Mumford (Fogarty, Kirwan) — *Geometric Invariant Theory* (Fast Track 3.31) — Audit + Gap Plan

Book: David Mumford, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 34 (Neue Folge), Springer-Verlag 1965; 3rd enlarged edition with John Fogarty and Frances Kirwan, 1994, xiv + 292 pp. ISBN 978-3-540-56963-3. Hereafter GIT (or MFK when the 3rd-edition material is being referenced).

Fast Track entry: 3.31, paired with Joe Harris Moduli of Curves (3.30) and Fulton Toric Varieties (3.32) as the moduli-and-quotients trio. GIT is the canonical text on constructing moduli spaces as quotients of reductive group actions — stability, semistability, the Hilbert-Mumford numerical criterion, the GIT quotient $X/{/}_{L}G=\mathrm{Proj}{⨁}_n{H}^0(X;L^{\otimes n}{)}^G$, and (in the third edition) Kirwan's stratification of the unstable locus linking GIT to symplectic reduction. The book Mumford was awarded the 1974 Fields Medal partly for.

PDF availability. No author-hosted PDF (commercial Springer Ergebnisse text; still in print). Not present in reference/textbooks-extra/, reference/fasttrack-texts/, or the newly-discovered reference/book-collection/free-downloads/ (which contains only Freed's CBMS / lecture notes). Springer SpringerLink (link.springer.com/book/10.1007/978-3-642-57916-5) gated behind institutional auth; Wikipedia and Newstead's freely-available Tata lectures used as cross-checks. This audit is REDUCED — produced from chapter structure as documented in MFK's introduction (1994), Newstead 1978, Hoskins's Moduli Problems and Geometric Invariant Theory lecture notes (Berlin/Bonn 2015), Mukai An Introduction to Invariants and Moduli (Cambridge 2003), Schmitt Geometric Invariant Theory and Decorated Principal Bundles (EMS 2008), and the Codex's already-shipped 04.10.02-git unit (which is the densest in-corpus reference for the book). Mark as REDUCED in the audit log; full P1 inventory at line-number granularity 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 and griffiths-harris-geometry-algebraic-curves.md.

The audit surface is focused — Codex's 04-algebraic-geometry/10-moduli/ chapter ships only two units (04.10.01-moduli-of-curves, 04.10.02-git), both of which already name-check every major GIT result. 04.10.02-git is a substantive master-tier unit and covers the Mumford 1965 GIT theorem statement, stability/semistability stratification, the Proj construction, the Hilbert-Mumford numerical criterion (worked for binary quartics), the Kempf-Ness theorem, Kirwan stratification (in the Master-tier Advanced Results section), variation of GIT, and Bridgeland stability. But every one of these is sketched, not unit-anchored. Hilbert-Mumford, Kempf-Ness, Kirwan stratification, moduli of vector bundles (slope stability), the Hilbert scheme, the reductivity / finite-generation theorem, and variation of GIT all deserve dedicated units; GIT (the book) cannot be considered FT-equivalent until they ship.

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§1 What GIT is for

GIT is the canonical text on constructing algebraic quotients of varieties by reductive group actions. Where naive set-theoretic quotients $X/G$ are usually non-Hausdorff, non-projective, and unhelpful for moduli theory, Mumford's framework introduces a linearisation — an ample line bundle $L$ on $X$ with a lift of the $G$-action — and defines the GIT quotient

$$X/{/}_{L}G:=\mathrm{Proj}\underset{n\ge 0}{⨁}{H}^0(X;L^{\otimes n}{)}^G.$$

The construction works exactly on the semistable locus $X^{ss}(L)$, which Mumford characterises by the existence of an invariant section non-vanishing at $x$. The stable locus $X^s(L)\subseteq X^{ss}(L)$ — orbits closed in $X^{ss}$ with finite stabiliser — gives a geometric quotient (set-theoretic orbit space); the full semistable locus gives only a categorical / good quotient (identifying points with intersecting orbit closures). The reductivity hypothesis on $G$ (Mumford: classical groups, semisimple Lie groups, tori; not unipotents) is essential for the finite generation of ${⨁}_n{H}^0(X;L^{\otimes n}{)}^G$ (Hilbert 1893 for classical groups, Nagata's 1959 counterexample for non-reductives, Mumford's general reductive case).

GIT is the construction tool that makes algebraic moduli spaces exist as schemes:

• Moduli of curves ${\mathcal{M}}_g$ (Mumford 1965) — GIT quotient of the Hilbert scheme of tri-canonically embedded curves by $\mathrm{PGL}$.
• Moduli of vector bundles $\mathcal{M}(r,d)$ on a curve (Mumford 1962, Seshadri 1967, Narasimhan-Seshadri 1965) — GIT quotient via slope stability $\mu (E)=\mathrm{deg}E/\mathrm{rank}E$.
• Moduli of abelian varieties (Mumford 1965, 1968) — GIT with theta-linearisation.
• Moduli of K-stable Fano varieties (Tian-Donaldson-Chen-Sun 2015) — infinite-dimensional GIT via test configurations.

Distinctive contributions, in roughly the order MFK develops them:

1. Reductive group actions and finite generation. Hilbert's classical-groups theorem extended to all reductive groups (over any characteristic, using good filtrations in char $p$ where the Reynolds operator fails). MFK Ch. 1, App. A.
2. Linearisation. A $G$-action on an ample line bundle $L$ covering the action on $X$; gives the $G$-graded section ring whose Proj is the GIT quotient. MFK Ch. 1.
3. Stable, semistable, unstable points. The stratification $X^s(L)\subseteq X^{ss}(L)\subseteq X$ via the existence and geometry of invariant sections. MFK Ch. 1, Defs. 1.7–1.8.
4. The GIT quotient theorem (Mumford 1965). Existence of $X/{/}_{L}G$ as a quasi-projective scheme (projective if $X$ is); categorical-quotient property on $X^{ss}$, geometric-quotient property on $X^s$. MFK Ch. 1 §4, Thm. 1.10.
5. Hilbert-Mumford numerical criterion. A point is unstable iff some one-parameter subgroup $\lambda :{\mathbb{G}}_m\to G$ destabilises it (positive weight at the limit). Reduces stability to a finite-checkable convex-geometric calculation. MFK Ch. 2, Thm. 2.1. Strengthened by Kempf 1978 (existence of a most destabilising 1-PS).
6. Applications: moduli of curves. Construction of ${\mathcal{M}}_g$ for $g\ge 2$ as a quasi-projective scheme via the tri-canonical Hilbert scheme. MFK Ch. 5.
7. Applications: moduli of vector bundles, Picard schemes, abelian varieties. MFK Chs. 6–7 (mostly Fogarty appendix in 2nd ed., absorbed into the main text in 3rd ed.).
8. Kirwan's stratification of the unstable locus (3rd edition appendix, from Kirwan 1984). Stratifies $X^{us}$ into finitely many $G$-invariant locally closed pieces matching the Morse-theoretic strata of the moment-map norm-squared $\Vert \mu {\Vert }^2$ — making the GIT picture symplectic via Kempf-Ness.

GIT is not a first introduction to algebraic geometry. It assumes Hartshorne-level scheme theory (Proj, ample line bundles, Hilbert scheme), basic algebraic-group theory (reductive groups, root systems, one-parameter subgroups), and (for the Kirwan appendix) basic symplectic geometry (moment maps, Marsden-Weinstein reduction). It is the canonical entry point to moduli theory in the algebraic-geometric tradition; the parallel symplectic-geometry tradition (Atiyah-Bott, Kirwan, Hitchin) is equivalent in content and complementary in style. The Fast Track explicitly chooses the Mumford / algebraic-geometric track, with Kirwan's symplectic viewpoint as a deepening (Cycle 5 Marsden-Weinstein connection already partially shipped at 05.04.01-moment-map and 05.04.02-symplectic-reduction).

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§2 Coverage table (Codex vs GIT)

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

GIT topic	Codex unit(s)	Status	Note
Reductive algebraic group; complete reducibility	—	✗	Gap. Mentioned in 04.10.02-git master-tier Ex. 4, but no anchor unit. Would live in 03-modern-geometry/03-lie-groups/ (03.03.0R or similar) or in algebraic-geometry chapter.
Hilbert-Nagata-Mumford finite generation of invariants	04.10.02-git (master tier, Advanced Results)	△	Mentioned in proof outline of GIT theorem; no dedicated unit on the finite-generation theorem itself.
$G$-linearised line bundle	04.10.02-git (formal definition)	△	Defined in passing; no anchor unit.
Stable / semistable / unstable points	04.10.02-git	✓	Covered in intermediate + worked example tier.
GIT quotient $X/{/}_{L}G=\mathrm{Proj}(R^G)$	04.10.02-git	✓	This is the main content of the existing unit.
Categorical vs geometric quotient	04.10.02-git	△	Stated but not given dedicated unit; standard moduli-theory distinction worth pulling out.
Hilbert-Mumford numerical criterion (1-PS)	04.10.02-git (Ex. 3 worked)	△	Worked for binary quartics; no anchor unit stating the general theorem with proof.
Kempf 1978: most-destabilising 1-PS	—	✗	Gap. Strengthening of HM; Master-tier deepening.
Hilbert scheme ${\mathrm{Hilb}}_{{\mathbb{P}}^N}^P$	04.10.01-moduli-of-curves (mentioned)	△	Cited as input to Mumford's ${\mathcal{M}}_g$ construction; no dedicated unit. Foundational for any moduli-via-GIT construction.
Moduli of curves ${\mathcal{M}}_g$ via GIT	04.10.01-moduli-of-curves	✓	Existing unit.
Deligne-Mumford stable-curve compactification $\overline{{\mathcal{M}}_g}$	04.10.01-moduli-of-curves	✓	Master tier.
Moduli of vector bundles on a curve; slope stability	04.10.02-git (Ex. 7 mentioned)	△	Foundational application of GIT; deserves dedicated unit.
Narasimhan-Seshadri theorem (stable bundles ↔ unitary reps)	—	✗	Gap. Bridge between GIT and Riemann-surface representations.
Gieseker stability for sheaves	—	✗	Gap. Higher-dim generalisation of slope stability.
Kempf-Ness theorem (GIT ↔ symplectic reduction)	04.10.02-git (Ex. 6 worked)	△	Stated; no anchor unit. Bridges to 05.04.02-symplectic-reduction.
Kirwan stratification of unstable locus	04.10.02-git (Advanced Results)	△	Mentioned; no anchor unit.
Variation of GIT (Dolgachev-Hu, Thaddeus)	04.10.02-git (Advanced Results)	△	Mentioned; no anchor unit.
K-stability and YTD conjecture	04.10.02-git (Advanced Results)	△	Mentioned at master tier; pointer-unit candidate.
Bridgeland stability conditions	04.10.02-git (Advanced Results)	△	Mentioned; categorical analogue of GIT.
Non-reductive GIT (Doran-Kirwan)	04.10.02-git (Advanced Results)	△	Mentioned; master-tier deepening.
Moment map for $G$-action on symplectic / Kähler manifold	05.04.01-moment-map	✓	Shipped via Cycle 5 (Cannas da Silva, Marsden-Weinstein).
Marsden-Weinstein symplectic reduction	05.04.02-symplectic-reduction	✓	Shipped via Cycle 5.
Atiyah-Bott Yang-Mills over Riemann surfaces	03.07.09-moduli-space-of-asd-connections-mathcal-m-k-s-4	△	Adjacent — moduli of ASD connections (4-manifolds), not Riemann-surface bundle case. Cycle 6 (Atiyah K-Theory) flagged.

Aggregate coverage estimate: ~25% of GIT's distinctive content has anchor-level coverage; ~70% has name-check coverage in 04.10.02-git. The existing GIT unit is dense but is doing the work of 6–8 units; the punch-list extracts those into dedicated anchor units so each is FT-equivalence-grade and can be cited cleanly from other moduli-using texts (Atiyah-Bott, Joe Harris, ACGH-II, Donaldson Floer Homology Groups in Yang-Mills, Hitchin Higgs bundles, etc.).

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

Priority 0 — soft prereqs already shipped or partial. Reductivity needs a $\ge$ Beginner/Intermediate anchor; Hilbert scheme needs a stub. Both can be Cycle-internal (algebraic-geometry chapter) deferrals.

Priority 1 — high-leverage, captures GIT's central content as dedicated anchor units (extracting from the existing 04.10.02-git catch-all):

1. 04.10.03 Hilbert-Mumford numerical criterion. Statement, sketch of proof via reduction to torus actions on ${\mathbb{A}}^N$, convexity / Newton-polytope picture, Kempf 1978 most-destabilising 1-PS strengthening (Master tier). Worked examples: binary $d$-forms (root-multiplicity criterion, generalising 04.10.02-git Ex. 3), $n$-point configurations on ${\mathbb{P}}^1$, Grassmannian Plücker stability. MFK Ch. 2 anchor. Three-tier, ~2000 words.
2. 04.10.04 Kempf-Ness theorem and the GIT–symplectic dictionary. Statement; proof sketch via norm-minimisation / convexity; explicit identification $X/{/}_{L}G\cong {\mu }^{-1}(0)/K$ with $K\subset G$ maximal compact. Pulls the existing 04.10.02-git Ex. 6 out into an anchor unit. Bridges to 05.04.02-symplectic-reduction (Marsden-Weinstein) — the Cycle 5 / Cannas-da-Silva moment-map / reduction batch is the symplectic-side prereq, now shipped. Kempf-Ness 1979 (LNM 732) originator citation. MFK 3rd ed. appendix (Kirwan) anchor. Three-tier, ~2000 words.
3. 04.10.05 Hilbert scheme ${\mathrm{Hilb}}_X^P$. Grothendieck 1961 FGA construction; representable functor parametrising closed subschemes with given Hilbert polynomial; relative version ${\mathrm{Hilb}}_{X/S}^P$; tangent space $H^0(N_{Y/X})$; smoothability and connectedness. Foundational input to any moduli-space construction via GIT (Mumford, Gieseker, Simpson). Worked example: ${\mathrm{Hilb}}^2({\mathbb{P}}^2)$ = blow-up of ${\mathrm{Sym}}^2{\mathbb{P}}^2$ at the diagonal. MFK Ch. 5 anchor; Grothendieck FGA originator. Three-tier, ~2000 words.
4. 04.10.06 Moduli of vector bundles on a curve; slope stability. Mumford's 1962 stability condition $\mu (E)=\mathrm{deg}E/\mathrm{rank}E$; construction of $\mathcal{M}(r,d)$ as a GIT quotient (Seshadri 1967, Newstead 1972); smoothness, dimension $r^2(g-1)+1$; Atiyah-Bott cohomology (1983); Narasimhan-Seshadri 1965 unitary-representations correspondence stated. Newstead 1978 Tata lectures anchor; MFK Ch. 6 / Seshadri 1967 Annals 85 originator. Three-tier, ~2200 words.
5. 04.10.07 Reductive algebraic groups and finite generation of invariants. Definition (complete reducibility / unipotent radical trivial); classical groups, semisimple groups, tori as examples; non-reductive examples (Borel, unipotent); Hilbert's 1893 finite-generation theorem for classical groups via the symbolic method; Nagata 1959 counterexample answering Hilbert 14 in the negative; Mumford / Haboush 1975 extension to all reductive groups in arbitrary characteristic (Reynolds operator in char 0; good filtrations / Haboush's theorem on geometric reductivity in char $p$). MFK Ch. 1 + App. A; Haboush 1975 Annals 102. Three-tier, ~1800 words. Bridges to 03-modern-geometry/03-lie-groups for the Lie-theoretic content; algebraic-geometry chapter is the anchor location given the GIT context.

Priority 2 — second-order content (Kirwan stratification, variation, categorical quotient):

6. 04.10.08 Kirwan stratification of the unstable locus. Kirwan 1984 Cohomology of Quotients in Symplectic and Algebraic Geometry (PUP). Stratification of $X^{us}$ into finitely many $G$-invariant locally closed pieces indexed by conjugacy classes of 1-parameter subgroups; matches Morse-theoretic strata of $\Vert \mu {\Vert }^2$ via Kempf-Ness; used to compute equivariant cohomology of GIT quotients $H_G^{\ast }(X^{ss})\to H^{\ast }(X//G)$ (Kirwan surjectivity). MFK 3rd ed. ch. 8 (Kirwan appendix) anchor. Master-tier-only, ~1500 words.
7. 04.10.09 Variation of GIT (VGIT). Different linearisations give different GIT quotients, related by flips / flops across walls. Dolgachev-Hu 1998 Publ. Math. IHES 87; Thaddeus 1996 J. Amer. Math. Soc. 9. The space of linearisations is a polytope in $\mathrm{NS}(X{)}^G\otimes \mathbb{R}$; quotients change as walls are crossed. Foundational for birational geometry of moduli spaces and the minimal model program. Intermediate + Master tier, ~1500 words.
8. 04.10.10 Categorical vs geometric quotient; good and geometric quotients. The Borel-Mumford notion of good quotient (surjective, $G$-invariant, affine pieces have invariant coordinate rings) and geometric quotient (fibres = $G$-orbits). Universal property; comparison with topological quotient. MFK Ch. 1 §0–1. Short Intermediate unit, ~1000 words. Could be folded into 04.10.02-git rather than a new unit; flagged as a P2 "extract from existing" item.

Priority 3 — modern deepenings (Master-tier, not strictly required for FT-equivalence but high-value for working algebraic geometers):

9. 04.10.11 Gieseker stability and moduli of sheaves. Gieseker 1977 Annals 106; stability via Hilbert polynomial (refining slope stability for higher-dim sheaves). Moduli space of Gieseker-stable sheaves on a projective surface / threefold; smoothness, virtual fundamental class, Donaldson-Thomas counts. Master-tier-only, ~1500 words.
10. 04.10.12 Bridgeland stability conditions. Bridgeland 2007 Annals 166; stability on triangulated categories ${\mathrm{D}}^b\mathrm{Coh}(X)$. The space of stability conditions $\mathrm{Stab}(X)$ is a complex manifold; foundational for mirror symmetry and derived-category moduli. Master-tier-only, ~1500 words.
11. 04.10.13 K-stability and the Yau-Tian-Donaldson conjecture. K-stability for Fano varieties via test configurations (infinite-dimensional GIT). Chen-Donaldson-Sun 2015 (three J. Amer. Math. Soc. papers): KE metric exists ⇔ K-stable. Master-tier-only, ~1500 words.

Priority 4 — survey pointers (optional, Master-only):

12. 04.10.14 Non-reductive GIT. Doran-Kirwan 2007; reductive envelopes; augmented quotient stacks. Pointer unit; ~1000 words.
13. 04.10.15 Derived GIT and magic windows. Halpern-Leistner 2014 Adv. Math.; lifts GIT to ${\mathrm{D}}^b\mathrm{Coh}$ with magic windows giving derived equivalences across walls. Pointer unit; ~1000 words.

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§4 Implementation sketch (P3 → P4)

For a full GIT coverage pass, items 1–5 are the minimum FT-equivalence set. Realistic production estimate (mirroring recent algebraic-geometry batches at 04.10.01-moduli-of-curves, 04.10.02-git, and the 04.04 Riemann-Roch units):

• ~3.5–4 hours per priority-1 unit (heavy on technical statements + worked examples; master tier requires careful HM-criterion / VGIT / slope-stability machinery and citation discipline).
• 5 priority-1 units × ~3.75 hours = ~18–19 hours of focused production.
• 3 priority-2 units × ~3 hours = ~9 hours.
• 5 priority-3/4 deepenings × ~2.5 hours = ~12–13 hours.
• Total ~40 hours for a complete pass; ~20 hours for the FT-equivalence minimum. Fits a focused 5-day window.

Originator-prose target. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, units 1, 2, 4, 5 (Hilbert-Mumford, Kempf-Ness, vector-bundle moduli, finite generation) should carry originator-prose treatment citing:

• Hilbert, D. (1893), Über die vollen Invariantensysteme, Math. Ann. 42, 313–373 — originator: finite generation of classical-group invariant rings via the symbolic method.
• Mumford, D. (1965), Geometric Invariant Theory, Springer Ergebnisse 34 — the book itself; originating the framework.
• Kempf, G. (1978), Instability in invariant theory, Annals of Math. 108, 299–316 — originator: most-destabilising 1-PS strengthening of Hilbert-Mumford.
• Kempf, G. & Ness, L. (1979), The length of vectors in representation spaces, in Algebraic Geometry, Copenhagen 1978, Springer LNM 732, 233–243 — originating the GIT ↔ symplectic reduction dictionary.
• Kirwan, F. (1984), Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes 31, Princeton University Press — originating the unstable-locus stratification (appended to the 3rd edition of MFK).
• Haboush, W. (1975), Reductive groups are geometrically reductive, Annals of Math. 102, 67–83 — originating finite generation in arbitrary characteristic.
• Mumford, D. (1962), Projective invariants of projective structures and applications, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962) — originating slope stability for vector bundles, pre-MFK.
• Seshadri, C. S. (1967), Space of unitary vector bundles on a compact Riemann surface, Annals of Math. 85, 303–336 — first rigorous GIT construction of $\mathcal{M}(r,d)$.

Notation crosswalk. MFK uses $X^{ss}(L)$, $X^s(L)$, $X^{us}(L)$ for the (semi)stable / unstable loci, $X/{/}_{L}G$ for the GIT quotient (or $X//G$ when the linearisation is implicit), ${\mu }^L(x,\lambda )$ for the Hilbert-Mumford function. Newstead and Hoskins follow MFK. The Codex's 04.10.02-git already adopts MFK notation throughout; the new units should preserve this. Kempf-Ness uses $\Phi$ or $\Vert v{\Vert }^2$ for the norm-squared moment map (real side); Kirwan uses $\mu :X\to {\mathfrak{k}}^{\ast }$ for the moment map (matching the Codex's 05.04.01-moment-map notation). Record in a §Notation paragraph of 04.10.03 and 04.10.04.

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§5 What this plan does NOT cover

• Joe Harris Moduli of Curves (FT 3.30). Deferred to its own audit pass. There is substantial overlap (Hilbert scheme; moduli of stable curves; tautological classes; Mumford-Morita-Miller $\kappa$-classes; Witten-Kontsevich) but Joe Harris's book is the curve-specific deformation-theory text, not the GIT framework text. Coordinate via cross-references once both plans are written.
• Symplectic / Marsden-Weinstein side beyond the Kempf-Ness bridge. The Cycle 5 (Cannas da Silva, Helgason) moment-map and symplectic-reduction units (05.04.01, 05.04.02) are the symplectic-side anchors; Atiyah-Bott Yang-Mills cohomology (FT 3.41 or similar) is the parallel symplectic-moduli text and has its own audit pass.
• A line-number-level inventory of every named theorem in MFK (full P1 audit; deferred — MFK is 292 pp. but technically dense, requires a PDF to do faithfully).
• Exercise-pack production. MFK has no formal exercise list; exercises in derived units should come from Newstead 1978 and Mukai 2003 (which both have explicit exercises).
• Cubical / derived / non-reductive deepenings beyond the pointer units 04.10.14 and 04.10.15. The Codex is not committing to a parallel derived-GIT track.
• K-stability analytic proof details (YTD conjecture). The pointer unit 04.10.13 states the theorem and sketches the test-configuration GIT picture; the analytic proof (Chen-Donaldson-Sun's three Ann. Math. papers, 200+ pp. total) is out of scope for FT-equivalence.

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§6 Acceptance criteria for FT equivalence (GIT)

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

• All 5 priority-1 units have shipped (04.10.03–04.10.07). This closes the central HM-criterion / Kempf-Ness / Hilbert-scheme / vector-bundle-moduli / reductivity gap that 04.10.02-git currently catches.
• ≥2 of the 3 priority-2 units have shipped (04.10.08 Kirwan stratification + at least one of 04.10.09 VGIT / 04.10.10 categorical quotient).
• At least 1 of the priority-3 units has shipped, with the others queued as Master-tier deepenings.
• All originator-prose citations (Hilbert 1893, Mumford 1965, Kempf 1978, Kempf-Ness 1979, Kirwan 1984, Haboush 1975, Seshadri
  1967. appear at least once in the corpus, with at least one unit per originator carrying the prose.
• ≥95% of GIT's named theorems in chapters 1–8 map to a Codex unit (currently ~25% anchor + ~70% mention; after priority-1 this rises to ~75% anchor; priority-1+2 to ~90% anchor; full ≥95% requires priority-1+2+3).
• ≥90% of GIT's worked examples (binary forms, $n$-points on ${\mathbb{P}}^1$, point-configurations on ${\mathbb{P}}^2$, moduli of curves, moduli of vector bundles, moduli of abelian varieties) have either a direct unit or a worked-example block in a covering unit.
• Notation decisions are recorded in 04.10.03 and 04.10.04.
• Pass-W weaving connects the new units laterally to 04.10.01-moduli-of-curves, 04.10.02-git, 05.04.01-moment-map, 05.04.02-symplectic-reduction, 03.07.09-moduli-space-of-asd-connections, and the (future) Joe Harris Moduli of Curves punch-list.

The 5 priority-1 units close most of the equivalence gap. Priority-2 closes the Kirwan / VGIT / categorical-quotient gap. Priority-3+4 are modern-research deepenings (Gieseker, Bridgeland, K-stability, non-reductive GIT, derived GIT) — high-value but not strictly required for FT-equivalence.

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§7 Sourcing

• Primary text. Mumford, Fogarty, Kirwan, Geometric Invariant Theory, 3rd enlarged edition, Springer Ergebnisse 34, 1994. No author-hosted PDF; commercial Springer text, still in print. Acquisition: BUY (per docs/catalogs/FASTTRACK_BOOKLIST.md entry 3.31). 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.
  • Newstead, P. E. (1978), Lectures on Introduction to Moduli Problems and Orbit Spaces, Tata Institute of Fundamental Research Lectures on Mathematics 51. Freely hosted at <https://www.math.tifr.res.in/~publ/ln/tifr51.pdf>. The canonical accessible GIT introduction; covers HM criterion and moduli-of-bundles in detail.
  • Hoskins, V. (2015), Moduli Problems and Geometric Invariant Theory, lecture notes (Berlin / Bonn). Freely hosted at <https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf>. Modern GIT lecture notes with explicit worked examples.
  • Mukai, S. (2003), An Introduction to Invariants and Moduli, Cambridge Studies in Advanced Mathematics 81. (No free PDF; supplementary reference cited from corpus.)
  • Schmitt, A. (2008), Geometric Invariant Theory and Decorated Principal Bundles, EMS Zürich Lectures in Advanced Mathematics. Modern reference for GIT applied to decorated bundles.
  • Thomas, R. P. (2006), Notes on GIT and Symplectic Reduction for Bundles and Varieties, Surveys in Differential Geometry X, 221–273. Freely available preprint <https://arxiv.org/abs/math/0512411>. The standard modern bridge between GIT and symplectic reduction.
  • Wikipedia, Geometric invariant theory. Cross-check for chapter-level structure of the 3rd edition.
• License. MFK is commercial; cite as Mumford, Fogarty, Kirwan, Geometric Invariant Theory, 3rd enlarged ed., Springer 1994. Newstead 1978 is freely hosted by TIFR; cite as Newstead, Lectures on Introduction to Moduli Problems and Orbit Spaces, TIFR Lectures on Mathematics 51, Springer 1978. Hoskins 2015 is author-hosted lecture notes.
• Local copy target. Once acquired (PDF or scan), add to reference/textbooks-extra/ as Mumford-Fogarty-Kirwan-GeometricInvariantTheory-3rdEd-1994.pdf to mirror the corpus pattern. Pre-acquisition: add Newstead-LecturesOnModuliProblemsAndOrbitSpaces-TIFR1978.pdf and Hoskins-ModuliProblemsAndGIT-2015-LectureNotes.pdf as freely-available proxies.

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Status: REDUCED audit (no PDF). Punch-list: 5 P1 + 3 P2 + 3 P3 + 2 P4 = 13 candidate units. Conversion to a full P1 audit deferred until MFK PDF acquired.