g y * *$ QH^(X)$.

Associativity is the deep statement: it is the WDVV equation derived from the linear equivalence of the three boundary points of $\overline{M}_{0, 4} ≅ P^{1}$ , lifted to the virtual class via the splitting axiom. For $X = P^{2}$ the structure constants are the numbers $N_{d}$ , and the associativity relation is precisely Kontsevich's recursion, which is therefore a theorem about $Q H^{*} (P^{2})$ rather than an ad-hoc count.

Theorem (Kontsevich's recursion for rational plane curves; Kontsevich 1995). Let $N_{d}$ denote the number of irreducible rational plane curves of degree $d$ through $3 d - 1$ general points of $P^{2}$ . Then $N_{1} = 1$ and $$ N_d = \sum_{d_1 + d_2 = d}, N_{d_1} N_{d_2}\left[ d_1^2 d_2^2 \binom{3d - 4}{3d_1 - 2} - d_1^3 d_2 \binom{3d - 4}{3d_1 - 1} \right]. $$

The recursion is the associativity relation in $Q H^{*} (P^{2})$ written out in the basis ${1, H, H^{2}}$ of hyperplane powers. The genus-zero invariants $⟨ H^{2}, \dots, H^{2} ⟩_{0, d} = N_{d}$ count rational curves through points, the divisor axiom reduces hyperplane insertions, and the WDVV equation applied to a four-point invariant with two point classes and two line classes produces the stated recursion after the binomial coefficients are unwound from the splitting sum. Kontsevich-Manin 1994 gave the quantum-cohomology derivation; Kontsevich 1995 also gave the independent torus-localisation computation on $\overline{M}_{0, n} (P^{2}, d)$ , which evaluates the same invariants as a sum over fixed loci of the $(C^{\times})^{3}$ -action, indexed by decorated trees.

Theorem (genus-zero invariants of the quintic; the A-model side). Let $X \subset P^{4}$ be a smooth quintic threefold, a Calabi-Yau threefold with $\int_{β} c_{1} (T_{X}) = 0$ for all $β$ . The expected dimension of $\overline{M}_{0, 0} (X, d)$ is zero, and the genus-zero degree- $d$ invariant $n_{d}^{0} = ⟨ ⟩_{0, d}$ is a virtual count of rational curves of degree $d$ . The numbers $n_{d}^{0}$ are the instanton numbers predicted by Candelas-de la Ossa-Green-Parkes from a period computation on the mirror.

Because $X$ is Calabi-Yau of dimension three, the virtual dimension of $\overline{M}_{0, 0} (X, d)$ is zero for every $d$ , so the invariant is a number with no insertions — the most basic curve count there is, and the one mirror symmetry was first tested against. The virtual class is essential: the actual moduli space has positive dimension (multiple covers and obstructed maps), and only the virtual count is deformation-invariant. The predicted values $n_{1}^{0} = 2875$ (lines on the quintic), $n_{2}^{0} = 609250$ (conics), and higher are recovered from the mirror B-model, the prediction that launched the subject.

Synthesis. The moduli of stable maps is the foundational reason enumerative geometry has a rigorous home, and the central insight is that the same compactification-by-degeneration strategy that produced $\overline{M}_{g, n}$ in 04.10.22 produces $\overline{M}_{g, n} (X, β)$ once maps to a target are admitted, with the stability condition migrating from the curve to the contracted components of the map. Putting these together, the virtual fundamental class is exactly what repairs the non-equidimensionality of the moduli space, and this is the bridge from a geometric object that cannot be naively integrated to the rational-number invariants that count curves: the obstruction theory generalises the deformation theory of stable curves, the evaluation morphisms generalise the marked-point structure organised by the forgetful and gluing morphisms of 04.10.26, and the splitting axiom is dual to those gluing morphisms, factoring the virtual class along the boundary. The quantum cohomology ring assembles the genus-zero three-point invariants into an associative deformation of the cup product, and its associativity is the WDVV equation. This pattern recurs across the subject: Kontsevich's recursion for $N_{d}$ is the associativity relation in $Q H^{*} (P^{2})$ , and the quintic instanton numbers are the genus-zero invariants whose prediction by mirror symmetry opened the modern era. The bridge is universal — from stable curves to stable maps, from fundamental class to virtual class — and it carries the enumerative content of 04.12.15 forward.

Full proof set Master

Proposition (the constant-map and zero-class reductions). Let $X$ be smooth projective. (i) For $β = 0$ and $2 g - 2 + n > 0$ , $\overline{M}_{g, n} (X, 0) ≅ \overline{M}_{g, n} \times X$ . (ii) For $(g, n) = (0, 3)$ , $\overline{M}_{0, 3} (X, 0) ≅ X$ .

Proof. A stable map in class $β = 0$ has $f_{*} [C] = 0$ , and since $f$ restricted to each irreducible component has image a point or a curve of positive class, every component maps to a point; connectedness of $C$ then forces $f$ to be constant with a single image point $x \in X$ . The datum reduces to a pointed curve $(C, p_{1}, \dots, p_{n})$ together with the point $x$ , and the stability condition for a constant map is exactly Deligne-Mumford stability of $(C, p_{1}, \dots, p_{n})$ (every genus-zero component carries at least three special points, every genus-one component at least one), because no component is non-constant. Hence the moduli is the product $\overline{M}_{g, n} \times X$ , which requires $2 g - 2 + n > 0$ for $\overline{M}_{g, n}$ to be defined. For $(g, n) = (0, 3)$ the curve moduli $\overline{M}_{0, 3}$ is a single point (three marked points on $P^{1}$ are rigid under $PGL_{2}$ ), so the product collapses to $X$ . $□$

Proposition (expected dimension via Riemann-Roch). For a stable map $(C, f)$ of genus $g$ to $X$ in class $β$ , the virtual dimension of $\overline{M}_{g, n} (X, β)$ is $(dim X - 3) (1 - g) + \int_{β} c_{1} (T_{X}) + n$ .

Proof. The deformation theory splits into deformations of the map with fixed source and deformations of the $n$ -pointed source. With fixed source, deformations are $H^{0} (C, f^{*} T_{X})$ and obstructions $H^{1} (C, f^{*} T_{X})$ , contributing $χ (C, f^{*} T_{X}) = \int_{β} c_{1} (T_{X}) + (dim X) (1 - g)$ by Hirzebruch-Riemann-Roch 04.05.10 for the rank- $(dim X)$ bundle $f^{*} T_{X}$ on the genus- $g$ curve, as in Exercise 3. Deformations of the $n$ -pointed source contribute $dim M_{g, n} = 3 g - 3 + n$ to the expected dimension (the deformations of the prestable pointed curve, with automorphisms of non-stable components already absorbed into the map-stability bookkeeping). Adding, $$ \mathrm{vdim} = \big[\int_\beta c_1(T_X) + (\dim X)(1 - g)\big] + (3g - 3 + n) = (\dim X - 3)(1 - g) + \int_\beta c_1(T_X) + n, $$ using $(dim X) (1 - g) + 3 g - 3 = (dim X) (1 - g) - 3 (1 - g) = (dim X - 3) (1 - g)$ . $□$

Proposition (genus-zero $P^{r}$ moduli is smooth of expected dimension). For $g = 0$ and $X = P^{r}$ , the stack $\overline{M}_{0, n} (P^{r}, d)$ is smooth of dimension $r - 3 + d (r + 1) + n$ , and its virtual class equals its ordinary fundamental class.

Proof. The tangent bundle $T_{P^{r}}$ is generated by global sections (the Euler sequence $0 \to O \to O (1)^{\oplus (r + 1)} \to T_{P^{r}} \to 0$ exhibits a surjection from a sum of line bundles of positive degree). For any genus-zero source $C$ and map $f$ , the pullback $f^{*} T_{P^{r}}$ is a quotient of $f^{*} O (1)^{\oplus (r + 1)}$ , a sum of line bundles of non-negative degree on each $P^{1}$ component, so $H^{1} (C, f^{*} T_{P^{r}}) = 0$ on a genus-zero source (using that $H^{1}$ of a non-negative-degree line bundle on $P^{1}$ vanishes, together with the normalisation sequence to pass from $C$ to its components). The obstruction space vanishes at every point, so the obstruction theory is a vector bundle, the intrinsic normal cone is its zero section, and the Gysin pullback returns the ordinary fundamental class; smoothness of dimension $vdim = r - 3 + d (r + 1) + n$ follows from the surjectivity of the deformation map and the vanishing of obstructions. $□$

Proposition (proper Deligne-Mumford property). For $X$ smooth projective and any $g, n, β$ , the stack $\overline{M}_{g, n} (X, β)$ is a proper Deligne-Mumford stack of finite type over $k$ .

Proof sketch (full proof in Fulton-Pandharipande 1997 §1-§3 and Behrend-Manin 1996). Boundedness: a stable map to $X$ of class $β$ embeds, after composing with a projective embedding $X ↪ P^{M}$ , into a fixed Hilbert scheme of maps, so the moduli is of finite type. Separatedness and properness are the valuative criterion: a family of stable maps over the generic point of a discrete valuation ring extends uniquely to the closed point, by the stable-reduction argument for maps — take any extension, resolve and semistably reduce the source, then contract destabilised components and stabilise the map; the extension is unique because stable maps have no infinitesimal automorphisms beyond the finite ones. The Deligne-Mumford property is finiteness of automorphism groups, which is the stability condition by construction: every contracted genus-zero component has at least three special points, so its automorphisms fixing them are finite, and non-contracted components are pinned by their image. $□$

Connections Master

Stable curve and Deligne-Mumford stability 04.10.22. The moduli of stable maps is the maps-to-a-target generalisation of the moduli of stable curves. Setting $β = 0$ and forgetting the target recovers $\overline{M}_{g, n}$ , and the stability condition migrates from the curve to the contracted components of the map: a genus-zero component crushed to a point must carry at least three special points, exactly the Deligne-Mumford rule for a rational component of a stable curve. The dualising-sheaf ampleness that characterises stability there becomes the finiteness-of-automorphisms condition here.
Forgetful and gluing morphisms on $\overline{M}_{g, n}$ 04.10.26. The evaluation morphisms $ev_{i}$ of stable-map moduli are the target-valued analogue of the marked-point structure organised by the forgetful and gluing morphisms. The splitting axiom for Gromov-Witten invariants is dual to the gluing morphisms: a boundary divisor where the source breaks into two pieces pulls the virtual class back to a product of virtual classes, summed over a basis of $H^{*} (X)$ inserted at the node, and this factorisation is the geometric origin of the WDVV equation and quantum-cohomology associativity.
Hirzebruch-Riemann-Roch, general dimension 04.05.10. The expected dimension of $\overline{M}_{g, n} (X, β)$ is computed by Hirzebruch-Riemann-Roch applied to $f^{*} T_{X}$ on the source curve: $χ (C, f^{*} T_{X}) = \int_{β} c_{1} (T_{X}) + (dim X) (1 - g)$ is the Euler characteristic of the deformation-obstruction complex of the map. The same theorem governs the rank of the perfect obstruction theory from which Behrend-Fantechi extract the virtual fundamental class.
Log Gromov-Witten invariants 04.12.15. Log Gromov-Witten theory builds the same obstruction-theory virtual class for log-smooth maps to a target with a divisor, refining the ordinary stable-map count to track tangency and degeneration data. It presupposes the ordinary Gromov-Witten theory constructed here, which it specialises and degenerates; the splitting axiom and the virtual-class formalism carry over with log-smooth deformation theory in place of the ordinary one.
ELSV formula — Hurwitz numbers as Hodge integrals 04.10.32. ELSV and Gromov-Witten theory both express enumerative counts as integrals over moduli of curves, and the Okounkov-Pandharipande proof of ELSV runs through virtual localisation on the stable-map moduli $\overline{M}_{g, n} (P^{1}, d)$ — the same space whose torus-fixed loci compute the Gromov-Witten invariants of $P^{1}$ . The Hodge integrals of ELSV reappear as the contributions of fixed loci in the localisation computation of Gromov-Witten invariants.

Historical & philosophical context Master

The enumerative questions that Gromov-Witten theory answers are among the oldest in algebraic geometry: the count of conics tangent to five conics (Steiner's 3264, corrected by Chasles), the lines on a cubic surface (Cayley-Salmon's 27), and Schubert's nineteenth-century calculus of incidence conditions all sought finite numbers attached to families of curves inside a fixed target. The recurring obstacle was rigour: the relevant parameter spaces were not compact, excess intersection produced spurious contributions, and "general position" arguments were hard to make precise. The twentieth-century resolution came from two directions at once — symplectic and algebraic.

On the symplectic side, Mikhail Gromov's Pseudo holomorphic curves in symplectic manifolds (Invent. Math. 82, 1985) ^{[Gromov 1985]} introduced $J$ -holomorphic curves and the compactness theorem that bears his name, showing that the space of such curves in a symplectic manifold, suitably compactified by bubbling, supports invariant counts. On the algebraic side, Maxim Kontsevich's Enumeration of rational curves via torus actions (in The Moduli Space of Curves, Birkhäuser 1995) ^{[Kontsevich 1995]} introduced the moduli space of stable maps $\overline{M}_{g, n} (X, β)$ as the algebraic compactification and used the torus-localisation theorem to compute the numbers $N_{d}$ of rational plane curves, turning a century-old enumerative problem into a recursion. Kontsevich and Yuri Manin's Gromov-Witten classes, quantum cohomology, and enumerative geometry (Comm. Math. Phys. 164, 1994) ^{[Kontsevich-Manin 1994]} axiomatised the invariants, introduced the quantum cohomology ring, and showed that the WDVV equations — named for Witten, Dijkgraaf, and the Verlinde brothers from two-dimensional topological field theory — encode the associativity that makes the recursion a theorem rather than a coincidence.

The construction was completed by the virtual fundamental class of Kai Behrend and Barbara Fantechi, The intrinsic normal cone (Invent. Math. 128, 1997) ^{[Behrend-Fantechi 1997]}, and independently by Jun Li and Gang Tian (J. Amer. Math. Soc. 11, 1998), which made the invariants rigorous for arbitrary smooth projective targets by extracting a class of the expected dimension from a perfect obstruction theory — resolving exactly the non-equidimensionality that had blocked naive counting. The whole apparatus was vindicated by physics: Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes, in A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory (Nuclear Phys. B 359, 1991) ^{[Candelas-de la Ossa-Green-Parkes 1991]}, predicted the genus-zero instanton numbers of the quintic threefold — 2875 lines, 609250 conics, and beyond — from a period computation on a mirror manifold, numbers later confirmed as Gromov-Witten invariants. The philosophical lesson is that a rigorous count sometimes requires enlarging the objects counted (from curves to stable maps) and replacing the count itself by a deformation-invariant virtual number, trading naive enumeration for an integral that is stable under the very degenerations that obstructed the original question.

Bibliography Master

@incollection{Kontsevich1995,
  author    = {Kontsevich, Maxim},
  title     = {Enumeration of rational curves via torus actions},
  booktitle = {The Moduli Space of Curves},
  editor    = {Dijkgraaf, Robbert H. and Faber, Carel and van der Geer, Gerard},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {129},
  year      = {1995},
  pages     = {335--368}
}

@article{KontsevichManin1994,
  author    = {Kontsevich, Maxim and Manin, Yuri},
  title     = {Gromov-Witten classes, quantum cohomology, and enumerative geometry},
  journal   = {Communications in Mathematical Physics},
  volume    = {164},
  year      = {1994},
  pages     = {525--562}
}

@article{BehrendFantechi1997,
  author    = {Behrend, Kai and Fantechi, Barbara},
  title     = {The intrinsic normal cone},
  journal   = {Inventiones Mathematicae},
  volume    = {128},
  year      = {1997},
  pages     = {45--88}
}

@article{Behrend1997,
  author    = {Behrend, Kai},
  title     = {Gromov-Witten invariants in algebraic geometry},
  journal   = {Inventiones Mathematicae},
  volume    = {127},
  year      = {1997},
  pages     = {601--617}
}

@article{LiTian1998,
  author    = {Li, Jun and Tian, Gang},
  title     = {Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties},
  journal   = {Journal of the American Mathematical Society},
  volume    = {11},
  year      = {1998},
  pages     = {119--174}
}

@incollection{FultonPandharipande1997,
  author    = {Fulton, William and Pandharipande, Rahul},
  title     = {Notes on stable maps and quantum cohomology},
  booktitle = {Algebraic Geometry---Santa Cruz 1995},
  publisher = {American Mathematical Society},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {62.2},
  year      = {1997},
  pages     = {45--96}
}

@article{BehrendManin1996,
  author    = {Behrend, Kai and Manin, Yuri},
  title     = {Stacks of stable maps and Gromov-Witten invariants},
  journal   = {Duke Mathematical Journal},
  volume    = {85},
  year      = {1996},
  pages     = {1--60}
}

@book{CoxKatz1999,
  author    = {Cox, David A. and Katz, Sheldon},
  title     = {Mirror Symmetry and Algebraic Geometry},
  publisher = {American Mathematical Society},
  series    = {Mathematical Surveys and Monographs},
  volume    = {68},
  year      = {1999}
}

@article{Gromov1985,
  author    = {Gromov, Mikhail},
  title     = {Pseudo holomorphic curves in symplectic manifolds},
  journal   = {Inventiones Mathematicae},
  volume    = {82},
  year      = {1985},
  pages     = {307--347}
}

@article{Candelas1991,
  author    = {Candelas, Philip and de la Ossa, Xenia C. and Green, Paul S. and Parkes, Linda},
  title     = {A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory},
  journal   = {Nuclear Physics B},
  volume    = {359},
  year      = {1991},
  pages     = {21--74}
}

@article{RuanTian1995,
  author    = {Ruan, Yongbin and Tian, Gang},
  title     = {A mathematical theory of quantum cohomology},
  journal   = {Journal of Differential Geometry},
  volume    = {42},
  year      = {1995},
  pages     = {259--367}
}

@book{McDuffSalamon2012,
  author    = {McDuff, Dusa and Salamon, Dietmar},
  title     = {$J$-holomorphic Curves and Symplectic Topology},
  publisher = {American Mathematical Society},
  series    = {Colloquium Publications},
  volume    = {52},
  edition   = {2},
  year      = {2012}
}

Prerequisites

04.10.22
04.10.26
04.05.10

Tier anchors

beginner: Kontsevich-Manin 1994 *Comm. Math. Phys.* 164 §0-§1 informal opening; Harris-Morrison *Moduli of Curves* (Springer GTM 187, 1998) Ch. 2 §2.E on maps to projective space
intermediate: Fulton-Pandharipande 1997 *Notes on stable maps and quantum cohomology* (Proc. Symp. Pure Math. 62.2) §0-§7; Cox-Katz *Mirror Symmetry and Algebraic Geometry* (AMS Surveys 68, 1999) Ch. 7; Behrend-Manin 1996 *Duke Math. J.* 85 §1-§5
master: Behrend-Fantechi 1997 *Invent. Math.* 128 (the intrinsic normal cone and the virtual fundamental class); Li-Tian 1998 *J. AMS* 11 (virtual classes for general targets); Kontsevich 1995 *Enumeration of rational curves via torus actions* (Birkhäuser, in *The Moduli Space of Curves*); Fulton-Pandharipande 1997; Cox-Katz 1999 Ch. 7-8; McDuff-Salamon *J-holomorphic Curves and Symplectic Topology* (AMS Colloq. 52, 2nd ed. 2012)

References

Kontsevich — Enumeration of rational curves via torus actions · in *The Moduli Space of Curves* (Dijkgraaf-Faber-van der Geer, eds.), Progress in Math. 129, Birkhäuser 1995, 335-368; introduces stable maps, the moduli space $\overline{\mathcal{M}}_{g,n}(X,\beta)$, and the torus-localisation computation of $N_d$
Kontsevich-Manin — Gromov-Witten classes, quantum cohomology, and enumerative geometry · Comm. Math. Phys. 164 (1994), 525-562; the axioms for Gromov-Witten classes, the WDVV / splitting axiom, and the quantum cohomology ring
Behrend-Fantechi — The intrinsic normal cone · Invent. Math. 128 (1997), 45-88; the virtual fundamental class $[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{vir}$ from a perfect obstruction theory and the intrinsic normal cone
Behrend — Gromov-Witten invariants in algebraic geometry · Invent. Math. 127 (1997), 601-617; the algebraic Gromov-Witten invariants via the virtual class and the splitting/forgetful behaviour
Li-Tian — Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties · J. Amer. Math. Soc. 11 (1998), 119-174; construction of the virtual fundamental class for arbitrary smooth projective targets
Fulton-Pandharipande — Notes on stable maps and quantum cohomology · Proc. Symp. Pure Math. 62.2 (1997), 45-96; the construction of $\overline{\mathcal{M}}_{g,n}(X,\beta)$, evaluation maps, the boundary, and the WDVV derivation of Kontsevich's recursion
Cox-Katz — Mirror Symmetry and Algebraic Geometry · AMS Mathematical Surveys and Monographs 68, 1999; Ch. 7 (Gromov-Witten invariants), Ch. 8 (quantum cohomology), Ch. 11 (the quintic A-model)
Behrend-Manin — Stacks of stable maps and Gromov-Witten invariants · Duke Math. J. 85 (1996), 1-60; the algebraic-stack foundations of $\overline{\mathcal{M}}_{g,n}(X,\beta)$ and the stable-modular-graph formalism for the boundary

Estimated time

beginner: 18m
intermediate: 48m
master: 85m