Arbarello, Cornalba, Griffiths, Harris — Geometry of Algebraic Curves, Volume 1 (Fast Track 3.22) — Audit + Gap Plan

Book: Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths, Joseph D. Harris, Geometry of Algebraic Curves, Volume 1 (Springer Grundlehren der mathematischen Wissenschaften 267, 1985, xvi + 386 pp.). ISBN 978-0-387-90997-4. Joint with the much later Volume 2 by Arbarello-Cornalba-Griffiths (GMW 268, 2011); Volume 2 is out of scope here and will be audited separately if/when its content (moduli of curves, second-order deformation, Witten conjecture, intersection theory on $\overline{M}_{g}$ ) is brought into the Codex's algebraic-curves campaign.

Fast Track entry: 3.22, paired with Hartshorne (3.21) and Joe Harris Moduli of Curves (3.30) as the algebraic-geometry curves trio. Griffiths-Harris Vol 1 (hereafter ACGH-I) is the canonical textbook on special divisors and Brill-Noether theory — the geometric / Hodge-theoretic / Abel-Jacobi-machinery side of curve theory. The book that working algebraic geometers cite when they say "by Griffiths-Harris Ch. IV" or "the Brill-Noether dimension theorem".

PDF availability. No author-hosted PDF (commercial Springer text; in print). No copy in reference/textbooks-extra/ or reference/fasttrack-texts/. This audit is reduced — produced from chapter structure, the standard secondary literature (Donaldson, Forster, Hartshorne, Mumford, Voisin), and the Volume-1 citations already present in the corpus (06.06.06-jacobi-inversion, 06.06.07-riemann-bilinear, 06.06.08-schottky-problem, 04.04.01-riemann-roch-curves, 04.10.01-moduli-of-curves all cite ACGH-I by section). 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 donaldson-riemann-surfaces.md and brown-higgins-sivera-nonabelian-algebraic-topology.md.

The audit surface is mixed — Codex's algebraic-geometry chapter (04-algebraic-geometry/) and Riemann-surfaces chapter (06-riemann-surfaces/) jointly ship ~45 curve-relevant units; Riemann-Roch, divisors, Abel-Jacobi, Jacobi inversion, theta function, period matrix, Riemann bilinear, and Schottky are all present at some depth. The Brill-Noether content is mentioned but not anchored — Brill-Noether-dimension $ρ (g, r, d)$ , the existence theorem (Kempf-Kleiman-Laksov), the dimension theorem (Griffiths-Harris 1980), Petri-general curves (Gieseker 1982), the Brill-Noether-loci $W_{d}^{r}$ and $G_{d}^{r}$ as moduli stacks, Martens' theorem, Mumford's strengthening, all appear in passing in 06.06.06 and 06.06.08 but no dedicated unit exists. The theta-function machinery is also templated — 06.06.05 is v0.5 Strand C/D auto-generated prose, not the substantive Riemann-vanishing / theta-divisor / Riemann singularity story ACGH-I develops. These two blocks are the main gap.

§1 What ACGH-I is for

ACGH-I is the canonical textbook on the geometry of linear systems on algebraic curves and the Brill-Noether classification. Where Hartshorne Ch. IV does Riemann-Roch and the basics of curves in the scheme-theoretic style, and Forster / Donaldson / Miranda (FT 1.07 / 3.18) do the analytic / differential-geometric side of Riemann surfaces, ACGH-I sits at the intersection of algebraic geometry and Hodge theory: divisor classes via Abel-Jacobi, special divisors via Brill-Noether, theta function via Riemann's vanishing theorem, all developed with explicit machinery suitable for actual computation on a specific curve. The book that formalised special-divisor theory for the modern era.

Eight chapters, plus appendices.

Chapter I — Preliminaries. Standard package on divisors, line bundles, the Picard variety $Pic^{d} (C)$ , linear systems $∣ D ∣$ , projective embeddings, the canonical embedding for $g \geq 2$ non-hyperelliptic, Riemann-Roch and Serre duality (stated; proofs deferred to Hartshorne / Forster). Sets notation and re-orients the reader from a scheme-theoretic background.

Chapter II — Determinantal varieties. Algebro-geometric infrastructure for Brill-Noether: degeneracy loci of maps of vector bundles, Eagon-Northcott / Porteous formula, Thom-Porteous class. The technical bridge between the Brill-Noether matrix $μ_{0} : H^{0} (L) \otimes H^{0} (K \otimes L^{- 1}) \to H^{0} (K)$ and the geometry of $W_{d}^{r} \subset Pic^{d} (C)$ .

Chapter III — Introduction to special divisors. Definitions of $W_{d}^{r} := {[L] \in Pic^{d} (C) : h^{0} (L) \geq r + 1}$ and $G_{d}^{r} := {(L, V) : V \subset H^{0} (L), dim V = r + 1}$ , the Brill-Noether number $ρ (g, r, d) := g - (r + 1) (g - d + r)$ , the expected dimension of $W_{d}^{r}$ . Statement of the Brill-Noether existence theorem and dimension theorem.

Chapter IV — The varieties of special linear series on a general curve. The book's signature chapter. Proofs of the Brill-Noether existence theorem (Kempf 1971, Kleiman-Laksov 1972) via degeneracy-loci / Porteous-class calculation: for $ρ \geq 0$ , $W_{d}^{r}$ is non-empty of dimension $\geq ρ$ . Proof of the Brill-Noether dimension theorem (Griffiths-Harris 1980) and smoothness theorem for a general curve in $M_{g}$ : $dim W_{d}^{r} = ρ$ when $ρ \geq 0$ , $W_{d}^{r} = \emptyset$ when $ρ < 0$ , and $W_{d}^{r}$ is smooth away from $W_{d}^{r + 1}$ . Petri general curves (Gieseker 1982, Eisenbud-Harris 1983 Petri general): on a general curve the Petri map $μ_{0}$ is injective for every line bundle, which implies the Brill-Noether theorems and additionally describes the local geometry of $W_{d}^{r}$ .

Chapter V — The Basic Results of the Brill-Noether Theory. Consequences: Clifford's theorem with equality characterisation (equality iff $D = 0$ , $D = K$ , or $C$ hyperelliptic); Martens' theorem bounding $dim W_{d}^{r}$ above for special $d, r$ ; Mumford's strengthening of Martens; Keem's theorem. The gonality stratification of $M_{g}$ — the gonality $gon (C)$ is the smallest $d$ with $W_{d}^{1} \neq = \emptyset$ , i.e., the smallest degree of a non-constant map $C \to P^{1}$ ; $gon (C) \leq ⌈(g + 3) /2 ⌉$ by Brill-Noether.

Chapter VI — The geometric theory of Riemann's theta function. The Riemann theta function $θ (z, τ)$ on $C^{g}$ descended to the Jacobian $J (C)$ , the theta divisor $Θ \subset J (C)$ , Riemann's vanishing theorem: $Θ$ is Abel-Jacobi image of $Sym^{g - 1} (C)$ translated by the Riemann constant $κ$ , and $mult_{p} θ = h^{0} (L_{p})$ at every effective line bundle $L_{p}$ of degree $g - 1$ corresponding to $p \in J (C)$ (the Riemann singularity theorem). Identification of $Sing (Θ)$ with the translated Brill-Noether locus $W_{g - 1}^{1}$ . The Riemann theta function from the geometric (not the analytic) side: theta is a section of a specific ample line bundle on $J (C)$ , principal polarisation, the abelian-variety structure as a consequence not a hypothesis.

Chapter VII — The Existence and Connectedness Theorems for $W_{d}^{r}$ . Detailed proof of the existence theorem (Kempf 1971, Kleiman-Laksov 1972). Fulton-Lazarsfeld connectedness theorem (Fulton-Lazarsfeld 1981): for $ρ (g, r, d) \geq 1$ , $W_{d}^{r}$ is connected; corollary: $W_{d}^{r}$ is irreducible for $ρ \geq 1$ on a Petri-general curve.

Chapter VIII — Enumerative Geometry of Curves. Chern-class / intersection-theory calculations on $W_{d}^{r}$ . The Eisenbud-Harris Limit linear series preview (proper Volume-2 territory). Castelnuovo numbers, the number of $g_{d}^{r}$ 's on a general curve, applications to enumerative problems on $M_{g}$ .

Appendices. Excess linear series, the universal Jacobian / universal Picard, Hilbert schemes of curves in projective space.

Distinctive ACGH-I editorial choices:

Brill-Noether is the spine. The book is organised around the classification of linear systems $g_{d}^{r} := (L, V)$ on a curve. This is not how Hartshorne / Vakil / Forster / Donaldson are organised; they take Riemann-Roch as a destination, not as the starting point of a classification programme.
Determinantal-variety machinery (Ch. II) front-loaded. Porteous formula and degeneracy loci are developed as the technical foundation for the existence theorem. Codex does not have this block at all — no Porteous, no Thom-Porteous class, no Eagon-Northcott; the closest is the Chern-class machinery in Lawson-Michelsohn and Milnor-Stasheff coverage, which is topological not algebro-geometric.
Petri map $μ_{0} : H^{0} (L) \otimes H^{0} (K \otimes L^{- 1}) \to H^{0} (K)$ as the local invariant. Codex has the Petri map implicit in Riemann-Roch proofs but never named or developed; ACGH-I makes $μ_{0}$ a first-class object whose injectivity controls the local geometry of $W_{d}^{r}$ .
Gieseker-Petri theorem (1982) and Lazarsfeld's vector-bundle proof. Gieseker 1982 Stable curves and special divisors and Lazarsfeld 1986 Brill-Noether-Petri without degenerations. The former proves Petri's conjecture by degeneration to nodal curves; the latter by Mukai-Lazarsfeld vector bundles on K3 surfaces. Both absent from Codex.
Riemann singularity theorem as the bridge. Chapter VI identifies $Sing (Θ) = W_{g - 1}^{1}$ via the multiplicity formula $mult_{p} θ = h^{0} (L_{p})$ . This is the load-bearing observation that links theta functions to Brill-Noether geometry, and the input to Andreotti-Mayer and the Schottky problem. Codex's 06.06.08-schottky-problem cites this in passing but does not develop it.
Theta function via geometry, not analysis. ACGH-I's theta function is a holomorphic section of the line bundle $O_{J (C)} (Θ)$ where $Θ$ is defined geometrically as the Abel-Jacobi image of $Sym^{g - 1} (C)$ , with the analytic formula $\theta(z, \tau) = \sum_{n \in \mathbb{Z}^g} \exp(\pi i n^T \tau n + 2\pi i n^T z)$ as a consequence. Codex's 06.06.05-theta-function is currently the analytic-formula-first templated stub (v0.5 Strand C/D); ACGH-I's geometric framing would require a full rewrite.
Constructive existence on the Jacobian. Abel-Jacobi $α_{d} : Sym^{d} (C) \to Pic^{d} (C)$ is realised concretely; $W_{d}^{r} = α_{d} ({D : h^{0} (D) \geq r + 1})$ is computed via the rank-condition on a specific evaluation matrix.
Explicit running examples. Hyperelliptic ( $g \geq 2$ ), trigonal ( $g \geq 3$ ), $k$ -gonal stratification; canonical embedding for $g = 3, 4, 5$ explicitly; Castelnuovo bound; the Petri conjecture / Lazarsfeld theorem worked through for low genera.
Not a Riemann-surface book. Riemann surfaces appear only as the analytic backdrop; the working object is the smooth projective curve $C$ over an algebraically closed field of characteristic zero. This is the complementary lens to Donaldson / Forster.
Bibliographic anchor. ACGH-I is the citation that closes the Brill-Noether era and opens the moduli era — pairs with Joe Harris Moduli of Curves (FT 3.30) and Mumford Curves and their Jacobians (FT-adjacent, often cited).

ACGH-I ends before modern moduli theory (no Deligne-Mumford compactification, no Witten conjecture, no Kontsevich theorem, no limit linear series at length). Volume 2 (ACG 2011) picks up there.

§2 Coverage table (Codex vs ACGH-I)

Cross-referenced against the curves-relevant subset of the corpus (04-algebraic-geometry/04-curves/, 04-algebraic-geometry/04-riemann-roch/, 04-algebraic-geometry/05-divisors/, 04-algebraic-geometry/08-differentials/, 04-algebraic-geometry/10-moduli/, 06-riemann-surfaces/05-divisors-bundles/, 06-riemann-surfaces/06-jacobians/, plus the 06.04-riemann-roch-rs/ and 06.04-cohomology/ units). ✓ = covered at ACGH-I-equivalent depth, △ = topic present but Codex unit shallower / different framing, ✗ = not covered.

Chapter I — Preliminaries

ACGH-I topic	Codex unit(s)	Status	Note
Divisor, line bundle on a curve, equivalence	`06.05.01-divisor-riemann-surface`; `06.05.02-holomorphic-line-bundle`; `04.05.01-weil-divisor`; `04.05.03-line-bundle`; `04.05.04-cartier-divisor`	✓	Shipped (some templated; Donaldson plan flags `06.05.01` for deepening).
Picard group $Pic (C)$ , $Pic^{d} (C)$ , degree map	`04.05.02-picard-group`	△	Shipped; structure $Pic^{d} ≅ Pic^{0} ≅ J (C)$ partial — Donaldson plan item 13 calls for deepening.
Linear system $	D	$, base-point-free, very ample	—
Projective embedding via $	D	$; very ample line bundles	`04.05.05-ample-line-bundle`; Donaldson plan item 19 (`06.05.04`)
Canonical embedding for $g \geq 2$ non-hyperelliptic	partial in `06.08.02-vhs-jacobian` and `04.08.02-canonical-sheaf`	△	Canonical sheaf present; the canonical map $C \to P^{g - 1}$ and its hyperelliptic-vs-non-hyperelliptic dichotomy is referenced only as commentary, not anchored.
Riemann-Roch theorem (statement)	`04.04.01-riemann-roch-curves`; `06.04.01-riemann-roch-compact-riemann-surfaces`	✓	Shipped both sides; Donaldson plan item 4 calls for proof-depth deepening on `06.04.01`.
Serre duality on curves	`06.04.04-serre-duality-curves`; `04.08.03-serre-duality`	△	Shipped both sides; depth varies (Donaldson plan item 3 covers `06.04.04`).
Hodge decomposition / period matrix	`06.06.02-period-matrix`; `06.06.07-riemann-bilinear`	✓	Shipped; `06.06.07` is at full Riemann-bilinear depth.
Hyperelliptic curve, $y^{2} = p (x)$	partial in `04.04.02-hurwitz-formula`; Donaldson plan item 18	△	Mentioned as example; no dedicated unit.

Chapter II — Determinantal varieties

ACGH-I topic	Codex unit(s)	Status	Note
Degeneracy loci of a map of vector bundles	—	✗	Gap. No dedicated unit anywhere in `04.` or `03.`.
Porteous formula / Thom-Porteous class	—	✗	Gap.
Eagon-Northcott complex	—	✗	Gap. Pointer-only acceptable.
Chern-class formalism for degeneracy loci	partial in `03.07.*` (gauge theory), `04.05.06-intersection-pairing`	△	Topological Chern classes shipped (Milnor-Stasheff anchor); the algebraic-geometry Chern-class / intersection-theory formalism on $Pic^{d}$ is not present.

Chapter III — Introduction to special divisors

ACGH-I topic	Codex unit(s)	Status	Note
Special divisor, $ℓ (K - D) > 0$	mentioned in `04.04.01-riemann-roch-curves` (Clifford prose)	△	Defined in passing in the RR unit; no dedicated unit.
Brill-Noether number $ρ (g, r, d) := g - (r + 1) (g - d + r)$	mentioned in `04.04.01`, `06.06.06`, `06.06.08`	△	Formula appears three times across the corpus but is never anchored in its own unit. Foundational symbol with no home.
$W_{d}^{r}$ , the Brill-Noether locus ${[L] \in Pic^{d} : h^{0} (L) \geq r + 1}$	mentioned in `06.06.06-jacobi-inversion`, `06.06.08-schottky-problem`	✗	Gap (high priority). No dedicated unit; the symbol is invoked by `06.06.08` (Schottky/Andreotti-Mayer) as if it were anchored.
$G_{d}^{r}$ , the variety of $g_{d}^{r}$ 's	—	✗	Gap.
Statement of Brill-Noether existence theorem	mentioned in `06.06.06-jacobi-inversion`	△	Statement present in the Jacobi-inversion unit's commentary; no dedicated theorem unit.
Statement of Brill-Noether dimension theorem (Griffiths-Harris 1980)	mentioned in `06.06.06-jacobi-inversion`, `06.06.08-schottky-problem`	△	Statement appears in commentary; the theorem itself has no anchoring unit. The Codex silently depends on this theorem via `06.06.08`'s singular-locus identification.

Chapter IV — Brill-Noether on a general curve

ACGH-I topic	Codex unit(s)	Status	Note
Brill-Noether existence theorem (Kempf 1971, Kleiman-Laksov 1972)	mentioned in `06.06.06-jacobi-inversion`	✗	Gap (high priority). Statement cited; proof / dedicated unit absent. The Porteous-class / degeneracy-locus proof is entirely absent (Ch. II gap above).
Brill-Noether dimension theorem (Griffiths-Harris 1980)	mentioned in `06.06.06`, `06.06.08`	✗	Gap (high priority). Codex's most-cited unanchored result — used silently by `06.06.08`.
Brill-Noether smoothness theorem	—	✗	Gap.
Petri map $μ_{0} : H^{0} (L) \otimes H^{0} (K \otimes L^{- 1}) \to H^{0} (K)$	—	✗	Gap. Foundational local invariant; absent.
Gieseker-Petri theorem (Gieseker 1982)	—	✗	Gap (high priority). Petri general curves; the modern proof of the Brill-Noether dimension theorem.
Lazarsfeld's vector-bundle proof (Lazarsfeld 1986)	—	✗	Gap. Master-tier deepening. K3-surface Mukai-Lazarsfeld bundles.

Chapter V — Basic results

ACGH-I topic	Codex unit(s)	Status	Note
Clifford's theorem	mentioned in `04.04.01-riemann-roch-curves`	△	Statement and hyperelliptic-equality case present in commentary; equality characterisation depth is templated.
Martens' theorem	—	✗	Gap.
Mumford's strengthening of Martens	—	✗	Gap.
Keem's theorem	—	✗	Gap (low priority — specialist).
Gonality $gon (C)$ , gonality stratification	mentioned in `06.06.08-schottky-problem` (trigonal/tetragonal Schottky)	✗	Gap. The concept is used in `06.06.08` as if anchored; no dedicated unit.
Bound $gon (C) \leq ⌈(g + 3) /2 ⌉$	—	✗	Gap. Brill-Noether corollary.
Hyperelliptic locus $H_{g} \subset M_{g}$	mentioned in `04.10.01-moduli-of-curves` (dimension $2 g - 1$ )	△	Dimension stated; structure as Brill-Noether stratum not developed.
Trigonal / $k$ -gonal loci	mentioned in `06.06.08`	✗	Gap.

Chapter VI — Geometric theory of theta

ACGH-I topic	Codex unit(s)	Status	Note
Riemann theta function on $J (C)$	`06.06.05-theta-function`	△	Shipped but v0.5 Strand C/D templated — analytic formula only, no geometric framing, no theta-divisor identification. High-priority rewrite.
Theta divisor $Θ \subset J (C)$	mentioned in `06.06.08-schottky-problem`, `06.08.02-vhs-jacobian`	△	Used by `06.06.08`'s Schottky / Andreotti-Mayer development; no dedicated unit anchoring the divisor itself.
Riemann's vanishing theorem	mentioned in `06.06.08`, Donaldson plan item 7	△	On Donaldson plan as deepening of `06.06.05` (item 7); not yet shipped.
Riemann singularity theorem $mult_{p} θ = h^{0} (L_{p})$	mentioned in `06.06.08-schottky-problem`	✗	Gap (high priority). Identifies $Sing (Θ) = W_{g - 1}^{1}$ ; the load-bearing bridge to Brill-Noether. Codex silently depends on this in `06.06.08`.
Riemann constant $κ$	—	✗	Gap.
Principal polarisation $Θ$ as a line bundle	mentioned in Donaldson plan item 20 (`06.06.09`)	△	Donaldson plan calls for `06.06.09`; not yet shipped.
Jacobi inversion via theta	`06.06.06-jacobi-inversion`	✓	Shipped at substantial depth (634 lines); proof via theta in Master tier.

Chapter VII — Existence / connectedness for $W_{d}^{r}$

ACGH-I topic	Codex unit(s)	Status	Note
Detailed proof of Brill-Noether existence (Kempf / Kleiman-Laksov via Porteous)	—	✗	Gap. Master-tier; depends on Ch. II determinantal infrastructure.
Fulton-Lazarsfeld connectedness theorem	—	✗	Gap (specialist).
Irreducibility of $W_{d}^{r}$ for $ρ \geq 1$ on Petri-general	—	✗	Gap.

Chapter VIII — Enumerative geometry

ACGH-I topic	Codex unit(s)	Status	Note
Chern classes on $W_{d}^{r}$	—	✗	Gap.
Castelnuovo number / number of $g_{d}^{r}$ 's	—	✗	Gap (specialist).
Limit linear series (preview)	—	✗	Defer to Eisenbud-Harris / Joe Harris audit.

Appendices

ACGH-I topic	Codex unit(s)	Status	Note
Universal Jacobian / universal Picard	mentioned in `06.08.02-vhs-jacobian`	△	Cited; not anchored.
Hilbert scheme of curves	—	✗	Gap. Defer to FT 3.30 (Joe Harris).
Excess linear series	—	✗	Gap (specialist).

Aggregate coverage estimate

Theorem layer: ~30% topic-level, ~15% ACGH-equivalent proof-depth. Gap concentrated in (a) the entire Chapter II determinantal infrastructure (absent), (b) the Chapter III–IV Brill-Noether-loci block (statements scattered as commentary across 06.06.06, 06.06.08, 04.04.01; no dedicated units; the Codex uses Brill-Noether $W_{d}^{r}$ in 06.06.08 without anchoring it), (c) the Chapter V Martens / Mumford / gonality block (absent), and (d) the Chapter VI Riemann-vanishing / Riemann-singularity / theta-divisor block (templated 06.06.05, no Riemann-singularity unit; bridge to Schottky / Andreotti-Mayer in 06.06.08 is load-bearing but unanchored). After priority-1: topic ~75%, proof-depth ~55%. After priority-1+2: ~92% topic-level, ~80% proof-depth.
Exercise layer: ACGH-I has long, intricate, often-multi-part exercises (especially Ch. III–V); Codex's templated 7-block is essentially zero overlap. Defer to dedicated exercise-pack pass.
Worked-example layer: ~40%. Codex covers low-genus hyperelliptic / elliptic / canonical examples; ACGH-I's running examples on Petri-general curves, trigonal curves, the Castelnuovo-Severi calculation, are absent.
Notation layer: ~70% aligned. ACGH-I writes $W_{d}^{r}$ , $G_{d}^{r}$ , $ρ (g, r, d)$ , $μ_{0}$ , $Θ$ , $κ$ , $θ$ , $C$ for the curve, $J (C)$ for the Jacobian; Codex uses $X$ / $C$ / $M$ for curves interchangeably (chapter-wide cleanup issue, not ACGH-I-specific). Recommend pinning $C$ for new ACGH-anchored units and consolidating the $ρ$ / $W_{d}^{r}$ symbols with a notation/griffiths-harris.md.
Sequencing layer: ~50%. Codex DAG has no Brill-Noether spine; the chain Petri map → Porteous → $W_{d}^{r}$ existence → dimension → Riemann singularity → theta-divisor needs to be built.
Intuition layer: ~30%. Special-divisor intuition (why Brill-Noether is the correct organising principle for curve theory) is essentially absent.
Application layer: ~40%. Gonality stratification, canonical embedding for non-hyperelliptic $g \leq 5$ , hyperelliptic / trigonal Schottky variants are referenced but not developed.

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

The Codex curves coverage is mixed: standard divisor / linear / Riemann-Roch / Abel-Jacobi / Jacobi-inversion content is present (some templated; Donaldson plan handles depth deepenings on the analytic side), but the Brill-Noether content and the Riemann-singularity / theta-divisor bridge are largely absent. The work below concentrates on closing those two blocks. Recommended slot range: 04.04.04-* and 04.04.05-* for new Brill-Noether units on the algebraic-geometry side, 06.06.10-* for the Riemann-singularity / theta-divisor / Brill-Noether-locus units on the Riemann-surfaces side, with cross-references.

Priority 1 — load-bearing Brill-Noether infrastructure and signature theorems

These items either anchor symbols already used elsewhere in the corpus ( $ρ$ , $W_{d}^{r}$ , $G_{d}^{r}$ , $μ_{0}$ , $gon$ ) or are ACGH-I's signature theorems (Brill-Noether existence, dimension, Gieseker-Petri, Riemann singularity). Without them the Brill-Noether / theta block cannot honestly claim ACGH-equivalence, and several already-shipped units (06.06.06, 06.06.08) cite unanchored results.

04.04.04 Linear system on a curve; $∣ D ∣$ , $g_{d}^{r}$ , base-point-free, very ample. ACGH-I Ch. I / Hartshorne IV.1 / Forster §16 anchors. Three-tier; ~2000 words. Foundational notation unit; defines $|D| = \mathbb{P}(H^0(\mathcal{O}(D))) $, t h e l in e a r ser i es$ g^r_d$ of dimension $r$ and degree $d$ , base-point-free / very-ample conditions. Worked examples: $g_{2}^{1}$ on a hyperelliptic curve, $g_{3}^{1}$ on a trigonal, canonical $g_{2 g - 2}^{g - 1}$ .
04.04.05 Brill-Noether number $ρ (g, r, d)$ and Brill-Noether loci $W_{d}^{r} \subset Pic^{d} (C)$ , $G_{d}^{r}$ . ACGH-I Ch. III anchor. Three-tier; ~2200 words. Highest priority — this is the unit 06.06.06, 06.06.08, and 04.04.01 silently cite. Master section: Brill-Noether number $ρ (g, r, d) := g - (r + 1) (g - d + r)$ as expected dimension; $W^r_d = {[L] \in \mathrm{Pic}^d(C) : h^0(L) \geq r + 1} $;$ G^r_d = {(L, V) : V \subset H^0(L), \dim V = r + 1} $w i t h f or g e t f u l ma p$ G^r_d \to W^r_d$; Brill-Noether matrix / Petri map. Worked examples: $W_{d}^{0}$ for small $d$ , $W_{2}^{1}$ on hyperelliptic.
04.04.06 Brill-Noether existence theorem (Kempf 1971; Kleiman-Laksov 1972). ACGH-I Ch. IV / VII anchor. Three-tier; ~1800 words. Master section: statement, the degeneracy-loci framing, sketch via Porteous formula (citing item 7 if shipped, else state Porteous as black-box). Originator prose: Severi 1915 (precursor), Kempf 1971, Kleiman-Laksov 1972.
04.04.07 Brill-Noether dimension theorem (Griffiths-Harris 1980; Gieseker 1982). ACGH-I Ch. IV anchor. Three-tier; ~2000 words. Master section: for a general curve $C \in \mathcal{M}_g $,$ \dim W^r_d = \rho $w h e n$ \rho \geq 0$ and $W_{d}^{r} = \emptyset$ when $ρ < 0$ ; smoothness of $W_{d}^{r}$ away from $W_{d}^{r + 1}$ . Originator prose: Griffiths-Harris 1980 On the variety of special linear systems on a general algebraic curve (Duke Math. J. 47, 233–272). Cross-reference Gieseker 1982 Stable curves and special divisors (Inventiones 66, 251–275) for the modern proof. The unit 06.06.06 and 06.06.08 already cite this as if anchored.
04.04.08 Petri map $μ_{0}$ and Gieseker-Petri theorem. ACGH-I Ch. IV anchor. Three-tier; ~1800 words. Master section: Petri map $\mu_0 : H^0(L) \otimes H^0(K \otimes L^{-1}) \to H^0(K)$; Petri's conjecture (Petri 1923); Gieseker 1982 proof (degeneration to nodal); Lazarsfeld 1986 Brill-Noether-Petri without degenerations (J. Differential Geom. 23, 299–307, K3-Mukai-Lazarsfeld bundles) as an alternative.
06.06.10 Riemann singularity theorem $\mathrm{mult}p \theta = h^0(L_p)$. ACGH-I Ch. VI anchor; Riemann 1857 originator. Three-tier; ~1800 words. Master section: identification $\mathrm{Sing}(\Theta) = W^1{g-1}$ (translated). The Codex load-bearer for 06.06.08's Andreotti-Mayer development. Three-tier; Master section gives the multiplicity formula via tangent-cone calculation; Intermediate gives the statement and worked $g = 4$ example.
Deepen 06.06.05-theta-function (geometric framing, theta divisor, Riemann's vanishing theorem). Replace the v0.5 Strand C/D templated unit with a substantive rewrite: theta function $\theta(z, \tau) = \sum_{n \in \mathbb{Z}^g} \exp(\pi i n^T \tau n + 2\pi i n^T z) $, t h e t a d i v i sor$ \Theta = \theta^{-1}(0)$ as Abel-Jacobi image of $Sym^{g - 1} (C)$ shifted by the Riemann constant $κ$ (Riemann's vanishing theorem), principal polarisation. Donaldson plan item 7 already calls for this deepening from the Donaldson side; ACGH-I gives the algebraic-geometry framing. Joint deepening with Donaldson item 7. No new unit ID; rewrite of Intermediate "Key theorem" and Master "Full proof" sections.

Priority 2 — Chapter V results and the determinantal-variety prelude

These items extend the priority-1 spine with consequences of Brill-Noether (Clifford / Martens / gonality) and the Chapter II algebro-geometric infrastructure (Porteous, determinantal varieties).

04.04.09 Clifford's theorem with equality. Statement + hyperelliptic-equality characterisation. ACGH-I Ch. V; Clifford 1878 originator. Three-tier; ~1500 words. Master section: full proof via $h^{0} (D) + h^{0} (K - D) \geq h^{0} (K) + 1$ exchange argument; equality case forces $D = 0$ , $D = K$ , or $C$ hyperelliptic. Currently in 04.04.01 commentary; extract to dedicated unit.
04.04.10 Martens' theorem and Mumford's strengthening. ACGH-I Ch. V; Martens 1967, Mumford 1974 originators. Three-tier; ~1700 words. Master section: $dim W_{d}^{r} \leq d - 2 r - 1$ for non-hyperelliptic $C$ (Martens) and the refinement for non-trigonal (Mumford).
04.04.11 Gonality of a curve. ACGH-I Ch. V anchor. Three-tier; ~1500 words. Master section: $\mathrm{gon}(C) := \min{d : W^1_d \neq \emptyset} $; b o u n d$ \mathrm{gon}(C) \leq \lceil (g + 3)/2 \rceil$ by Brill-Noether existence; gonality stratification of $M_{g}$ . Worked examples: hyperelliptic ( $gon = 2$ ), trigonal ( $gon = 3$ ), canonical curve cases.
04.04.12 Petri-general curve and the open Petri locus $M_{g}^{Petri}$ . ACGH-I Ch. IV / V. Three-tier; ~1500 words. The generic locus of $M_{g}$ where Petri's conjecture holds and Brill-Noether is governed by $ρ$ .
04.04.13 Determinantal varieties and the Porteous formula. ACGH-I Ch. II anchor; Porteous 1971 originator (Liverpool Singularities Symposium I). Three-tier; ~1800 words. Master section: $\mathrm{Z}_k(\varphi) := {x : \mathrm{rk}(\varphi_x) \leq k} $f or ama p o f v ec t or b u n d l es$ \varphi : E \to F$; expected codimension $(e - k) (f - k)$ ; Porteous-class formula $[Z_{k}] = Δ_{(e - k)^{f - k}} (c (F - E))$ as a Schur-polynomial determinant in Chern classes. Setup for item 3 (Brill-Noether existence proof). Or include as a Master section of item 3; do not require as standalone unit if 03.* Chern-class infrastructure remains light.
Deepen 04.04.01-riemann-roch-curves (link to Brill-Noether block). Add cross-references from the Brill-Noether prose in 04.04.01 to the new units 2-5 above. Replace the templated "Synthesis" paragraph with substantive cross-references. No new unit ID.
Deepen 06.06.08-schottky-problem (cross-reference Riemann singularity and Brill-Noether anchors). Update the 06.06.08 Andreotti-Mayer development to cite the new 06.06.10 (Riemann singularity) and 04.04.07 (Brill-Noether dimension) units. No new unit ID; bibliography + prose update.

Priority 3 — Chapter VII–VIII deepenings and Lazarsfeld

These items round out the existence-theorem proof, the connectedness theorem, and the Lazarsfeld vector-bundle alternative proof.

04.04.14 Brill-Noether existence theorem — full Porteous-class proof. Master-only deepening of item 3 once item 12 ships. ~1500 words. The detailed Kempf-Kleiman-Laksov proof on the universal Picard.
04.04.15 Fulton-Lazarsfeld connectedness theorem. ACGH-I Ch. VII; Fulton-Lazarsfeld 1981 On the connectedness of degeneracy loci and special divisors (Acta Math. 146). Master-only; ~1200 words. Statement + connectedness corollary for $W_{d}^{r}$ when $ρ \geq 1$ .
04.04.16 Lazarsfeld's K3-vector-bundle proof of Petri. Lazarsfeld 1986; Mukai 1989 Curves, K3 surfaces and Fano 3-folds of genus $\leq 10$ . Master-only; ~1500 words. The Mukai-Lazarsfeld bundle on a K3 surface containing $C$ ; Petri's conjecture as a vector-bundle stability statement.

Priority 4 — Survey / pointer items, optional

04.04.17 Survey: enumerative geometry of $W_{d}^{r}$ and $g_{d}^{r}$ 's on a general curve. Master-only; ~900 words. ACGH-I Ch. VIII pointer; Castelnuovo number, the number of $g_{d}^{r}$ 's on a Petri-general curve as a Schubert calculation. Defer unless Codex commits to the enumerative-curves track.
04.04.18 Universal Jacobian / universal Picard variety $J_{g} \to M_{g}$ . Master-only; ~1000 words. ACGH-I Appendix. Bridge to Joe Harris (FT 3.30) / Volume 2 audit. Defer.
Notation crosswalk notation/griffiths-harris.md. ~500 words. Pin $C$ as the curve, $W_{d}^{r}$ / $G_{d}^{r}$ / $ρ$ / $μ_{0}$ / $Θ$ / $κ$ symbol conventions. Worth producing if priority-1+2 batch ships; otherwise inline in unit Master sections.

§4 Implementation sketch (P3 → P4)

Minimum ACGH-I-equivalence batch = priority 1 only (items 1–7): 6 new units (04.04.04, 04.04.05, 04.04.06, 04.04.07, 04.04.08, 06.06.10) plus 1 deepening (06.06.05). Realistic production estimate (mirroring earlier Cannas / Donaldson / Lawson-Michelsohn batches):

~3 hours per typical new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose).
~4 hours for the Brill-Noether-loci unit (item 2 — large and load-bearing, multiple symbols anchored).
~3.5 hours for the Brill-Noether-dimension-theorem unit (item 4 — originator-prose for Griffiths-Harris 1980 mandatory).
~2 hours for the theta-function rewrite (item 7 — joint with Donaldson plan item 7; one production pass serves both).
Priority-1 totals: 1 large × 4 h + 4 typical × 3 h + 1 originator- heavy × 3.5 h + 1 deepening × 2 h = ~21.5 hours.
Priority-1+2 totals: priority-1 + items 8–14 × ~2.5 h average = ~21.5 + ~18 = ~40 hours.

At 3–5 production agents in parallel, priority-1 fits in a 1–2 day window with one integration agent stitching outputs. Priority-1+2 together fits a 3–4 day campaign.

Batch structure.

Batch A (Brill-Noether spine, items 1, 2, 3, 4, 5, ~13.5 h): opens new sections 04.04.04-08. Load-bearing; items 3-5 depend on items 1-2. The signature theorems; closes the most-cited unanchored block in the corpus.
Batch B (Theta / Riemann singularity, items 6, 7, ~5.5 h): joint with Donaldson plan item 7. Opens 06.06.10. Depends on Batch A item 2 (the $W_{g - 1}^{1}$ symbol).
Batch C (Chapter V consequences, items 8, 9, 10, 11, ~6 h): extends 04.04.*. Depends on Batch A. Items 8-11 are independent of each other and can run in parallel.
Batch D (Chapter II infrastructure + deepenings, items 12-14, ~6 h): opens 04.04.13 (Porteous / determinantal) if Chern-class infrastructure is judged ready; otherwise leave as inline Master section. Items 13-14 are bibliography + prose updates on existing units.
Optional Batch E (priority-3+4, items 15-20, ~10 h): after priority-1+2 lands.

Originator-prose targets (each priority-1 unit's Master section cites originator + ACGH-I):

Linear system (item 1): Brill-Noether 1873 Über die algebraischen Functionen und ihre Anwendung in der Geometrie (Math. Ann. 7, 269–310) as the originating linear-systems framework; Riemann 1857 Theorie der Abel'schen Functionen (Crelle 54, 115–155) for the divisor / linear-system pre-history.
Brill-Noether loci (item 2): Brill-Noether 1873; Severi 1915 Sulla classificazione delle curve algebriche (precursor existence claim).
Brill-Noether existence (item 3): Severi 1915 (precursor); Kempf 1971 Schubert methods with an application to algebraic curves (Stichting Math. Centrum Amsterdam) and Kleiman-Laksov 1972 On the existence of special divisors (Amer. J. Math. 94, 431–436). Originator-prose mandatory.
Brill-Noether dimension theorem (item 4): Griffiths-Harris 1980 On the variety of special linear systems on a general algebraic curve (Duke Math. J. 47, 233–272). **Phillip Griffiths
- Joseph Harris are the originators**; originator-prose mandatory. Cite Gieseker 1982 Stable curves and special divisors (Inventiones 66, 251–275) for the modern proof of Petri.
Petri map / Gieseker-Petri (item 5): Petri 1923 Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen (Math. Ann. 88); Gieseker 1982; Lazarsfeld 1986 Brill-Noether-Petri without degenerations (J. Differential Geom. 23, 299–307).
Riemann singularity theorem (item 6): Riemann 1866 Bemerkungen über die Integration der Differentialgleichungen (Crelle 65); Kempf 1973 On the geometry of a theorem of Riemann (Ann. Math. 98, 178–185) for the modern proof. Originator voice mandatory.
Theta function rewrite (item 7): Riemann 1857; Jacobi 1832/1834; Mumford Tata Lectures on Theta I-II (Birkhäuser PM 28/43); Fay 1973 Theta Functions on Riemann Surfaces (Springer LNM 352).
Clifford (item 8): Clifford 1878 On the classification of loci (Phil. Trans. Roy. Soc. 169).
Martens / Mumford (item 9): Martens 1967 Über den Dimensionssatz im Raume der speziellen Divisoren (Crelle 233); Mumford 1974 Prym varieties I (in Contributions to Analysis).
Gonality (item 10): bound via Brill-Noether 1873 + Kleiman-Laksov 1972.
Petri general (item 11): Petri 1923; Gieseker 1982.
Porteous (item 12): Porteous 1971 Simple singularities of maps (Liverpool Singularities Symposium I, Springer LNM 192); Fulton 1984 Intersection Theory §14 for the modern presentation.

Notation crosswalk. Recommend new notation/griffiths-harris.md (item 20) consolidating $W_{d}^{r}$ , $G_{d}^{r}$ , $ρ (g, r, d)$ , $μ_{0}$ , $Θ$ , $κ$ , $gon (C)$ , $g_{d}^{r}$ , $∣ D ∣$ . Pin $C$ for new ACGH-anchored units (the chapter-wide $M / X /Σ/ C$ issue Donaldson plan flags is a separate maintenance pass).

DAG edges to add. New prerequisites for the priority-1+2 batch:

04.04.04 (linear system) ← {04.05.01-weil-divisor, 04.05.03-line-bundle, 06.05.01-divisor-riemann-surface}
04.04.05 (Brill-Noether loci) ← {04.04.04, 04.05.02-picard-group, 04.04.01-riemann-roch-curves}
04.04.06 (Brill-Noether existence) ← 04.04.05
04.04.07 (Brill-Noether dimension) ← {04.04.05, 04.04.06}
04.04.08 (Petri map) ← {04.04.05, 04.04.07}
06.06.10 (Riemann singularity) ← {04.04.05, 06.06.05-theta-function, 06.06.06-jacobi-inversion}
04.04.09 (Clifford) ← 04.04.05
04.04.10 (Martens / Mumford) ← 04.04.09
04.04.11 (gonality) ← {04.04.05, 04.04.07}
04.04.12 (Petri general) ← {04.04.07, 04.04.08}
04.04.13 (Porteous) ← Chern-class infrastructure (TBD)
04.04.07 → 06.06.08-schottky-problem (close the silent dependency)
06.06.10 → 06.06.08-schottky-problem (close the silent dependency)

Joint deepening with Donaldson. Donaldson plan item 7 (Riemann's vanishing theorem, deepening of 06.06.05) overlaps with this plan's item 7 (theta function geometric rewrite). One production pass on 06.06.05 serves both equivalences. Schedule in the same campaign window as the Donaldson priority-1 batch.

Joint coverage with Hartshorne. Hartshorne IV does Riemann-Roch, Hurwitz, and curves embedded in projective space; this overlap is already shipped via 04.04.01, 04.04.02-hurwitz-formula, and the canonical-embedding deepening on the Donaldson plan punch-list. ACGH-I picks up where Hartshorne IV ends: special divisors, Brill-Noether. The priority-1 batch here does not duplicate Hartshorne — these are genuinely new units the Hartshorne audit doesn't claim.

§5 What this plan does NOT cover

Volume 2 (Arbarello-Cornalba-Griffiths 2011) — moduli of curves, second-order deformation, Witten conjecture, intersection theory on $\overline{M}_{g}$ . Defer to a separate ACG-II audit when Volume 2 enters the campaign.
Joe Harris, Moduli of Curves (FT 3.30) — Hilbert schemes of curves, Deligne-Mumford compactification, tautological ring, Kontsevich theorem, Witten's conjecture, ELSV formula. Defer to the Harris-Moduli audit. Item 19 above (universal Jacobian) bridges the two.
Limit linear series (Eisenbud-Harris 1986). ACGH-I Ch. VIII previews; the full theory belongs to a dedicated EH audit, not shipped here.
Higher rank Brill-Noether (Bertram-Feinberg, Mukai). Out of scope.
Line-number-level inventory of every theorem / exercise across ACGH-I's eight chapters. Reduced audit; defer to a full P1 pass once a PDF is acquired.
ACGH-I's extensive exercise sets (especially Ch. III–V). Defer to dedicated exercise-pack pass.
The Chapter II determinantal-variety infrastructure beyond the Porteous-formula pointer (item 12). Eagon-Northcott, full Schubert calculus on $Gr (k, n)$ , and the Macaulay-Bayer algebraic-combinatorics input are deferred to a Fulton Intersection Theory / Griffiths-Harris Principles audit.
Algebraic-curves-over-non-algebraically-closed-fields content. ACGH-I is char-0, $k = \overline{k}$ ; arithmetic Brill-Noether (Coleman, Faltings, Voloch) is out of scope.
The Schottky problem in genus $\geq 4$ beyond what 06.06.08 already ships. ACGH-I touches the Schottky-Jung framework in Ch. VI; full coverage belongs to a Mumford Tata Lectures on Theta II or Beauville-Debarre audit.
Lean formalisation of Brill-Noether existence / dimension theorems. None exist in Mathlib; Lean stubs will mark the formalisation as TBD.

§6 Acceptance criteria for FT equivalence (ACGH-I)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

≥95% of ACGH-I's named theorems in chapters I–VI map to Codex units at ACGH-I-equivalent proof depth (currently ~15%; after priority-1 this rises to ~55%; after priority-1+2 to ~82%; after priority-3 to ~92%+; full ≥95% requires priority-1+2+3 + selective priority-4).
≥80% of ACGH-I's exercises have a Codex equivalent (currently ~5%; closing this requires the dedicated ACGH-I-exercise-pack pass per §5).
≥90% of ACGH-I's worked examples (hyperelliptic, trigonal, canonical, Castelnuovo) are reproduced in some Codex unit (currently ~40%; the priority-1+2 batch's required worked-example rewrites bring this to ~75%; full ≥90% requires the Donaldson plan item 18 "standard examples" unit plus the new Brill-Noether spine).
The silent dependencies of 06.06.08-schottky-problem on Brill-Noether $W_{g - 1}^{1}$ and the Riemann singularity theorem are closed by anchoring 04.04.07 (Brill-Noether dimension) and 06.06.10 (Riemann singularity).
The 06.06.05-theta-function templated stub is replaced with the ACGH-anchored geometric-framing rewrite (joint with Donaldson plan item 7).
The notation alignment is recorded inline or via the optional notation/griffiths-harris.md (item 20).
For every chapter dependency in ACGH-I (Ch. III → Ch. IV → Ch. V; Ch. VI → Ch. VII), there is a corresponding prerequisites arrow chain in Codex's DAG.
Pass-W weaving connects the new units (04.04.04-18, 06.06.10) to the existing 04.04.01 (Riemann-Roch), 04.04.02 (Hurwitz), 04.04.03 (elliptic), 04.05.* (divisors / Picard), 06.06.* (Jacobian / Abel-Jacobi / theta / Schottky) via lateral connections.

The 7 priority-1 items close the load-bearing Brill-Noether spine and the Riemann-singularity / theta-divisor bridge. Priority-2 closes the Chapter V consequences (Clifford, Martens, gonality, Petri general) and the Chapter II algebro-geometric prelude. Priority-3 closes the existence-theorem detailed proof and Lazarsfeld's alternative. Priority-4 is depth-completion and survey pointers.

Composite ACGH-I + Donaldson + Forster batch. ACGH-I Ch. VI (theta function, Riemann vanishing, Riemann singularity), Donaldson Ch. 11 (theta function with Jacobi inversion, Riemann's bilinear relations), and Forster Ch. III–V (Čech cohomology + Riemann-Roch proof) all share the Jacobian-side theta machinery. Producing the ACGH-I priority-1 + Donaldson priority-1 batches together yields a ~13-unit composite closing the Brill-Noether spine, theta-divisor bridge, and the Donaldson PDE / cohomology infrastructure simultaneously.

Honest scope. Mixed equivalence gap: curve basics shipped across 04.04-*, 04.05-*, 06.04-*, 06.05-*, 06.06-*; Brill-Noether content almost entirely absent despite being silently cited by 06.06.06, 06.06.08, 04.04.01, and 04.10.01. Work concentrated in two new sub-chapters: 04.04.04-18 (Brill-Noether spine + Chapter V consequences + Chapter II prelude) and 06.06.10 plus deepening of 06.06.05 (Riemann singularity / theta-divisor bridge). No infrastructure chapter (Chern-class / intersection-theory on Picard variety) is mandatory — Porteous can be inline.

Largest single ACGH-I-distinctive gap: the Brill-Noether spine (items 1–5: linear systems, $ρ$ , $W_{d}^{r}$ , existence, dimension, Petri). Without these, the algebraic-curves section of Codex carries unanchored symbols that already appear in shipped units. Closing this is the highest-leverage curve-theory audit deliverable available in the campaign window.

Unusual finding. 06.06.08-schottky-problem (743 lines, mature Strand A) depends on the Riemann singularity theorem and the Brill-Noether dimension theorem as if they were anchored — multiple in-text statements (" $dim Sing (Θ) = g - 4$ generically by Griffiths-Harris 1980, cf. [06.06.06]") cite a result that is not in any dedicated unit. 06.06.06-jacobi-inversion similarly states the Brill-Noether existence theorem and the dimension theorem in its Master commentary, citing Griffiths-Harris 1980 and Gieseker 1982, but the theorems themselves have no homes. The Codex therefore has a load-bearing citation chain that resolves to no theorem unit — a documentation-integrity defect surfaced by this audit. Closing it via priority-1 items 2, 4, and 6 is the cleanest available fix.

§7 Sourcing

No free PDF. ACGH-I is a Springer Grundlehren volume in active commercial print; no author-hosted copy, no open-access Springer release. Anna's-Archive availability fluctuates but is not relied on for this audit.
No local copy. Not present in reference/textbooks-extra/ or reference/fasttrack-texts/. Acquisition required for a full P1 line-number-level pass; the present audit is REDUCED and works from chapter structure plus the citations already embedded in the corpus.
License. Springer commercial. Cite as Arbarello, Cornalba, Griffiths, Harris, Geometry of Algebraic Curves, Volume 1, Grundlehren der mathematischen Wissenschaften 267, Springer-Verlag 1985.
Peer sources used for this reduced audit.
- Hartshorne, Algebraic Geometry (Springer GTM 52, 1977), Chapter IV — Riemann-Roch / curves; FT 3.21 sibling. plans/fasttrack/hartshorne-algebraic-geometry.md.
- Forster, Lectures on Riemann Surfaces (Springer GTM 81, 1981), §§16–30 — Riemann-Roch, Jacobi inversion, Abel. plans/fasttrack/forster-riemann-surfaces.md.
- Donaldson, Riemann Surfaces (OUP OGTM 22, 2011), Ch. 8–11 — sheaf cohomology, Serre duality, theta function, Jacobi inversion, bilinear relations. FT 1.07. plans/fasttrack/donaldson-riemann-surfaces.md.
- Mumford, Curves and Their Jacobians (Univ. of Michigan Press, 1975, reissued as the second half of The Red Book) — Jacobian-as-Picard, theta divisor, Schottky pointer. Originator citations for many Brill-Noether and theta results.
- Voisin, Hodge Theory and Complex Algebraic Geometry I (Cambridge SAM 76, 2002), Ch. 7 — Abel-Jacobi, period matrix, Hodge decomposition for curves. FT-adjacent; already audited (plans/fasttrack/voisin-hodge-theory-volume-1.md).
- Mumford, Tata Lectures on Theta I-II (Birkhäuser PM 28/43, 1983/1984) — theta function with characteristics, Schottky-Jung, Riemann's vanishing. Originator-citation anchor for theta-side units.
- Eisenbud-Harris, 3264 and All That (CUP 2016) §14 — determinantal varieties / Porteous formula. Modern alternative to Fulton Intersection Theory for item 12.
- Fulton-Lazarsfeld 1981, On the connectedness of degeneracy loci and special divisors (Acta Math. 146) — connectedness theorem originator. Item 16.
- Griffiths-Harris 1980, On the variety of special linear systems on a general algebraic curve (Duke Math. J. 47, 233–272) — Brill-Noether dimension theorem originator paper; Phillip Griffiths and Joseph Harris are the originators of the result the book formalises. Item 4 originator-prose.
- Gieseker 1982, Stable curves and special divisors (Inventiones 66, 251–275) — modern proof of Petri's conjecture / Brill-Noether-Petri theorem. Item 5 originator-prose alternative.
- Lazarsfeld 1986, Brill-Noether-Petri without degenerations (J. Differential Geom. 23, 299–307) — K3-Mukai-Lazarsfeld bundle proof of Petri. Item 17.
- Kempf 1971, Schubert methods with an application to algebraic curves (Math. Centrum Amsterdam preprint) and Kleiman-Laksov 1972 On the existence of special divisors (Amer. J. Math. 94, 431–436) — Brill-Noether existence originator papers. Item 3.
- Riemann 1857/1866 Theorie der Abel'schen Functionen / Bemerkungen über die Integration — originator-citation for Riemann-Roch (in its original Riemann-only form), theta function, Riemann singularity, Riemann's vanishing.

Recommendation: acquire a Springer-licensed copy of ACGH-I before the priority-1 batch enters production; the chapter-section correspondence in §2 above is from secondary literature and may mis-number specific results. A full P1 line-number pass should follow acquisition.

Arbarello, Cornalba, Griffiths, Harris — *Geometry of Algebraic Curves, Volume 1* (Fast Track 3.22) — Audit + Gap Plan