Hartshorne — Algebraic Geometry (Fast Track reference anchor) — Audit + Gap Plan

Book: Robin Hartshorne, Algebraic Geometry (Springer GTM 52, 1977). The canonical English-language graduate textbook for scheme-theoretic algebraic geometry; the de-facto syllabus that every PhD student in the subject is expected to have worked through (or, more honestly, fought with) by the end of their second year.

Fast Track entry: not numbered — Hartshorne sits in the Fast Track booklist as a peer reference anchor alongside Vakil (The Rising Sea), Eisenbud-Harris (The Geometry of Schemes), and Mumford (The Red Book). It is cited by 23 of the 27 algebraic-geometry units in Codex (and by several outside the AG section — Chern-Weil, spin structures, sphere bundles all cross-reference Hartshorne) but has never been audited for equivalence coverage. Treated here as a peer-anchor parallel to Hatcher, with a numbered slot deferred until the booklist is revised.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units so that Hartshorne is covered to the equivalence threshold (≥95%; see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Mirrors the Hatcher plan in structure. Not a full P1 audit (which would inventory every theorem and exercise across ~470 pages and ~600 problems at line level); works from chapter structure and editorial choices.

§1 What Hartshorne's book is for

Hartshorne is the Anglophone graduate textbook on scheme-theoretic algebraic geometry. Vakil's Rising Sea is "foundations from scratch with motivation everywhere"; Eisenbud-Harris stays intuitive and classical; Hartshorne is terse, theorem-driven, uncompromising — the book that codified scheme theory + sheaf cohomology + Riemann-Roch

curves + surfaces as a single graduate package. The source working algebraic geometers cite when they say "in Hartshorne III.5".

Five chapters, one classical-varieties prelude, three appendices.

Chapter I: Varieties. Classical algebraic geometry over an algebraically closed field $k$ . Affine and projective varieties, Nullstellensatz, Zariski topology, regular and rational maps, dimension, nonsingular varieties, Zariski tangent space, intersection in projective space, blow-up on a surface. This chapter is deliberately lightweight — a ~50-page motivation chapter so the student knows what schemes generalise. Not load-bearing; a runway.

Chapter II: Schemes. The technical heart of the book. Sheaves (§II.1), schemes (§II.2), first properties — locally Noetherian, irreducible, reduced, integral (§II.3), separated and proper morphisms (§II.4), quasi-coherent and coherent sheaves (§II.5), divisors — Weil and Cartier, the Picard group (§II.6), projective morphisms — Proj, twisting sheaves $O (n)$ , ample/very-ample (§II.7), differentials — Kähler differentials, the sheaf of differentials, smoothness, cotangent and canonical sheaves (§II.8), formal schemes (§II.9, often skipped).

Chapter III: Cohomology. The chapter where most students actually live. Derived functors (§III.1), cohomology of sheaves (§III.2), Serre's vanishing on a Noetherian affine scheme (§III.3), Čech cohomology (§III.4), cohomology of projective space — Serre's computation of $H^i(\mathbb{P}^n, \mathcal{O}(d))$ (§III.5), Ext groups and sheaves (§III.6), Serre duality (§III.7), higher direct images (§III.8), flat morphisms (§III.9), smooth morphisms (§III.10), formal functions (§III.11), semicontinuity (§III.12). Sections §III.1–§III.5 are core; §III.6–§III.12 are advanced and selectively skipped.

Chapter IV: Curves. Smooth projective curves over algebraically closed $k$ . Riemann-Roch (§IV.1), Hurwitz (§IV.2), embeddings in projective space (§IV.3), elliptic curves (§IV.4), canonical embedding (§IV.5), classification in $P^{3}$ (§IV.6). Every section is a long worked example.

Chapter V: Surfaces. Smooth projective surfaces. Intersection pairing, adjunction formula, Riemann-Roch for surfaces (§V.1), ruled surfaces (§V.2), monoidal transformations / blow-ups (§V.3), 27 lines on the cubic (§V.4), Castelnuovo's contractibility (§V.5), Enriques-Kodaira classification sketch (§V.6). §V.1 is foundational; the rest is selective.

Appendices. Survey-only — A: general intersection theory / Chow. B: GAGA, Hodge, Kähler. C: étale cohomology, Weil conjectures.

Distinctive Hartshorne choices:

Schemes first, varieties as motivation. The book's most famous editorial choice. Mumford's Red Book takes the opposite approach; Vakil splits the difference; Eisenbud-Harris stays mostly classical. Hartshorne's "schemes first" is now the modern default.
Sheaves before schemes. §II.1 develops sheaves before a single scheme appears. Schemes then become a one-paragraph construction (locally ringed space + $Spec$ ).
Coherent sheaves are the central object. §II.5 + §III throughout: "algebraic geometry = the cohomology of coherent sheaves." Serre's vanishing (affine) and finiteness (projective) anchor the chapter.
Projective space as the universe. §II.7 + §III.5: the computation of $H^{i} (P^{n}, O (d))$ via Čech is the single most-cited calculation in the book. Chapters IV and V route through it.
Čech for computation, derived functors for the definition. §III.1–§III.2 give derived functors (so Serre duality is cleanly stateable); §III.4 gives Čech (for computation); Hartshorne proves agreement in the cases that matter (Noetherian separated, acyclic covers).
Serre duality as the central duality. §III.7. The clean form $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \omega_X \otimes \mathcal{F}^\vee)^\vee $f or s m oo t h p r o j ec t i v e$ X/k$. Riemann-Roch (Ch IV) and adjunction (Ch V) are corollaries.
Riemann-Roch worked, not stated. §IV.1 states it; the rest of Chapter IV applies it. Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch are deferred to Appendix A.
Intersection on surfaces, not in general. §V.1: pairing, adjunction, Hodge index. Higher-dimensional intersection theory is Appendix A only.
Exercises as theorems. ~600 exercises, many non-trivial results used in later text without proof. The exercises are how one learns the material — they are not optional.
Notation: scheme-theoretic. $O_{X}$ , $F$ , $Spec A$ , $Proj S$ , $O_{X} (D)$ , $Ω_{X / Y}$ , $ω_{X}$ — the conventions every subsequent textbook now uses.

§2 Coverage table (Codex vs Hartshorne)

Cross-referenced against the current 27-unit 04-algebraic-geometry/ section. ✓ = covered, △ = partial / different framing or depth, ✗ = not covered. Bott-Tu-overlap flags items where the gap also appears in the Bott-Tu plan punch-list (sheaf cohomology, Čech, etc).

Chapter I — Varieties

Hartshorne topic	Codex unit(s)	Status	Note
Affine varieties, Zariski topology	—	✗	Gap. No standalone unit on classical affine varieties. Codex jumps directly to schemes via `04.02.01`.
Projective varieties	`04.07.01` projective-space	△	Projective space exists; "projective variety" as a classical-geometry object does not. Hartshorne §I.2.
Nullstellensatz	—	✗	Gap (high priority — foundational). Hartshorne §I.1 anchor. The classical bridge from algebra to geometry. Used implicitly throughout `04.02.*`.
Regular and rational maps	`04.02.04` morphism-of-schemes	△	Covered scheme-theoretically; classical regular/rational map (rational function field) not explicitly introduced.
Dimension, Krull dimension	—	✗	Gap. Hartshorne §I.1, §I.7. Used silently in `04.02.`, `04.05.`.
Nonsingular varieties, Zariski tangent space	`04.08.01` sheaf-of-differentials (uses)	△	The cotangent module is in the differentials unit; the Zariski tangent space at a point and the Jacobian criterion for nonsingularity are not their own unit. Hartshorne §I.5.
Blowing up a point on a surface	`04.07.02` blowup	✓	Codex has the blowup unit (Cannas-aligned framing). Hartshorne §I.4 + §V.3 reference.

Chapter II — Schemes

Hartshorne topic	Codex unit(s)	Status	Note
Presheaves, sheaves	`04.01.01` sheaf	✓	Covered. Hartshorne §II.1 reference.
Stalks	`04.01.02` stalk	✓	Covered. Hartshorne §II.1 reference.
Sheafification	`04.01.03` sheafification	✓	Covered. Hartshorne §II.1 reference.
Direct image, inverse image	`04.01.04` direct-inverse-image	✓	Covered. Hartshorne §II.1 reference.
Kernel, cokernel, exact sequences of sheaves	`04.01.01`, `04.01.03` (uses)	△	Used in the sheaf and sheafification units; no standalone treatment of the abelian category $Sh (X)$ .
$Spec A$ as a topological space	`04.02.02` affine-scheme	△	The affine scheme unit covers $Spec$ as the underlying space + structure sheaf together; the construction is present but the topological-space-only construction (prime spectrum with Zariski topology) is folded in. Hartshorne §II.2 reference. Adequate.
Affine scheme, structure sheaf	`04.02.02` affine-scheme	✓	Covered.
Scheme (general)	`04.02.01` scheme	✓	Covered.
Morphism of schemes	`04.02.04` morphism-of-schemes	✓	Covered.
Locally Noetherian, integral, reduced, irreducible	—	✗	Gap. Hartshorne §II.3. The basic adjectives on schemes (locally Noetherian, integral, etc.) are used but not catalogued.
Separated and proper morphisms	`04.02.04` (Master, mention)	△	Mentioned in the morphism-of-schemes unit; not a standalone unit. Hartshorne §II.4 is the canonical reference. Gap (medium priority).
Quasi-coherent sheaves	`04.06.01` quasi-coherent-sheaf	✓	Covered. Hartshorne §II.5.
Coherent sheaves	`04.06.02` coherent-sheaf	✓	Covered. Hartshorne §II.5.
Sheaf $M$ from a module	`04.06.01` (uses)	△	The construction $M \mapsto M$ on $Spec A$ is in the quasi-coherent-sheaf unit. Adequate.
Projective scheme, $Proj S$	`04.02.03` projective-scheme	✓	Covered. Hartshorne §II.2, §II.7 reference.
Twisting sheaf $O (n)$ on $P^{n}$	`04.05.05` ample-line-bundle (uses); `04.07.01` (uses)	△	Gap (high priority). Used in the ample-line-bundle and projective-space units, not its own unit. Hartshorne §II.5, §II.7 are core references. The very-ample / ample correspondence with closed embeddings is also missing as a standalone item.
Ample, very ample line bundles	`04.05.05` ample-line-bundle	✓	Covered. Hartshorne §II.7 reference.
Weil divisors	`04.05.01` weil-divisor	✓	Covered. Hartshorne §II.6.
Cartier divisors	`04.05.04` cartier-divisor	✓	Covered. Hartshorne §II.6.
Picard group	`04.05.02` picard-group	✓	Covered. Hartshorne §II.6. Hartshorne's depth on the divisor-line-bundle correspondence + the Cartier-Weil agreement on locally factorial schemes is a deepening opportunity, not a gap.
Line bundle, invertible sheaf	`04.05.03` line-bundle	✓	Covered.
Sheaf of differentials, $Ω_{X / Y}$	`04.08.01` sheaf-of-differentials	✓	Covered. Hartshorne §II.8 reference.
Canonical sheaf $ω_{X}$	`04.08.02` canonical-sheaf	✓	Covered. Hartshorne §II.8 reference.
Smooth, étale, unramified morphisms	—	✗	Gap (high priority). Hartshorne §III.10 (smoothness via the differentials), Appendix C (étale). No Codex unit exists. Used implicitly in spin-geometry and moduli units.
Kähler differentials (commutative-algebra side)	`04.08.01` (uses)	△	The module-theoretic Kähler differentials $Ω_{B / A}$ is in the sheaf-of-differentials unit; not its own algebra unit. Adequate for the AG section; could deepen via a `03-01-*` (commutative algebra) cross-link.
Formal schemes	—	✗	Hartshorne §II.9. Specialty — defer to Tier-γ. Not load-bearing for the rest of the curriculum.

Chapter III — Cohomology

Hartshorne topic	Codex unit(s)	Status	Note
Derived functors	—	✗	Gap (medium priority). Hartshorne §III.1. The general derived-functor formalism is used in `04.03.01` but not standalone. (Cross-section with category theory: belongs in `01-foundations/05-category-theory/` as a homological-algebra unit.)
Sheaf cohomology (definition)	`04.03.01` sheaf-cohomology	✓	Covered. Hartshorne §III.2 reference.
Cohomology of an affine Noetherian scheme — Serre's vanishing	`04.06.01` (uses); `04.03.01` (mention)	△	Gap (high priority). Theorem $H^{i} (Spec A, M) = 0$ for $i > 0$ on Noetherian affine schemes is the Serre vanishing theorem (not Kodaira vanishing). Currently mentioned but no standalone unit. Hartshorne §III.3.
Čech cohomology of a scheme	`03.04.11` cech-de-rham (analogue)	△	Gap (high priority). The de Rham version exists in the Bott-Tu strand. The scheme-theoretic Čech cohomology, with the comparison theorem to derived-functor cohomology on Noetherian separated schemes, is missing. Hartshorne §III.4.
Cohomology of projective space — $H^{i} (P^{n}, O (d))$	`04.03.01` (mention); `04.07.01` (mention)	△	Gap (high priority — load-bearing). The foundational computation of algebraic geometry. Currently scattered as mentions; no dedicated unit walks through the calculation. Hartshorne §III.5 is the canonical worked reference.
Serre's finiteness theorem (cohomology of coherent sheaves on projective schemes)	—	✗	Gap (medium priority). Hartshorne §III.5, Theorem 5.2. Not its own unit. Pairs with Serre vanishing as the FAC ("Faisceaux algébriques cohérents", Serre 1955) anchor pair.
Ext groups, Ext sheaves	—	✗	Gap. Hartshorne §III.6. Used in Serre duality without being introduced.
Serre duality	`04.08.03` serre-duality	✓	Covered. Hartshorne §III.7 reference.
Higher direct images $R^{i} f_{*}$	—	✗	Hartshorne §III.8. Specialty — used in advanced moduli / cohomology and base change. Gap (medium priority).
Flat morphisms	—	✗	Hartshorne §III.9. Gap (medium priority). Used silently in moduli theory.
Smooth morphisms (cohomological side)	—	✗	Hartshorne §III.10. Same gap as the geometric-side smooth-morphisms entry above.
Theorem on formal functions	—	✗	Hartshorne §III.11. Specialty — defer to Tier-γ.
Semicontinuity theorem	—	✗	Hartshorne §III.12. Specialty — defer to Tier-γ. Used in moduli theory.

Chapter IV — Curves

Hartshorne topic	Codex unit(s)	Status	Note
Riemann-Roch theorem for curves	`04.04.01` riemann-roch-curves	✓	Covered. Hartshorne §IV.1. Hartshorne's depth — the cohomology proof via Serre duality, the genus formula, the application to embeddings — is a deepening opportunity.
Hurwitz's theorem (ramification, genus formula for covers)	—	✗	Gap (medium priority). Hartshorne §IV.2 anchor. Foundational for curve theory; used implicitly in the moduli-of-curves unit.
Embeddings in projective space (curves)	`04.05.05` (uses)	△	Covered as part of the ample-line-bundle unit; not its own unit on the embedding criterion. Hartshorne §IV.3.
Elliptic curves	—	✗	Gap (medium priority). Hartshorne §IV.4 anchor. Codex has nothing dedicated to elliptic curves — a glaring absence given how central they are to number theory, complex geometry, and the moduli-of-curves story.
Canonical embedding, canonical curve	`04.08.02` (uses)	△	The canonical sheaf is covered; the canonical-embedding theorem (degree $2 g - 2$ embedding into $P^{g - 1}$ for non-hyperelliptic $g \geq 3$ ) is not. Hartshorne §IV.5.
Classification of curves in $P^{3}$	—	✗	Hartshorne §IV.6. Specialty — defer.

Chapter V — Surfaces

Hartshorne topic	Codex unit(s)	Status	Note
Intersection pairing on a surface, $D \cdot D^{'}$	—	✗	Gap (high priority — load-bearing). Hartshorne §V.1 anchor. The intersection number on a smooth projective surface; the pairing $Pic (X) \times Pic (X) \to Z$ . Used implicitly in `04.07.02` blowup, `04.05.02` picard-group, and the spin-geometry strand.
Adjunction formula, $K_C = (K_X + C)\big	_C$	`04.08.02` (mention)	△
Riemann-Roch theorem for surfaces	—	✗	Gap (medium priority). Hartshorne §V.1 anchor. The surface form: $χ (L) = χ (O_{X}) + \frac{1}{2} (D \cdot (D - K_{X}))$ .
Hodge index theorem	—	✗	Gap (medium priority). Hartshorne §V.1 anchor. The signature $(1, h^{1, 1} - 1)$ on $NS (X) \otimes R$ for a smooth projective surface.
Ruled surfaces	—	✗	Hartshorne §V.2. Specialty — defer.
Monoidal transformations (blow-up of a point on a surface)	`04.07.02` blowup	✓	Covered. Hartshorne §V.3 reference.
27 lines on the cubic surface	—	✗	Hartshorne §V.4. Specialty (but a famous example) — defer to Tier-γ worked-examples pack.
Castelnuovo's contractibility criterion	—	✗	Hartshorne §V.5. Specialty — defer.
Enriques-Kodaira classification (sketch)	—	✗	Hartshorne §V.6. Specialty — defer.

Appendices A, B, C

Hartshorne topic	Codex unit(s)	Status	Note
Chow groups, intersection theory (general)	—	✗	Hartshorne Appendix A. Survey-level in Hartshorne; defer. Tier-γ if at all.
GAGA	—	✗	Hartshorne Appendix B. Specialty — defer. Bridges algebraic and complex-analytic geometry.
Hodge theory, Kähler manifolds	`04.09.01` hodge-decomposition	△	Codex has the Hodge-decomposition unit (complex-geometry framing). Hartshorne Appendix B. Adequate for the AG section.
Étale cohomology, Weil conjectures	—	✗	Hartshorne Appendix C. Defer to Tier-γ. Étale-morphisms entry above is the prerequisite.

Aggregate coverage estimate

Theorem layer: ~50% map cleanly; ~25% partial or used implicitly (Serre vanishing, projective-space cohomology, intersection pairing, adjunction, Hurwitz); ~25% absent (Nullstellensatz, dimension theory, étale/smooth/flat morphisms, surfaces theory, elliptic curves, derived functors, Ext). After the priority-1+2 punch-list, ~88%.

Exercise layer: <1%. Codex AG section has zero exercise packs. ~600 Hartshorne exercises — the second-largest exercise-layer gap on the Fast Track after Hatcher.

Worked-example layer: ~35%. The cohomology-of-projective-space computation (§III.5.1) is the single most-cited example in the book and currently lives only as a mention.

Notation layer: ~80% aligned. Codex uses Hartshorne's conventions directly. Gaps: $Cl (X)$ vs $Pic (X)$ distinction, $Ω_{X / k}$ versus $Ω_{X}$ implicit-base, closed-immersion $i : Z ↪ X$ notation. A notation/hartshorne.md crosswalk is warranted.

Sequencing layer: ~85%. Codex DAG follows Hartshorne's flow (sheaves → schemes → cohomology → divisors → differentials → Riemann-Roch). Hartshorne's "schemes-first" choice is mirrored: Codex has no classical varieties chapter.

Intuition layer: ~65%. Codex captures the scheme + cohomology framings cleanly; the gap is in the density of Hartshorne's worked calculations, not in conceptual reach.

Application layer: ~70%. Riemann-Roch and spin-geometry strands fold in Hartshorne's applications; the surface-theory applications (intersection, adjunction, Hodge index) are absent.

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

The Cannas plan punch-list (blow-up, GIT, moduli) closed three units in the algebraic-geometry section: 04.07.02 blowup, 04.10.02 git, 04.10.01 moduli-of-curves. The Bott-Tu plan punch-list closed the de-Rham-side Čech (03.04.11) but not the scheme-side. The remaining Hartshorne gaps are therefore largely new — they are not duplicated elsewhere in the existing Fast Track audits.

Hartshorne-distinctive priority 1 — fills load-bearing scheme + cohomology gaps that the rest of the AG curriculum currently uses silently:

04.03.03 Čech cohomology of schemes. Hartshorne §III.4 anchor. The scheme-theoretic Čech complex on an open cover, the comparison theorem to derived-functor cohomology (Noetherian separated case), the Leray theorem for acyclic covers. Three-tier; ~1500 words. Couples to 03.04.11 cech-de-rham via a §Bridge cross-link. Needed for the priority-1 item 2 (projective-space cohomology) below.
04.03.04 Cohomology of projective space — $H^i(\mathbb{P}^n, \mathcal{O}(d))$. Hartshorne §III.5 anchor. The canonical computation of algebraic geometry. Master section gives the full Čech calculation; intermediate gives the dimension formulas; beginner gives the genus-of-curves application. Three-tier; ~2000 words. Load-bearing for Riemann-Roch deepening (item 11 below) and the surface-theory units.
04.03.05 Serre's vanishing and finiteness theorems. Hartshorne §III.3, §III.5 anchors. FAC (Serre 1955) as originator-text. The pair: vanishing $H^{i} (Spec A, M) = 0$ for $i > 0$ and Noetherian $A$ ; finiteness $dim_{k} H^{i} (X, F) < \infty$ for $X$ projective and $F$ coherent. Three-tier; ~1500 words. Originator-prose: Serre's 1955 framing, the FAC paper as foundational text.
04.02.05 Smooth, étale, and unramified morphisms. Hartshorne §III.10 + Appendix C anchors; SGA 1 as originator-text reference. The Jacobian criterion, the differential characterisation ( $Ω_{X / Y}$ locally free for smooth, zero for étale), the formal-smoothness lifting characterisation. Three-tier; ~2000 words. Load-bearing — every reference to "smooth scheme" in the spin- geometry strand and the moduli units currently uses the word without a Codex anchor.
04.07.03 Twisting sheaves $O (n)$ on projective space and the very-ample / closed-embedding correspondence. Hartshorne §II.5, §II.7 anchors. The Serre twists, the global sections $H^{0} (P^{n}, O (d)) =$ degree- $d$ homogeneous polynomials, the line-bundle / closed-embedding equivalence. Three-tier; ~1500 words. Couples to 04.05.05 ample-line-bundle as the concrete projective-space case.

Hartshorne-distinctive priority 2 — fills surface-theory + curve-theory gaps:

04.05.06 Intersection pairing on surfaces. Hartshorne §V.1 anchor. The pairing $\mathrm{Pic}(X) \times \mathrm{Pic}(X) \to \mathbb{Z}$ on a smooth projective surface, intersection multiplicity at a point, the projection formula, intersection of a curve with itself via blow-up. Three-tier; ~1800 words. Load-bearing for adjunction (item 7) and the spin-geometry / Riemann-surfaces strands.
04.05.07 Adjunction formula. Hartshorne §V.1 anchor. $K_{C} = (K_{X} + C)_{C}$ for a smooth curve $C$ on a smooth surface $X$ ; equivalent statement via canonical sheaves. Three-tier; ~1200 words. Closes the gap currently held only as a §V.1 mention in 04.08.02 canonical-sheaf.
04.05.08 Riemann-Roch theorem for surfaces. Hartshorne §V.1 anchor. $\chi(\mathcal{L}) = \chi(\mathcal{O}_X) + \tfrac{1}{2}D \cdot (D - K_X) $f or a d i v i sor$ D$ on a smooth projective surface. Three-tier; ~1500 words. Couples to the curves-Riemann-Roch unit and the eventual Hirzebruch-Riemann-Roch deepening.
04.05.09 Hodge index theorem. Hartshorne §V.1 anchor. The signature $(1, ρ - 1)$ on the Néron-Severi group of a smooth projective surface (where $ρ$ is the Picard number). Three-tier; ~1200 words. Foundational for the spin-geometry strand (Atiyah-Singer on 4-manifolds) and for Hodge theory.
04.04.02 Hurwitz theorem and ramification. Hartshorne §IV.2 anchor. Riemann-Hurwitz formula $2g_X - 2 = n(2g_Y - 2) + \sum_P (e_P - 1) $f or a f ini t e m or p hi s m$ f: X \to Y$ of smooth projective curves. Three-tier; ~1200 words.
04.04.03 Elliptic curves. Hartshorne §IV.4 anchor; Silverman Arithmetic of Elliptic Curves as cross-anchor. Smooth projective curves of genus 1 with a marked point, Weierstrass form, group law, $j$ -invariant, the moduli space $M_{1, 1}$ . Three-tier; ~2500 words. A glaring absence in the current Codex. Used as motivation in 04.10.01 moduli-of-curves and in number-theory cross-references; deserves a dedicated unit.

Hartshorne-distinctive priority 3 — Hartshorne-depth deepenings on existing units, plus medium-priority foundations:

Deepening of 04.04.01 riemann-roch-curves: add a Master section on the cohomology proof via Serre duality (Hartshorne §IV.1 actual proof, not the analytical sketch). The current unit is solid on the statement and the genus-formula corollary; the modern proof (line bundles, $H^{0}$ / $H^{1}$ via Serre duality, Riemann-Roch as $χ (L) = de g L + 1 - g$ ) is the deepening.
Deepening of 04.05.02 picard-group: add a Master section on the Cartier-Weil agreement on locally factorial schemes (Hartshorne §II.6). The current unit treats Pic abstractly; the explicit "on a smooth variety, Weil and Cartier give the same group" statement is a one-paragraph §III deepening.
04.02.06 Properties of schemes — Noetherian, integral, reduced, irreducible, separated, proper. Hartshorne §II.3, §II.4 anchors. A consolidation unit covering the basic adjectives in one place (currently scattered across morphism-of-schemes and used silently elsewhere). Three-tier; ~1500 words.
04.02.07 Nullstellensatz and dimension theory. Hartshorne §I.1, §I.7 anchors. The bridge from commutative algebra to classical geometry; Krull dimension; transcendence-degree characterisation for varieties over $k$ . Three-tier; ~1200 words. Useful as a Chapter-I-style runway for classical-variety intuition.
Deepening of 04.08.01 sheaf-of-differentials: add a Master section on the Jacobian criterion for nonsingularity at a point (Zariski tangent space + maximal-ideal cotangent space agreement). Hartshorne §I.5 / §II.8.

Hartshorne-distinctive priority 4 — survey-level pointers, exercise pack, optional:

04.03.06 Derived functors and Ext. Hartshorne §III.1, §III.6. A consolidation unit on the homological-algebra side: derived functors, Ext groups, Ext sheaves. Better placed in 01-foundations/05-category-theory/ as a homological-algebra cross-section unit; noted here as a pointer.
04.03.07 Higher direct images and base change. Hartshorne §III.8, §III.9. Survey-level Master-only unit, ~1000 words. Specialty — defer.
04.04.E1 Curves exercise pack — Hartshorne Chapter IV exercises as graded problems. ~30 problems across Riemann-Roch applications, Hurwitz computations, elliptic-curve $j$ -invariants, canonical embeddings.
04.05.E1 Divisors and surfaces exercise pack — Hartshorne Chapter II §6 + Chapter V §1 exercises. ~25 problems on Cartier-Weil, intersection numbers, adjunction.
04.03.E1 Cohomology exercise pack — Hartshorne Chapter III exercises. ~30 problems. The cohomology-of-projective-space computation, Čech-vs-derived comparison, Serre duality applications.

§4 Implementation sketch (P3 → P4)

For a full Hartshorne coverage pass, items 1–11 are the minimum priority-1+2 set (11 new units). Production estimate (mirroring earlier batches):

~3 hours per unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis paragraphs in real prose, not the templated form).
11 priority-1+2 units × 3 hours = ~33 hours of focused production. At 4–6 production agents in parallel, this fits in a 4-day window with an integration agent stitching outputs.
Priority-3 deepenings (items 12–16) = ~12 additional hours.
Priority-4 exercise packs (items 19–21) = ~24 additional hours (each pack is ~8 hours by Lawson-Michelsohn batch-2 calibration).

Total Hartshorne campaign: ~70 hours of focused production work, spanning ~6 days in parallel-agent mode. This is comparable to the Bott-Tu campaign (14 new units, ~121 hours estimated) and larger than the Lawson-Michelsohn campaign (12 new units, ~104 hours).

Composite batching with Vakil and Eisenbud-Harris. A composite Hartshorne+Vakil batch could close both books' equivalence gaps simultaneously — particularly on items 1, 2, 3 (Čech, projective-space cohomology, Serre vanishing/finiteness), where all three books anchor. Defer to a follow-up Vakil overlay audit; this plan covers Hartshorne in isolation.

Originator-prose targets. Hartshorne is textbook synthesis, not originator-text. Citations for priority-1+2 units:

Schemes, $Spec$ , $Proj$ : Grothendieck-Dieudonné EGA I-IV (1960–1967).
Sheaf cohomology, Serre vanishing/finiteness, twisting sheaves: Serre 1955, Faisceaux algébriques cohérents (Annals of Math).
Étale morphisms: SGA 1 (1960–61); SGA 4 for étale cohomology.
Surfaces intersection theory: Zariski 1958 (Publ. Math. Soc. Japan).
Riemann-Roch: Riemann 1857, Roch 1865; modern form Hirzebruch 1956.
Hodge index: Hodge 1937.
Adjunction, Hurwitz: classical (Riemann-Hurwitz 1857; adjunction traces to Picard 1897).

Each priority-1+2 unit's Master section cites the originator paper in addition to Hartshorne.

Notation crosswalk. Distinctions to record in a notation/hartshorne.md crosswalk:

$Cl (X) ≅ Pic (X)$ on locally factorial schemes but NOT in general.
$Ω_{X / k}$ versus $Ω_{X}$ — Hartshorne writes $Ω_{X}$ when the base is an implicit algebraically closed field. Codex should be explicit.
Closed immersion $i : Z ↪ X$ with conormal sheaf $N_{Z / X}^{\lor} = i^{*} I_{Z} / I_{Z}^{2}$ — this notation flows into the adjunction formula.
Derived-functor versus Čech — Hartshorne uses $H^{i}$ for both; Codex should write $\overset{ˇ}{H}^{i}$ until agreement is invoked.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem and exercise. Deferred unless the priority-1+2 punch-list expands.
Full exercise-pack coverage. Items 19–21 sketch three per-chapter packs; full ~600-problem Hartshorne coverage requires ~5 dedicated *.E* passes — second-largest exercise commitment on the Fast Track.
§II.9 (formal schemes), §III.11 (formal functions), §III.12 (semicontinuity) — Tier-γ; specialty, not load-bearing.
§V.4 (27 lines), §V.5 (Castelnuovo contractibility), §V.6 (Enriques-Kodaira) — Tier-γ. The 27-lines result deserves eventual inclusion in a worked-examples pack.
Appendix A (general intersection theory) — requires Fulton as canonical reference; defer to Tier-γ.
Appendix B (GAGA, transcendental methods) — partially covered by 04.09.01 hodge-decomposition; GAGA bridge is a separate audit.
Appendix C (Weil conjectures, étale cohomology) — Tier-γ; the étale-morphisms entry (priority-1 item 4) is prerequisite.
Vakil and Eisenbud-Harris cross-coverage. Composite batching is noted in §4; a Vakil audit is the right forum.

§6 Acceptance criteria for FT equivalence (Hartshorne)

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

≥95% of Hartshorne's named theorems map to Codex units (currently ~50%; after priority-1 units this rises to ~75%; after priority-1+2 to ~88%; after priority-3 deepenings + foundations to ~95%).
≥80% of Hartshorne's exercises have a Codex equivalent (currently <1%; closing this requires the dedicated exercise-pack pass per §5 and items 19–21).
≥90% of Hartshorne's worked examples are reproduced in some Codex unit (currently ~35%; the priority-1+2 batch — particularly items 2 and 11 — brings this to ~70%; the remainder requires worked-example densification in 04.04.01 Riemann-Roch curves and the new surface-theory units).
A notation/hartshorne.md crosswalk exists.
For every chapter dependency in Hartshorne (Ch I → Ch II → Ch III → Ch IV → Ch V), there is a corresponding prerequisites arrow chain in Codex's DAG.
Pass-W weaving connects the new units to the existing 04-algebraic-geometry/ units, plus the spin-geometry strand (03-09-spin-geometry), the differential-forms / cohomology strand (03-04-differential-forms), and the moduli units (04-10-moduli) via lateral connections.

The 11 priority-1+2 units close most of the theorem-layer gap; priority-3 closes the depth gap on Picard, Riemann-Roch, schemes properties, and Nullstellensatz/dimension; priority-4 are pointers and exercise packs.

Recommended sequencing. Priority-1 (items 1–5) as one batch (5 agents × ~3h = ~3h wall + ~30m integration); priority-2 surface-theory bundle (items 6–9, interlinked) as a second batch; priority-2 curves bundle (10–11) + priority-3 (12–16) as a third batch; exercise packs as a fourth pass. This matches Hartshorne's chapter dependencies (Ch II + III foundations → Ch V surfaces → Ch IV curves → exercises) and keeps each integration to <12 units, in line with Lawson-Michelsohn and Bott-Tu calibrated batch sizes.

Composite Vakil+Hartshorne batch. After priority-1 ships, trigger a Vakil overlay audit to identify Vakil-specific framings (functor of points, Yoneda-style scheme definitions, the "rising sea") that need their own units versus Hartshorne units that already suffice. Avoids redundant production across the two anchor textbooks.

Plan v1. To be refined after the priority-1 batch ships and the Vakil overlay audit completes.

Hartshorne — *Algebraic Geometry* (Fast Track reference anchor) — Audit + Gap Plan