Harris, Morrison — *Moduli of Curves* (Fast Track 3.30) — Audit + Gap Plan

Book: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag 1998, xiv + 366 pp. ISBN 978-0-387-98429-2 / 978-0-387-98438-4. Hereafter HM.

Fast Track entry: 3.30, paired with Hartshorne (3.21), ACGH-I (3.22), and Mumford-Fogarty-Kirwan GIT (3.31) as the algebraic-geometry curves / moduli quartet. HM is the canonical textbook on the moduli space ${\mathcal{M}}_g$ itself — the space whose points are isomorphism classes of smooth projective genus-$g$ curves and whose Deligne-Mumford compactification ${\overline{\mathcal{M}}}_g$ adds stable curves with at- worst-nodal singularities. Where ACGH-I (FT 3.22, audited Cycle 7) studies linear systems and Brill-Noether theory on a fixed curve, and MFK (FT 3.31, audited Cycle 8) is the abstract quotient framework, HM is the book on the moduli space itself, its boundary, its tautological classes, and its birational geometry. The book working algebraic geometers cite when they say "by Harris-Morrison Ch. 6" or "the $3\lambda -\delta$ slope inequality" or "the Mumford relation."

PDF availability. No author-hosted PDF (commercial Springer GTM, still in print). Not present in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/ (the last contains only Freed's notes). Springer SpringerLink (link.springer.com/book/10.1007/b98867) gated behind institutional auth; Anna's Archive not reached within the time budget. This audit is REDUCED — produced from HM's standard chapter structure as documented across the algebraic-geometry community (MathSciNet review by Vakil 1999, Vakil's own Moduli of Curves Park-City lectures, Mumford 1983 Towards an enumerative geometry of the moduli space of curves, Deligne-Mumford 1969 Publ. Math. IHES 36, ACGH-II = Arbarello-Cornalba-Griffiths 2011 Geometry of Algebraic Curves II GMW 268, the Faber-Pandharipande tautological-relations literature, and Hain's Lectures on Moduli Spaces of Riemann Surfaces PCMI 2011), together with the Codex's already-shipped 04.10.01-moduli-of-curves and 06.08.03-moduli-of-riemann-surfaces units (which already name-check every major HM topic). Mark as REDUCED in the audit log; a full P1 inventory at line-number granularity is deferred until a PDF is acquired.

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite per orchestration protocol). Punch-list of new units + deepenings to reach the equivalence threshold (docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Mirrors the structure of brown-higgins-sivera-nonabelian-algebraic-topology.md, griffiths-harris-geometry-algebraic-curves.md, and mumford-geometric-invariant-theory.md.

The audit surface is highly focused — Codex's 04-algebraic-geometry/10-moduli/ chapter ships only two units (04.10.01-moduli-of-curves, 04.10.02-git), and 06-riemann-surfaces/08-vhs/06.08.03-moduli-of-riemann-surfaces.md mirrors the 04.10.01 content from the Riemann-surfaces side. Both 04.10.01 and 06.08.03 are substantive units that name-check $\psi$, $\kappa$, $\lambda$, the Hodge bundle, Deligne-Mumford compactification, stable curves, Mumford conjecture, Madsen-Weiss, Witten-Kontsevich, and the $3g-3$ dimension count. Every single one of those topics is sketched, not unit-anchored. The tautological-ring machinery (Mumford 1983 $\kappa$ classes, Mumford's relations, Faber-Pandharipande / Pixton relations), the boundary-divisor / dual-graph stratification of ${\overline{\mathcal{M}}}_g$, deformation theory of nodal curves, Hurwitz numbers and admissible covers, the Harris-Mumford "of general type for $g\ge 24$" theorem, the $3\lambda -\delta$ slope inequalities, and the modern ${\overline{\mathcal{M}}}_{g,n}$ marked-point story are load-bearing for the existing units but have no anchor units of their own. This is the largest single-book gap in the algebraic-geometry chapter.

Coordination with Mumford GIT (Cycle 8). Substantial overlap on the Hilbert-scheme / tri-canonical / GIT-quotient construction of ${\mathcal{M}}_g$ (HM Ch. 4, MFK Ch. 5). The Cycle-8 punch-list at plans/fasttrack/mumford-geometric-invariant-theory.md already proposes 04.10.05 Hilbert scheme and 04.10.07 reductive groups / finite generation; these are upstream prerequisites for the HM punch-list and must not be double-counted. HM-specific units below assume those ship first and pick up at deformation-theory of curves / boundary divisors / tautological classes, which are curve-specific material that MFK does not cover.

---

§1 What HM is for

HM is the canonical textbook on the moduli space of curves ${\mathcal{M}}_g$ and its Deligne-Mumford compactification ${\overline{\mathcal{M}}}_g$, treated as geometric objects whose internal structure (boundary, tautological classes, birational invariants, deformation theory) one wants to compute and use. Where Mumford-Fogarty- Kirwan GIT (FT 3.31) gives the abstract quotient-construction machinery, and ACGH-I (FT 3.22) gives the fixed-curve Brill-Noether and Abel-Jacobi machinery, HM is the book on the moduli space itself as the central object of study — written by working algebraic geometers for working algebraic geometers, with extensive low-genus (${\mathcal{M}}_2$, ${\mathcal{M}}_3$) hands-on calculations and a clear problem-set-driven pedagogical style. The book that made ${\overline{\mathcal{M}}}_g$ accessible to a generation of algebraic geometers and bridged the 1969 Deligne-Mumford construction to the 1990s tautological-ring boom.

Six chapters, with the first half developing the geometry of curves and the second half developing the moduli space.

Chapter 1 — Parameter spaces: constructions and examples. Hilbert schemes, Chow varieties, parameter-space constructions in general (Mumford's pluri-canonical for curves, Grothendieck's $\mathrm{Quot}$ and $\mathrm{Hilb}$). Motivating low-dimensional examples: the modular curve ${\overline{\mathcal{M}}}_{1,1}$ as $\mathbb{H}/{\mathrm{SL}}_2(\mathbb{Z})$, the moduli of hyperelliptic curves as $\mathrm{Sym}^{2g + 2}(\mathbb{P}^1)/ \mathrm{PGL}_2$,the$\mathcal{M}_2$ explicit model. Sets the framing question: what does it mean for a space to "be" a moduli space (fine vs coarse vs stack).

Chapter 2 — Basic facts about moduli spaces of curves. ${\mathcal{M}}_g$ as a coarse moduli space; the moduli functor; representability failure because of automorphisms; introduction to the Deligne-Mumford stack ${\overline{\mathcal{M}}}_g$; the $3g-3$ dimension count via deformation theory (Kodaira-Spencer, $H^1(C;T_C)$); the Teichmüller-theoretic and Hodge-theoretic alternative constructions; the universal property sketches; smoothness of the stack and how it interacts with the non-smoothness of the coarse space along the hyperelliptic locus.

Chapter 3 — Techniques. Deformation theory of smooth and nodal curves: ${\mathrm{Ext}}^1({\Omega }_C,{\mathcal{O}}_C)$ as the tangent space to deformations; the local-to-global spectral sequence; deformations of nodes vs of smoothings; logarithmic deformation theory; Schlessinger's deformation framework. The technical chapter underpinning the rest of the book.

Chapter 4 — Construction of ${\overline{\mathcal{M}}}_g$. The tri-canonical embedding, Hilbert scheme, $\mathrm{PGL}$ quotient via GIT (citing Mumford 1965); the Deligne-Mumford 1969 stable-reduction construction; stable curves and their automorphism groups; the boundary $\partial\overline{\mathcal{M}}_g = \overline{\mathcal{M}}_g \setminus \mathcal{M}g$ as a normal-crossing divisor with components $\Delta_0, \Delta_1, \ldots, \Delta{\lfloor g/2 \rfloor}$; the dual graph of a stable curve as the combinatorial stratification device for boundary strata; irreducibility of ${\overline{\mathcal{M}}}_g$ (Deligne-Mumford).

Chapter 5 — Limit linear series and Brill-Noether theory. Eisenbud- Harris 1986 limit linear series — extending Brill-Noether existence / non-emptiness theorems to nodal / boundary curves via degenerations. Brill-Noether-loci $W_d^r$ as moduli problems over ${\overline{\mathcal{M}}}_g$. Connects HM directly to ACGH-I (which does the fixed-curve story).

Chapter 6 — Geometry of moduli spaces: selected topics. The tautological classes: ${\psi }_i=c_1({\mathbb{L}}_i)$ on ${\overline{\mathcal{M}}}_{g,n}$, ${\kappa }_a={\pi }_{\ast }{\psi }^{a+1}$ (Mumford- Morita-Miller), ${\lambda }_a=c_a(\mathbb{E})$ (Hodge bundle), boundary classes ${\delta }_i$. Mumford's GRR relations (1983 Towards an enumerative geometry): the explicit formula for $\lambda$ in terms of ${\kappa }_1$ and $\delta$ via Grothendieck-Riemann-Roch applied to the universal curve. The slope inequality $s(\overline{\mathcal{M}}_g) \leq \frac{12}{g + 1}$ and the Harris-Mumford 1982 theorem (${\overline{\mathcal{M}}}_g$ is of general type for $g\ge 24$). Pointers forward to Mumford's conjecture (Madsen-Weiss 2007) and Witten-Kontsevich (1992) intersection numbers.

Distinctive contributions of HM, in roughly the order they are developed:

1. The moduli space as a geometric object to compute with, not just to construct. HM's signature pedagogical move.
2. Hands-on ${\mathcal{M}}_2$, ${\mathcal{M}}_3$ models via hyperelliptic and plane-quartic descriptions, with explicit Igusa / Clebsch invariants for ${\mathcal{M}}_2$ and the Aronhold model for ${\mathcal{M}}_3$.
3. Deformation theory of nodal curves as the technical foundation. ${\mathrm{Ext}}^1({\Omega }_C,{\mathcal{O}}_C)$, smoothings of nodes, the versal deformation space, Schlessinger.
4. The dual graph stratification of $\partial {\overline{\mathcal{M}}}_g$, making the boundary explicitly combinatorial and computable.
5. Tautological classes $\psi ,\kappa ,\lambda$ and the Hodge bundle $\mathbb{E}$ treated as concrete computable invariants, with explicit Chern-class computations and the Mumford relation $12\lambda ={\kappa }_1+\delta$ on ${\overline{\mathcal{M}}}_g$ from Mumford 1983.
6. Limit linear series (Eisenbud-Harris) bridging fixed-curve Brill-Noether (ACGH-I) to families over ${\overline{\mathcal{M}}}_g$.
7. The Harris-Mumford / Eisenbud-Harris slope inequalities and the $g\ge 24$ general-type theorem: birational classification of ${\overline{\mathcal{M}}}_g$ via canonical-class positivity of $K_{{\overline{\mathcal{M}}}_g}=13\lambda -2\delta$.
8. Hurwitz numbers and admissible covers: counting branched covers $C\to {\mathbb{P}}^1$ with prescribed ramification, via the Harris-Mumford compactification of the Hurwitz scheme by admissible covers. Connects to the ELSV formula (1999) and Witten-Kontsevich.
9. Pointers to the modern theory: Mumford's conjecture / Madsen- Weiss, Witten-Kontsevich KdV hierarchy, Faber-Pandharipande tautological-ring conjectures, the moduli stack vs coarse space distinction.

HM is not a first introduction to algebraic geometry, scheme theory, or moduli theory. It assumes Hartshorne-level scheme theory (Ch. II–III), basic curve theory (Riemann-Roch, canonical embedding, hyperelliptic involution), and Mumford-level GIT (the construction of ${\mathcal{M}}_g$ by tri-canonical Hilbert + $\mathrm{PGL}$ quotient is stated, not re-proved). It is the canonical entry point to moduli of curves in the algebraic-geometric tradition; the parallel topological / mapping-class- group tradition (Farb-Margalit, Hain PCMI 2011, Madsen-Weiss) is equivalent in target but very different in style. The Fast Track explicitly chooses HM, with mapping-class-group pointers as a Master-tier deepening.

Peer / originator literature cited.

• Mumford, D. (1965), Geometric Invariant Theory, Springer Ergebnisse 34 (3rd ed. with Fogarty & Kirwan 1994). The construction of ${\mathcal{M}}_g$ as a quasi-projective scheme via tri-canonical GIT. Fast Track 3.31, audited Cycle 8 at plans/fasttrack/mumford-geometric-invariant-theory.md.
• Deligne, P. & Mumford, D. (1969), The irreducibility of the space of curves of given genus, Publ. Math. IHES 36, 75–109. The stable-curves compactification ${\overline{\mathcal{M}}}_g$ and the proof of geometric irreducibility in arbitrary characteristic.
• Mumford, D. (1983), Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry II, Birkhäuser, 271–328. Originating the tautological ring and the $\psi ,\kappa ,\lambda$ classes, and the Grothendieck-Riemann-Roch derivation of Mumford's relation $12\lambda ={\kappa }_1+\delta$.
• Arbarello, E., Cornalba, M., Griffiths, P. & Harris, J. (1985), Geometry of Algebraic Curves I, Springer Grundlehren 267. Fast Track 3.22, audited Cycle 7 at plans/fasttrack/griffiths-harris-geometry-algebraic-curves.md. Fixed-curve Brill-Noether and Abel-Jacobi machinery; HM extends to families over ${\overline{\mathcal{M}}}_g$.
• Arbarello, E., Cornalba, M. & Griffiths, P. (2011), Geometry of Algebraic Curves II, Springer Grundlehren 268. The moduli-theoretic companion volume; mapping class group, Teichmüller, period mapping, tautological intersection theory. Not yet audited; queued behind HM.
• Harris, J. & Mumford, D. (1982), On the Kodaira dimension of the moduli space of curves, Invent. Math. 67, 23–86. Originating the "of general type for $g\ge 24$" theorem.
• Vakil, R. (2003), The moduli space of curves and its tautological ring, Notices Amer. Math. Soc. 50, 647–658. Free survey at <https://math.stanford.edu/~vakil/files/curvesNotices.pdf>. The best concise modern introduction to HM-style material.
• Vakil, R. (2017), Foundations of Algebraic Geometry. Free at <https://math.stanford.edu/~vakil/216blog/>. The accessible scheme-theoretic prereq for HM; chapter on moduli of curves is effectively a digest of HM Ch. 2–4.

---

§2 Coverage table (Codex vs HM)

Cross-referenced against the current corpus. ✓ = covered, △ = partial / mentioned-but-not-anchored, ✗ = not covered.

HM topic	Codex unit(s)	Status	Note
Moduli space ${\mathcal{M}}_g$ as coarse moduli (functor + GIT)	04.10.01-moduli-of-curves, 06.08.03-moduli-of-riemann-surfaces	✓	Both units are substantive (master-tier). Riemann's $3g-3$ count covered.
${\mathcal{M}}_g$ as Deligne-Mumford stack	04.10.01 (caveat box), 06.08.03 (master tier)	△	Mentioned; no dedicated unit on Deligne-Mumford stacks. Foundational gap.
Tri-canonical Hilbert / $\mathrm{PGL}$-quotient construction	04.10.01 (proof outline), 04.10.02-git (referenced)	△	The proof outline in 04.10.01 is the entire treatment; the Hilbert scheme itself has no anchor (already on the Cycle-8 MFK punch-list as 04.10.05).
Deformation theory of smooth curves; $H^1(C;T_C)$ tangent space	—	✗	Gap. Mentioned in passing in 04.10.01 proof outline. Foundational for HM Ch. 3. Would live in 04.10 or 04-algebraic-geometry/02-schemes/.
Deformation theory of nodal curves; smoothing nodes	—	✗	Gap. HM Ch. 3 core content; needed for any boundary-divisor / stable-reduction treatment.
Schlessinger's criteria; versal deformations	—	✗	Gap. Foundational deformation-theory machinery; broader than HM but anchored here.
Deligne-Mumford stable-curves compactification ${\overline{\mathcal{M}}}_g$	04.10.01 (master tier), 06.08.03 (master tier)	△	Stated, definition given; no proof of properness / projectivity, no anchor on stable-reduction theorem.
Stable curve: nodal singularities, finite $\mathrm{Aut}$, ${\omega }_C$ ample	04.10.01 (master tier)	△	Defined in passing; deserves dedicated unit.
Boundary divisors ${\Delta }_0,{\Delta }_1,\dots ,{\Delta }_{\lfloor g/2\rfloor }$	04.10.01 (master tier paragraph)	△	Listed but not unfolded; no dimension / glueing / monodromy computation.
Dual graph of a stable curve; boundary stratification	—	✗	Gap. Foundational combinatorial device for $\partial {\overline{\mathcal{M}}}_g$.
Irreducibility of ${\mathcal{M}}_g$ (Deligne-Mumford 1969)	06.08.03 (cited)	△	Cited in references; no anchor unit stating the theorem with proof sketch.
Hodge bundle $\mathbb{E}={\pi }_{\ast }{\omega }_{\mathcal{C}/\mathcal{M}}$	04.10.01 (Exercise 4 with proof)	△	Defined in an exercise; deserves its own master-tier unit (Hodge bundle is the central rank-$g$ vector bundle on ${\overline{\mathcal{M}}}_g$).
$\psi$-classes ${\psi }_i=c_1({\mathbb{L}}_i)$ on ${\overline{\mathcal{M}}}_{g,n}$	04.10.01 (master-tier paragraph)	△	Named, not anchored. Load-bearing for Witten-Kontsevich exercise in 04.10.01.
$\kappa$-classes ${\kappa }_a={\pi }_{\ast }{\psi }^{a+1}$ (Mumford-Morita-Miller)	04.10.01 (master-tier paragraph)	△	Named, not anchored. Load-bearing for Mumford-conjecture exercise in 04.10.01.
$\lambda$-classes ${\lambda }_a=c_a(\mathbb{E})$ (Hodge classes)	04.10.01 (master-tier paragraph)	△	Named, not anchored. Load-bearing for the Picard-group statement in 04.10.01.
Mumford's relation $12\lambda ={\kappa }_1+\delta$	—	✗	Gap. Mumford 1983 GRR derivation; foundational tautological identity.
Tautological ring $R^{\ast }({\overline{\mathcal{M}}}_{g,n})$	04.10.01 (master tier, Advanced Results)	△	Defined in one paragraph; deserves a dedicated unit with Faber-Pandharipande / Pixton-relations pointer.
Faber's intersection conjecture; Gorenstein-with-explicit-socle	04.10.01 (master tier paragraph)	△	Mentioned; pointer-unit candidate.
Pixton's tautological relations	04.10.01 (master tier paragraph)	△	Mentioned; pointer-unit candidate.
Marked-point moduli ${\overline{\mathcal{M}}}_{g,n}$; $\dim =3g-3+n$	04.10.01 (formal-definition paragraph)	△	Dimension formula stated, no unit dedicated to the marked-point construction or its boundary / forgetful / gluing morphisms.
Forgetful morphism $\pi :{\overline{\mathcal{M}}}_{g,n+1}\to {\overline{\mathcal{M}}}_{g,n}$	—	✗	Gap. Foundational structural morphism; used to define $\kappa$ classes.
Gluing / clutching morphisms ${\overline{\mathcal{M}}}_{g_1,n_1+1}\times {\overline{\mathcal{M}}}_{g_2,n_2+1}\to {\overline{\mathcal{M}}}_{g_1+g_2,n_1+n_2}$	—	✗	Gap. The morphisms defining the boundary strata.
Limit linear series (Eisenbud-Harris 1986)	—	✗	Gap. HM Ch. 5 core content; bridges ACGH-I Brill-Noether to families. Also on ACGH-I punch-list.
Brill-Noether loci $W_d^r$ over ${\overline{\mathcal{M}}}_g$	—	✗	Gap. On ACGH-I punch-list as well; coordinate.
Harris-Mumford 1982: ${\overline{\mathcal{M}}}_g$ of general type for $g\ge 24$	04.10.01 (master tier paragraph)	△	Mentioned; no anchor with $K=13\lambda -2\delta$ canonical-class computation.
Slope inequality / slope of ${\overline{\mathcal{M}}}_g$	04.10.01 (master tier paragraph)	△	Mentioned via Harris-Mumford; no anchor.
Hurwitz numbers; Hurwitz scheme; admissible covers	04.04.02-hurwitz-formula (Riemann-Hurwitz only)	△	Riemann-Hurwitz formula covered; Hurwitz numbers (count of branched covers) and the admissible-covers compactification are gaps.
Kontsevich's ribbon graphs / matrix Airy function	04.10.01 (Exercise 7 sketch)	△	Stated as theorem statement; no anchor on the ribbon-graph / matrix-model proof technique.
Witten conjecture $F$ satisfies KdV	04.10.01 (Exercise 7)	△	Statement only; no anchor unit.
Mumford conjecture $\lim H^{\ast }({\mathcal{M}}_g)=\mathbb{Q}[{\kappa }_1,{\kappa }_2,\dots ]$	04.10.01 (Exercise 6)	△	Statement + Madsen-Weiss outline; no anchor unit.
Madsen-Weiss theorem (stable mapping class group ≃ ${\Omega }^{\infty }\mathbf{MTSO}(2)$)	04.10.01 (Exercise 6 sketch)	△	Stated; no anchor — substantial topology prereq (Galatius-Madsen-Tillmann-Weiss).
ELSV formula (Hurwitz numbers ↔ Hodge-integrals)	—	✗	Gap. Ekedahl-Lando-Shapiro-Vainshtein 2001; not yet on any punch-list.
Igusa / Clebsch invariants for ${\mathcal{M}}_2$	04.10.01 (Exercise 5 sketch)	△	Hyperelliptic-quotient construction sketched; no explicit Igusa-invariant computation.
Plane-quartic / Aronhold model for ${\mathcal{M}}_3$	—	✗	Gap. HM's signature low-genus example.
Teichmüller space ${\mathcal{T}}_g$; mapping class group ${\mathrm{Mod}}_g$	06.08.03 (master tier paragraph)	△	Mentioned; no anchor. Cross-reference to Farb-Margalit (not in Fast Track at present).
Moduli of abelian varieties ${\mathcal{A}}_g$; Torelli morphism ${\mathcal{M}}_g\to {\mathcal{A}}_g$	06.06.03-jacobian-variety, 06.06.08-schottky-problem	△	Jacobian unit covers single-curve case; Torelli morphism + Torelli theorem has no anchor unit. Schottky problem unit references it.
Modular forms ↔ ${\overline{\mathcal{M}}}_{1,1}$	04.10.01 (master tier paragraph)	△	Mentioned; pointer to number theory / Wiles.
Mirzakhani's hyperbolic-geometry proof of Witten-Kontsevich	04.10.01 (master tier paragraph)	△	Mentioned; pointer only.

Aggregate coverage estimate: Two substantive units (04.10.01-moduli-of-curves, 06.08.03-moduli-of-riemann-surfaces) already exist and name-check essentially every HM topic. But anchor-level coverage of HM-distinctive material (deformation theory, boundary divisors, dual graphs, tautological classes, Mumford's relation, limit linear series, Harris-Mumford general-type theorem, Hurwitz numbers, admissible covers) is roughly 5% — only the foundational ${\mathcal{M}}_g$ definition + worked exercises are anchored. The existing 04.10.01 unit is doing the work of 8–10 units; the punch-list extracts the silently-load-bearing pieces into dedicated anchor units. This is the largest single-book gap in the algebraic-geometry chapter and silently undermines the existing units' citations.

Critically: the Mumford-conjecture exercise (04.10.01 Ex. 6), the Witten-Kontsevich exercise (Ex. 7), and the Hodge-bundle exercise (Ex. 4) all reference $\psi$, $\kappa$, $\lambda$ classes as named objects, but those classes have no anchor units. Any reader following the exercise chain will hit a dead-end. Same for the Picard-group statement in the master-tier prose. Closing this gap is a corpus- integrity priority, not just an FT-equivalence priority.

---

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — soft prereqs shared with Mumford GIT plan. The Cycle-8 MFK punch-list (plans/fasttrack/mumford-geometric-invariant-theory.md §3) already proposes:

• 04.10.05 Hilbert scheme ${\mathrm{Hilb}}_X^P$ (foundational input).
• 04.10.07 Reductive groups + finite generation.

These are shared prerequisites with the HM punch-list and must ship once, not twice. The HM-specific units below assume they exist or are queued.

Priority 1 — high-leverage, captures HM's central content as dedicated anchor units that are silently load-bearing for 04.10.01 and 06.08.03:

1. 04.10.20 Deformation theory of smooth curves. Tangent space $T_{[C]}\mathcal{M}_g = H^1(C; T_C) = \mathrm{Ext}^1(\Omega_C, \mathcal{O}_C)$;obstructionin$H^2(C; T_C) = 0$ for curves; Kodaira-Spencer map; $\dim H^1(C;T_C)=3g-3$ via Riemann-Roch on $T_C={\omega }_C^{-1}$. Worked example: deformation space of an elliptic curve is 1-dimensional. Schlessinger's criteria stated. HM Ch. 3 anchor; Kodaira-Spencer 1958 J. Math. Mech. 8 originator. Three-tier, ~2000 words. Critical for the $3g-3$ dimension count to be more than a slogan in 04.10.01.
2. 04.10.21 Deformation theory of nodal curves; smoothings of nodes. Local deformation of a node $xy=0$ has 1-parameter smoothing $xy=t$; global picture via ${\mathrm{Ext}}^1$; logarithmic deformation theory for stable curves; $\dim T_{[C_0]}{\overline{\mathcal{M}}}_g=3g-3$ holds at nodal curves with the node-smoothing parameters; the boundary divisor ${\Delta }_i$ as the locus where one node persists. HM Ch. 3 anchor; Deligne-Mumford 1969 Publ. Math. IHES 36 originator citation. Three-tier, ~2200 words. Critical for the ${\overline{\mathcal{M}}}_g$ master-tier prose in 04.10.01 to be more than a description.
3. 04.10.22 Stable curve and Deligne-Mumford stability. Definition (nodal, finite $\mathrm{Aut}$, ${\omega }_C$ ample; equivalently every smooth rational component meets the rest in $\ge 3$ points); dual graph; arithmetic genus; the three equivalent characterisations; examples and non-examples; the moduli stack ${\overline{\mathcal{M}}}_g$ as a Deligne-Mumford stack. HM Ch. 4 + Deligne-Mumford 1969 anchor. Three-tier, ~1800 words. Anchor for stable-curve content that 04.10.01 currently states inline.
4. 04.10.23 Boundary divisors ${\Delta }_i$ and the dual-graph stratification of $\partial {\overline{\mathcal{M}}}_g$. The irreducible boundary components ${\Delta }_0$ (irreducible nodal genus $g-1$) and ${\Delta }_i$ ($1\le i\le g/2$, two components of genera $i$ and $g-i$ glued at a node); the normal-crossing structure; the dual graph as the combinatorial index of boundary strata of ${\overline{\mathcal{M}}}_g$. The gluing morphisms $\xi_i : \overline{\mathcal{M}}{i, 1} \times \overline{\mathcal{M}}{g - i, 1} \to \Delta_i \subset \overline{\mathcal{M}}_g$. HM Ch. 4 anchor. Three-tier, ~2000 words.
5. 04.10.24 Tautological classes I: $\psi$-classes on ${\overline{\mathcal{M}}}_{g,n}$. Definition $\psi_i = c_1 (\mathbb{L}_i)$,where$\mathbb{L}_i$ is the cotangent line at the $i$-th marked point; behaviour under forgetful and gluing morphisms; the string equation and dilaton equation; $\psi$-class intersection numbers and the Witten-Kontsevich generating function (pointer to KdV). Mumford 1983 + Witten 1990 originators. Three-tier, ~2000 words.
6. 04.10.25 Tautological classes II: $\kappa$- and $\lambda$- classes; Hodge bundle. ${\kappa }_a={\pi }_{\ast }({\psi }_{n+1}^{a+1})$ (Mumford-Morita-Miller); ${\lambda }_a=c_a(\mathbb{E})$ where $\mathbb{E}={\pi }_{\ast }{\omega }_{\mathcal{C}/\mathcal{M}}$ is the Hodge bundle; Mumford's relation $12\lambda = \kappa_1 + \delta$on$\overline{\mathcal{M}}_g$ via GRR applied to ${\omega }_{\mathcal{C}/\mathcal{M}}$; the tautological ring $R^{\ast }({\overline{\mathcal{M}}}_{g,n})\subset {\mathrm{CH}}^{\ast }$ as the subring generated by these classes plus boundary classes. Mumford 1983 originator. Three-tier, ~2200 words. Closes the $\kappa ,\lambda$ gap that 04.10.01 opens but does not fill.

Priority 2 — important second-order content (forgetful / gluing morphisms, Mumford relations, Harris-Mumford):

7. 04.10.26 Forgetful and gluing morphisms on ${\overline{\mathcal{M}}}_{g,n}$. $\pi_i : \overline{\mathcal{M}}{g, n + 1} \to \overline{\mathcal{M}}{g, n}$forgettingthe$i$-th marked point; the universal curve $\overline{\mathcal{C}}{g, n} \cong \overline{\mathcal{M}}{g, n + 1}$ via stabilisation; gluing morphisms producing boundary divisors; behaviour of $\psi$ / $\kappa$ / $\lambda$ classes under pullback. HM Ch. 6 anchor. Intermediate + Master tier, ~1500 words.
8. 04.10.27 Mumford-Morita-Miller relations and the tautological ring. Beyond Mumford's $12\lambda ={\kappa }_1+\delta$: Faber-Pandharipande 1999 conjecture (Gorenstein with explicit socle); Pixton's relations 2012 (conjectural complete set); Janda 2018 (proof of $3$-spin / Pixton-DR side); the DR cycles = double-ramification cycles. Master-tier-only, ~1800 words.
9. 04.10.28 Harris-Mumford theorem: ${\overline{\mathcal{M}}}_g$ is of general type for $g\ge 24$. Canonical class $K_{{\overline{\mathcal{M}}}_g}=13\lambda -2\delta$ via the adjunction-type formula and Mumford's relation; positivity for $g\ge 24$ via slope inequalities on Brill-Noether divisors; subsequent improvements (Eisenbud-Harris-Mumford, Farkas $g=22,23$). Harris-Mumford 1982 Invent. Math. 67; Farkas 2010s for boundary cases. Master-tier-only, ~1800 words.
10. 04.10.29 Limit linear series (Eisenbud-Harris). Extending Brill-Noether $W_d^r$ to nodal / boundary curves; limit-linear-series on chains of components; smoothability criteria; applications to families of curves. Coordinate with ACGH-I punch-list (which proposes a fixed-curve Brill-Noether anchor first; HM extends to families). HM Ch. 5 anchor; Eisenbud-Harris 1986 Invent. Math. 85. Master-tier-only, ~1800 words.

Priority 3 — Hurwitz / admissible covers / explicit-low-genus (distinctive HM material, master-tier):

11. 04.10.30 Hurwitz numbers and the Hurwitz scheme. Count of degree-$d$ branched covers $C\to {\mathbb{P}}^1$ with prescribed ramification profiles; the Hurwitz scheme as a parameter space; monodromy / character-theoretic formulas (Burnside, Frobenius); connection to symmetric-group representation theory. HM Ch. 1 + 6 references; ELSV 1999 originator (pointer for the formula). Three-tier, ~2000 words. Bridges to 04.04.02-hurwitz-formula (Riemann-Hurwitz, already shipped) and 07-representation-theory/.
12. 04.10.31 Admissible covers (Harris-Mumford compactification of the Hurwitz scheme). Compactification of the Hurwitz scheme by admissible covers of nodal curves; the morphism to ${\overline{\mathcal{M}}}_g$; used in the Harris-Mumford 1982 proof of the general-type theorem. HM Ch. 4 + Harris-Mumford 1982 anchor. Master-tier-only, ~1500 words.
13. 04.10.32 ELSV formula: Hurwitz numbers as Hodge integrals. Ekedahl-Lando-Shapiro-Vainshtein 2001 Invent. Math. 146 formula expressing simple Hurwitz numbers as integrals of $\lambda$ / $\psi$ classes over ${\overline{\mathcal{M}}}_{g,n}$. Connection to Gromov-Witten of ${\mathbb{P}}^1$; Okounkov-Pandharipande proof of Witten via ELSV + matrix models. Master-tier-only, ~1500 words.
14. 04.10.33 Low-genus models: ${\mathcal{M}}_2$ (Igusa-Clebsch) and ${\mathcal{M}}_3$ (Aronhold / plane quartic). Explicit Igusa-Clebsch invariants $I_2,I_4,I_6,I_{10}$ giving ${\mathcal{M}}_2$ as a weighted projective variety; the Aronhold invariant identifying ${\mathcal{M}}_3$ with the moduli of plane quartics minus the hyperelliptic locus. HM Ch. 1 + 2 explicit examples. Igusa 1960 Amer. J. Math. 82 originator for ${\mathcal{M}}_2$. Intermediate + Master tier, ~1800 words.

Priority 4 — modern-research pointer units (Master-only, optional but high-value):

15. 04.10.34 Torelli morphism and Torelli theorem. The morphism $\tau :{\mathcal{M}}_g\to {\mathcal{A}}_g$ sending $C \mapsto (\mathrm{Jac}(C), \Theta)$; Torelli theorem (1913, rigorous proofs by Andreotti 1958, Matsusaka 1958, Weil 1957): $\tau$ is generically injective; the Schottky problem (already covered at 06.06.08-schottky-problem). Master-tier pointer; ~1200 words.
16. 04.10.35 Witten-Kontsevich-Mirzakhani: the KdV hierarchy on ${\overline{\mathcal{M}}}_{g,n}$. Pointer unit unfolding the statement of 04.10.01 Exercise 7. Witten 1990 conjecture; Kontsevich 1992 matrix-Airy proof; Mirzakhani 2007 hyperbolic proof; Okounkov-Pandharipande proof via ELSV. Master-tier-only; ~1500 words. Closes the dead-end exercise reference in 04.10.01.
17. 04.10.36 Mumford's conjecture and the Madsen-Weiss theorem (pointer). Pointer unit unfolding 04.10.01 Exercise 6. Stable mapping class group ${\Gamma }_{\infty }$; group completion $B{\Gamma }_{\infty }^{+}\simeq {\Omega }^{\infty }\mathbf{MTSO}(2)$; cohomology calculation giving $\mathbb{Q}[\kappa_1, \kappa_2, \ldots]$. Galatius-Madsen-Tillmann-Weiss generalisation. Master-tier-only; ~1500 words. Significant topology prereq (cobordism categories) — flag for Cycle-coordination with the topology track.

---

§4 Implementation sketch (P3 → P4)

For a full HM coverage pass, items 1–6 are the minimum FT-equivalence set (and the corpus-integrity minimum, closing the silently load- bearing dead-end references in 04.10.01 and 06.08.03). Realistic production estimate (mirroring the algebraic-geometry batches at 04.10.01-moduli-of-curves, 04.10.02-git, and the 04.04 Riemann- Roch units):

• ~4 hours per priority-1 unit (heavy on technical statements + worked computations; master tier requires careful ${\mathrm{Ext}}^1$ / GRR / Chern-class / Mumford-relation machinery, citation discipline for originator prose, and consistency-checking against the existing 04.10.01 and 06.08.03 units).
• 6 priority-1 units × ~4 hours = ~24 hours of focused production.
• 4 priority-2 units × ~3.25 hours = ~13 hours.
• 4 priority-3 units × ~3 hours = ~12 hours.
• 3 priority-4 pointer units × ~2 hours = ~6 hours.
• Total ~55 hours for a complete pass; ~24 hours for the FT-equivalence + corpus-integrity minimum. Fits a focused 6–7 day window.

Originator-prose target. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, units 2, 3, 4, 6, 9, 11 (nodal deformation, stable curves, boundary divisors, $\kappa$-classes / Mumford relation, Harris-Mumford general-type, Hurwitz numbers) should carry originator-prose treatment citing:

• Deligne, P. & Mumford, D. (1969), The irreducibility of the space of curves of given genus, Publ. Math. IHES 36, 75–109 — originating ${\overline{\mathcal{M}}}_g$, the stable-curve definition, the stable-reduction theorem.
• Mumford, D. (1983), Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry II, Birkhäuser, 271–328 — originating $\kappa$-classes, the tautological ring, Mumford's relation $12\lambda ={\kappa }_1+\delta$.
• Witten, E. (1990), Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry 1, 243–310 — conjecturing the KdV-hierarchy structure of $\psi$-class intersection numbers.
• Kontsevich, M. (1992), Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147, 1–23 — proving Witten via ribbon graphs and matrix integrals.
• Madsen, I. & Weiss, M. (2007), The stable moduli space of Riemann surfaces: Mumford's conjecture, Annals of Math. 165, 843–941 — proving Mumford's 1983 stable-cohomology conjecture.
• Harris, J. & Mumford, D. (1982), On the Kodaira dimension of the moduli space of curves, Invent. Math. 67, 23–86 — originating the general-type theorem for $g\ge 24$.
• Eisenbud, D. & Harris, J. (1986), Limit linear series: basic theory, Invent. Math. 85, 337–371 — originating limit linear series.
• Kodaira, K. & Spencer, D. C. (1958), On deformations of complex analytic structures I, II, Annals of Math. 67, 328–401, 403–466 — originating deformation theory + Kodaira-Spencer map.
• Ekedahl, T., Lando, S., Shapiro, M. & Vainshtein, A. (2001), Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146, 297–327 — ELSV formula.

Notation crosswalk. HM uses ${\mathcal{M}}_g$ and ${\overline{\mathcal{M}}}_g$ for the coarse moduli and its DM compactification, ${\mathcal{M}}_{g,n}$ and ${\overline{\mathcal{M}}}_{g,n}$ for the $n$-marked-point versions, ${\Delta }_i$ for the boundary divisors, $\lambda$ for the first Chern class of the Hodge bundle (sometimes ${\lambda }_1$ for it, ${\lambda }_a$ for higher Chern classes), ${\kappa }_a$ for the Mumford-Morita-Miller classes, ${\psi }_i$ for the cotangent classes at marked points, $\delta =\sum {\Delta }_i$ for the total boundary class, $\mathbb{E}$ for the Hodge bundle. ACGH-II follows the same convention. The Codex's existing 04.10.01 and 06.08.03 units already adopt this notation throughout; the new HM units must preserve consistency. Record in a §Notation paragraph of 04.10.24 and 04.10.25.

Coordination notes.

• MFK Cycle 8 (plans/fasttrack/mumford-geometric-invariant-theory.md): the Hilbert-scheme unit 04.10.05 and reductivity unit 04.10.07 are shared prerequisites. HM punch-list reserves IDs 04.10.20–36 to leave 04.10.03–19 available for the MFK side.
• ACGH-I Cycle 7 (plans/fasttrack/griffiths-harris-geometry- algebraic-curves.md): the limit-linear-series unit 04.10.29 here builds on the fixed-curve Brill-Noether anchor that ACGH-I punch-list proposes. Cross-reference both plans.
• Riemann-surface side (06.08.03-moduli-of-riemann-surfaces.md): the existing unit at 06.08.03 is a Riemann-surface-side mirror of 04.10.01. Treat it as a complementary anchor (analytic / Hodge- theoretic phrasing) rather than rewriting; lateral connections via pass-W.

---

§5 What this plan does NOT cover

• Madsen-Weiss proof internals (04.10.36 is a pointer-only unit). Cobordism categories, scanning, group-completion theorems for mapping class group spaces, the Galatius-Madsen-Tillmann-Weiss identification — these require a topology-side audit that is not yet in the Fast Track queue. Defer.
• Witten-Kontsevich / Mirzakhani proof internals (04.10.35 is a pointer-only unit). The matrix-Airy / ribbon-graph proof, Mirzakhani's hyperbolic-volume proof, and the Okounkov-Pandharipande ELSV-based proof are each substantial papers; only statements and high-level outlines are in scope. Defer the proofs.
• ACGH-II (Arbarello-Cornalba-Griffiths 2011 GMW 268). This is the moduli-theoretic companion to ACGH-I and overlaps HM substantially on tautological classes, intersection theory on ${\overline{\mathcal{M}}}_{g,n}$, period mapping, and the Hodge bundle. Audited separately if/when needed; coordinate via cross- references once both plans are written.
• Mapping class group ${\mathrm{Mod}}_g$ machinery (Dehn-Lickorish generators, Birman exact sequence, the Nielsen-Thurston classification, Farb-Margalit). Touched in 06.08.03 master-tier prose and 04.10.36; full treatment defers to a future Farb-Margalit Fast Track addition (not yet listed).
• Teichmüller theory (extremal quasi-conformal maps, Teichmüller's existence + uniqueness theorems, Teichmüller geodesics, Bers's embedding). Touched in 06.08.03 master-tier; full treatment defers to a future Teichmüller-theory audit (Imayoshi-Taniguchi, Hubbard, etc.).
• GIT-side construction details beyond 04.10.01 proof outline. The tri-canonical / Hilbert / $\mathrm{PGL}$ quotient is the Mumford GIT plan's responsibility (04.10.05); HM Ch. 4 cites the construction and develops the boundary / deformation / tautological-class picture, which is what the HM punch-list anchors.
• A line-number-level inventory of every named theorem in HM (full P1 audit; deferred — HM is 366 pp. and technically dense, requires a PDF to do faithfully).
• Exercise-pack production from HM's problem sets. HM exercises are extensive and explicit; an exercise-pack pass should follow the priority-1 unit batch once PDF acquisition makes line-number citation possible.
• K-stability / KSBA / KSB moduli of higher-dimensional varieties. Out of scope; pointer in 04.10.02-git master-tier already.

---

§6 Acceptance criteria for FT equivalence (HM)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

• All 6 priority-1 units have shipped (04.10.20–04.10.25). This closes the central deformation-theory / stable-curve / boundary- divisor / tautological-class gap that 04.10.01 and 06.08.03 currently catch silently.
• ≥3 of the 4 priority-2 units have shipped, with Mumford's relation 04.10.27 and Harris-Mumford general-type 04.10.28 mandatory.
• ≥2 of the 4 priority-3 units have shipped, with the Hurwitz- numbers unit 04.10.30 mandatory (closing the 04.04.02 Riemann- Hurwitz → Hurwitz-numbers gap).
• At least 1 of the priority-4 pointer units has shipped, with Witten-Kontsevich 04.10.35 and Mumford-conjecture 04.10.36 the priorities (they close the dead-end exercise references in 04.10.01).
• All originator-prose citations (Deligne-Mumford 1969, Mumford 1983, Witten 1990, Kontsevich 1992, Madsen-Weiss 2007, Harris-Mumford 1982, Eisenbud-Harris 1986, Kodaira-Spencer 1958, ELSV 2001) appear at least once in the corpus with at least one unit per originator carrying the full prose.
• ≥95% of HM's named theorems in chapters 1–6 map to a Codex unit (currently ~5% anchor + ~90% mention; after priority-1 this rises to ~55% anchor; priority-1+2 to ~80% anchor; full ≥95% requires priority-1+2+3).
• ≥90% of HM's worked low-genus examples (${\mathcal{M}}_2$ via Igusa-Clebsch, ${\mathcal{M}}_3$ via Aronhold, ${\overline{\mathcal{M}}}_2$ boundary description, ${\overline{\mathcal{M}}}_3$ canonical-class computation) have either a direct unit (Priority-3 04.10.33) or a worked-example block in a covering unit.
• Notation decisions are recorded in 04.10.24 and 04.10.25.
• Pass-W weaving connects the new units laterally to 04.10.01-moduli-of-curves, 04.10.02-git, 04.04.01-riemann-roch-curves, 04.04.02-hurwitz-formula, 04.08.02-canonical-sheaf, 04.09.01-hodge-decomposition, 06.03.01-riemann-surface, 06.06.03-jacobian-variety, 06.06.08-schottky-problem, 06.08.03-moduli-of-riemann-surfaces, and the (existing) Mumford GIT punch-list shared units 04.10.05 (Hilbert scheme) and 04.10.07 (reductive groups).
• Corpus-integrity audit: every named reference to $\psi$, $\kappa$, $\lambda$, Hodge bundle, Deligne-Mumford stack, boundary divisor, Mumford conjecture, Madsen-Weiss, Witten-Kontsevich in 04.10.01 and 06.08.03 is updated to cite the new anchor unit by ID and slug. No more dead-end name-checks.

The 6 priority-1 units close the corpus-integrity gap and most of the FT-equivalence gap. Priority-2 closes Harris-Mumford / tautological- relations / limit-linear-series. Priority-3 closes Hurwitz / admissible covers / low-genus examples. Priority-4 closes the dead-end pointer references in the existing units.

---

§7 Sourcing

• Primary text. Harris, J. & Morrison, I., Moduli of Curves, Springer GTM 187, 1998, xiv + 366 pp. ISBN 978-0-387-98429-2 / 978-0-387-98438-4. No author-hosted PDF; commercial Springer GTM, still in print. Acquisition: BUY (per docs/catalogs/FASTTRACK_BOOKLIST.md entry 3.30). Pre-acquisition the audit is REDUCED; post-acquisition a P1 full audit at line- number granularity should follow.
• Free secondary sources used in this reduced audit.
  • Vakil, R. (2003), The moduli space of curves and its tautological ring, Notices Amer. Math. Soc. 50, 647–658. Free at <https://math.stanford.edu/~vakil/files/curvesNotices.pdf>. The best concise modern introduction to HM-style material; covers ${\mathcal{M}}_g$, ${\overline{\mathcal{M}}}_g$, tautological ring, Witten-Kontsevich, ELSV, Pixton's relations.
  • Vakil, R. (2017), Foundations of Algebraic Geometry. Free at <https://math.stanford.edu/~vakil/216blog/>. Chapter on moduli of curves is effectively a digest of HM Ch. 2–4 with the scheme-theoretic prerequisites built in. Used as the chapter-structure crosscheck for HM Ch. 2–4 in this reduced audit.
  • Hain, R. (2011), Lectures on Moduli Spaces of Riemann Surfaces, Park City IAS / PCMI 2011 Lectures. Free preprint <https://arxiv.org/abs/1403.6911> and at <https://math.duke.edu/~hain/papers/PCMI.pdf>. Modern lecture- notes survey of ${\mathcal{M}}_g$, ${\overline{\mathcal{M}}}_g$, the tautological ring, mapping class group, Teichmüller, period mapping. The Riemann-surface-side companion to HM.
  • Mumford, D. (1983), Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry II, Birkhäuser, 271–328. The originating paper for tautological classes and Mumford's relation; library access typically required.
  • Deligne, P. & Mumford, D. (1969), The irreducibility of the space of curves of given genus, Publ. Math. IHES 36, 75–109. Free at <http://www.numdam.org/item/PMIHES_1969__36__75_0/>. The originating compactification paper.
  • Faber, C. & Pandharipande, R. (2000), Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48, 215–252; and subsequent papers in the Faber-Pandharipande series. Free arXiv preprints.
  • Cornalba, M. & Harris, J. (1988), Divisor classes associated to families of stable varieties, Ann. Sci. ENS 21, 455–475. Free at numdam.org. The slope-inequality calibration paper.
  • Arbarello, E., Cornalba, M. & Griffiths, P. (2011), Geometry of Algebraic Curves II, Springer GMW 268. Same status as HM: commercial Springer text, no free PDF, used here only via its standard citation pattern in the secondary literature. Will be audited separately.
• License. HM is commercial; cite as Harris, J. & Morrison, I., Moduli of Curves, Springer GTM 187, 1998. Pre-acquisition the audit relies on the free Vakil / Hain / numdam sources above plus the standing citations in 04.10.01-moduli-of-curves and 06.08.03-moduli-of-riemann-surfaces.
• Local copy. Once acquired, add to reference/textbooks-extra/ as Harris-Morrison-ModuliOfCurves.pdf to mirror the pattern of other commercial Fast Track texts; promote to reference/fasttrack-texts/ once an FT-tier acquisition pass is run.
• Audit status: REDUCED. Flag in logs/audit-cycle-N.md (this cycle) for post-acquisition P1 promotion.