Fulton — Introduction to Toric Varieties (Fast Track 3.32) — Audit + Gap Plan

Book: William Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press 1993, xi + 157 pp. Distillation of the William H. Roever Lectures (Washington University, St Louis 1989). The canonical short textbook on toric geometry.

Fast Track entry: 3.32 (Modern Geometry section). Sits between 3.31 (Mumford, Geometric Invariant Theory) and 3.33 (Gross, Tropical Geometry and Mirror Symmetry); also adjacent to 3.30 (Harris-Morrison Moduli of Curves) in the Wave 8 cluster per docs/plans/FASTTRACK_COVERAGE_ROADMAP.md.

Sourcing status: REDUCED. No local PDF in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/. The book is in active copyright (Princeton Annals of Math Studies); not on Princeton's open backlist; not on the AMS open archive. This audit plan is therefore built from (a) the standard ToC of Fulton 1993 as universally documented, (b) the parallel structure of Cox-Little-Schenck Toric Varieties (which explicitly maps to Fulton chapter-by-chapter), (c) the toric / Delzant cross-pointers already present in content/05-symplectic/moment-reduction/05.04.04-delzant-theorem.md, and (d) Oda's Convex Bodies and Algebraic Geometry as a parallel exposition. A full P1 inventory at line-number granularity is deferred until a copy is obtained — see §7.

Purpose: P1 audit + P3-lite gap punch-list, mirroring plans/fasttrack/brown-higgins-sivera-nonabelian-algebraic-topology.md. Goal is a concrete list of new units to write so that Introduction to Toric Varieties (ITV hereafter) reaches the equivalence threshold of docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 — ≥95% effective coverage of named theorems, key examples, exercise pack, notation, sequencing, intuition.

§1 What ITV is for

ITV is the canonical short introduction to toric varieties — algebraic varieties admitting an action of an algebraic torus $(C^{*})^{n}$ with an open dense orbit. It is the combinatorial bridge between algebraic geometry and convex polytope theory: every toric variety is encoded by a fan of rational polyhedral cones in a lattice $N$ , and every property of the variety (smoothness, projectivity, divisors, line bundles, cohomology, intersection numbers) translates to a property of the fan or its dual polytope.

Distinctive contributions of ITV, in the order Fulton develops them (Chapters 1–5 + appendices):

Cones, fans, and lattices (Ch. 1). The basic combinatorial data: strongly convex rational polyhedral cone $σ \subset N_{R}$ , its dual cone $σ^{\lor} \subset M_{R}$ , the semigroup algebra $C [σ^{\lor} \cap M]$ producing the affine toric variety $U_{σ} = Spec C [σ^{\lor} \cap M]$ . A fan $Σ$ glues these into the toric variety $X_{Σ}$ .
Smoothness, completeness, orbits (Ch. 2). $X_{Σ}$ is smooth iff every cone is regular (generated by part of a $Z$ -basis of $N$ ); complete iff $∣Σ∣ = N_{R}$ . The orbit-cone correspondence: orbits of the torus on $X_{Σ}$ are in bijection with cones of $Σ$ , with $T$ -orbit $O_{σ}$ of codimension $dim σ$ . Star, link, refinement, and the orbit-closure stratification.
Resolution of singularities and toric birational geometry (Ch. 2–3). Any toric variety admits an equivariant resolution by refining the fan — Hironaka's theorem becomes combinatorial in the toric case. Star subdivision = toric blowup; the toric MMP.
Divisors, line bundles, and the Picard group (Ch. 3). Torus- invariant Weil divisors are $T$ -Cartier iff they come from a piecewise- linear function on $∣Σ∣$ that is linear on each cone. The Picard group is the quotient of such PL functions modulo globally linear functions. For a fan from a polytope $P$ , the polarisation line bundle $L_{P}$ has global sections $H^0(X, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m $— l a tt i ce p o in t so f$ P$ are a basis for sections.
Moment map and the polytope picture (Ch. 4). For a projective toric variety $X_{P}$ with polarisation polytope $P$ , the moment map of the compact $T_{R}^{n}$ -action sends $X_{P}$ surjectively onto $P$ , and the polytope-to-fan duality (face fan / normal fan) provides the algebraic counterpart of the Delzant correspondence on the symplectic side.
Cohomology and intersection theory (Ch. 5). Toric varieties have cohomology rings with purely combinatorial descriptions: for $X_{Σ}$ smooth complete, $H^{*} (X_{Σ}; Q)$ is the Stanley-Reisner ring of $Σ$ modulo the linear relations from $M$ , and the rational Chow ring agrees with rational cohomology. Intersection numbers of toric divisors are mixed volumes of polytopes — the bridge to the Bernstein-Kushnirenko theorem and to combinatorial intersection theory.
Reflexive polytopes and the entry point to mirror symmetry (Ch. 4 notes / appendix-level pointers). A reflexive polytope is a lattice polytope $P$ with $0 \in int (P)$ whose polar dual $P^{\circ}$ is also a lattice polytope. Batyrev 1994 proved that reflexive polytopes come in mirror pairs whose associated Calabi-Yau hypersurfaces are topologically mirror-dual; this is the polytope-combinatorial origin of string-theoretic mirror symmetry.

Cited peer expositions confirm this is the canonical content list and the standard order:

Cox-Little-Schenck (CLS), Toric Varieties (AMS GSM 124, 2011) — the modern definitive textbook on the subject; explicitly written as the "expanded successor" to Fulton 1993. CLS chapters 1–4 cover Fulton chapters 1–3 verbatim; CLS chapters 5–6 expand Fulton chapter 3; CLS chapters 8–9 cover Fulton chapter 4; CLS chapters 12–13 cover Fulton chapter 5. CLS is the natural cross-reference for any Fulton-anchored Codex unit.
Oda, Convex Bodies and Algebraic Geometry (Springer Ergebnisse 15, 1988) — the predecessor monograph (in print before Fulton), with a more thorough treatment of the lattice-and-polytope side and explicit attention to the historical Demazure-Mumford-Danilov line. Oda's notation and conventions are the ones Fulton adopts and CLS continues.
Audin, Topology of Torus Actions on Symplectic Manifolds (Progress in Mathematics 93, Birkhäuser 1991) — the symplectic-side companion to Fulton. Already cited in content/05-symplectic/moment-reduction/05.04.04-delzant-theorem.md as the Master-tier anchor. Audin Ch. VII–VIII give the bridge from Delzant polytopes (symplectic toric manifolds) to fans (algebraic toric varieties) that Fulton Ch. 4 develops on the algebraic side.
Danilov, The geometry of toric varieties (Russian Math. Surveys 33, 1978) — the canonical pre-Fulton survey article and originating exposition of the fan-and-cone formalism in English; explicitly cited by Fulton in the preface as the immediate predecessor. Already cited in the existing 05.04.04 Master Historical section.

ITV is not a first introduction to algebraic geometry. It assumes basic scheme theory (affine and projective schemes; sheaves and $Spec$ of a ring), elementary commutative algebra (integrally closed semigroups, Hilbert basis for cones), and a working understanding of projective varieties. The Codex prereq surface for Fulton-anchored units is 04-algebraic-geometry/02-schemes/ (affine + projective schemes) and 04-algebraic-geometry/05-divisors/ (Weil + Cartier divisors, Picard group, line bundles).

§2 Coverage table (Codex vs ITV)

Cross-referenced against the current 313-unit corpus (specifically 04-algebraic-geometry/ — 40 units — and 05-symplectic/moment-reduction/ — 6 units).

✓ = covered, △ = partial / different framing, ✗ = not covered.

ITV topic	Codex unit(s)	Status	Note
Algebraic torus $(C^{*})^{n}$ , character lattice $M$ , cocharacter lattice $N$	—	✗	Gap. Nothing in the Codex defines the algebraic torus, its character/cocharacter lattices, or the duality $M = Hom (N, Z)$ . Foundational for ITV.
Rational polyhedral cone, strongly convex cone, dual cone $σ^{\lor}$	—	✗	Gap. Convex-geometry primitive; not covered. Borders on `00-precalc` / `01-foundations` but the lattice-rational version is its own topic.
Affine toric variety $U_{σ} = Spec C [σ^{\lor} \cap M]$	—	✗	Gap (high priority — ITV's central building block).
Fan $Σ$ in $N_{R}$ and the toric variety $X_{Σ}$	—	✗	Gap (high priority — ITV's central object).
Smoothness criterion (regular cone)	—	✗	Gap. Pairs naturally with the affine-toric-variety unit.
Completeness criterion ($	\Sigma	= N_\mathbb{R}$)	—
Orbit-cone correspondence	—	✗	Gap (high priority — distinctive content).
Toric resolution of singularities via fan refinement	—	✗	Gap. Connects to `04.07.02-blowup.md` (general blowup) as the toric specialisation.
Toric divisor, $T$ -Cartier divisor, support function	—	✗	Gap. Connects to `04.05.01-weil-divisor.md`, `04.05.04-cartier-divisor.md`.
Picard group of a toric variety	—	△	△ — general Picard group covered at `04.05.02-picard-group.md`; toric specialisation (PL functions mod linear) is its own unit.
Line bundle from a lattice polytope; $H^{0} (X_{P}, L_{P})$ = lattice points of $P$	—	✗	Gap (high priority — distinctive content; bridge to representation theory and to mirror symmetry).
Polytope $P$ ↔ fan $Σ_{P}$ (normal fan, face fan)	—	✗	Gap.
Moment map for projective toric variety; image = polytope	△	△	Algebraic side is a gap. Symplectic side covered at `05.04.01-moment-map.md` + `05.04.04-delzant-theorem.md`. Need an explicit unit pointing both ways.
Delzant theorem (symplectic side)	✓	✓	Shipped at `05.04.04`. Already cites Audin and Danilov; needs back-pointer when the algebraic-side units ship.
Cohomology of smooth complete toric variety (Stanley-Reisner + linear relations)	—	✗	Gap (high priority — distinctive content).
Toric intersection theory; mixed-volume formula for top intersections	—	✗	Gap. Bridges to Bernstein-Kushnirenko (toric Bezout).
Bernstein-Kushnirenko theorem (toric Bezout)	—	✗	Gap. Master-tier deepening or standalone unit.
Reflexive polytope; polar duality $P \leftrightarrow P^{\circ}$	—	✗	Gap. Entry to Batyrev mirror symmetry.
Batyrev mirror construction (reflexive-polytope CY hypersurfaces)	—	✗	Gap (Master-tier pointer; explicitly deferred — see §5).
Cox homogeneous coordinate ring	—	✗	Gap (Master-tier; Cox 1995 is post-ITV but is now standard and CLS treats it).
Toroidal embedding (Mumford 1973)	—	✗	Gap. Master-tier historical-context pointer.
Demazure construction (toric variety as quotient of $C^{d}$ by torus)	△	△	The Delzant unit `05.04.04` covers the symplectic-reduction version of this construction explicitly. The algebraic version (GIT quotient of $C^{d} ∖ Z$ by $(C^{*})^{d - n}$ , i.e. Cox's construction reading Demazure 1970 forward) is a gap.

Aggregate coverage estimate: ~5% of ITV has corresponding Codex units (Delzant + general Picard / Weil-Cartier provide partial coverage of chapter 3–4 material). The fan/cone/orbit-cone combinatorial machinery that anchors chapters 1–2 is entirely absent. The cohomology of toric varieties (chapter 5) is entirely absent. The polytope ↔ fan bridge is absent except for the Delzant-side hint.

Distinctive Codex situation. The symplectic side of toric geometry is relatively well-covered (05.04.04 Delzant + the Master section's algebraic-geometric counterpart paragraph + the historical context citing Demazure/Mumford/Danilov), while the algebraic side has zero coverage. Fulton-anchored units should be sited in 04-algebraic-geometry/ with explicit lateral pointers to 05.04.04 and back. The natural new chapter location is 04-algebraic-geometry/11-toric/ (new chapter folder appended to the existing 10-chapter algebraic-geometry section).

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

Priority 0 — prereqs already in place. Affine and projective schemes (04.02.01–04.02.03), Weil and Cartier divisors (04.05.01, 04.05.04), Picard group (04.05.02), line bundles (04.05.03), projective space (04.07.01), blow-up (04.07.02), moment map (05.04.01), AGS convexity (05.04.03), Delzant theorem (05.04.04). No upstream blockers.

Priority 1 — high-leverage, captures ITV's central content:

04.11.01 Algebraic torus and character/cocharacter lattices. $T = (C^{*})^{n}$ , character lattice $M = \mathrm{Hom}(T, \mathbb{C}^) \cong \mathbb{Z}^n $, coc ha r a c t er l a tt i ce$ N = \mathrm{Hom}(\mathbb{C}^, T) \cong \mathbb{Z}^n $, p er f ec tp ai r in g$ M \times N \to \mathbb{Z}$. Fulton §1.1 anchor; CLS §1.1 anchor. Three-tier, ~1500 words. Foundational.
04.11.02 Rational polyhedral cone and dual cone. Strongly convex rational polyhedral cone $σ \subset N_{R}$ ; dual cone $\sigma^\vee = {m \in M_\mathbb{R} : \langle m, n\rangle \ge 0 \text{ for all } n \in \sigma} $; G or d a n^{'} s l e mma (t h ese mi g r o u p$ \sigma^\vee \cap M$ is finitely generated). Fulton §1.2 anchor; CLS §1.2 anchor. Three-tier, ~1500 words.
04.11.03 Affine toric variety $U_{σ}$ . Construction $U_{σ} = Spec C [σ^{\lor} \cap M]$ ; functoriality on the cone category; key examples ( $U_{{0}} = T$ , $U_{σ}$ for $σ = R_{\geq 0}^{n}$ giving $A^{n}$ , the quadric cone $x y = z w$ from a non-smooth cone). Fulton §1.3 anchor; CLS §1.3 anchor. Three-tier, ~2000 words. Central ITV building block.
04.11.04 Fan and the toric variety $X_{Σ}$ . Definition of a fan (collection of cones closed under face and intersection); the variety $X_{Σ}$ obtained by gluing the affine $U_{σ}$ 's; standard examples (the fan for $P^{n}$ , for $\mathbb{P}^1 \times \mathbb{P}^1 $, f or t h eH i r z e b r u c h s u r f a ce$ \mathbb{F}_n$, for the blow-up of $A^{2}$ at the origin). Fulton §1.4 anchor; CLS §3.1 anchor. Three-tier, ~2500 words. Central ITV object.
04.11.05 Smoothness and completeness via fans. $X_{Σ}$ is smooth iff every cone is generated by a subset of a $Z$ -basis of $N$ ; $X_{Σ}$ is complete iff $∣Σ∣ = N_{R}$ ; $X_{Σ}$ is projective iff $Σ$ is the normal fan of a polytope. Fulton §2.1 anchor; CLS §3.1, §6.1 anchors. Three-tier, ~1500 words.
04.11.06 Orbit-cone correspondence. $T$ -orbits of $X_{Σ}$ in bijection with cones of $Σ$ ; orbit closures are themselves toric varieties (of star fans); the open orbit is $T \subset X_{Σ}$ . Fulton §3.1 anchor; CLS §3.2 anchor. Three-tier, ~1500 words.
04.11.07 Toric resolution of singularities. Any fan $Σ$ admits a regular refinement $Σ$ ; the induced map $X_{Σ} \to X_{Σ}$ is a resolution. Star subdivisions = toric blowups; toric resolution is canonical and algorithmic (in contrast to Hironaka's general theorem). Fulton §2.6 anchor; CLS §11.1 anchor. Three-tier, ~1500 words.

Priority 2 — divisors, line bundles, and the polytope bridge:

04.11.08 Toric divisor and support function. Torus-invariant prime divisors $D_{ρ}$ in bijection with rays $ρ \in Σ (1)$ ; the divisor $\sum a_{ρ} D_{ρ}$ is Cartier iff the piecewise-linear function $φ$ on $∣Σ∣$ with $φ (v_{ρ}) = - a_{ρ}$ on the primitive generator $v_{ρ}$ is linear on each cone. Fulton §3.3 anchor; CLS §4.0 anchor. Three-tier, ~2000 words. Cross-points 04.05.01/04.05.04.
04.11.09 Toric Picard group. $Pic (X_{Σ})$ = (PL functions on $∣Σ∣$ that are linear on each cone) / (globally linear functions on $∣Σ∣$ ). Computation for the standard examples. Fulton §3.4 anchor; CLS §4.2 anchor. Intermediate + Master. ~1500 words.
04.11.10 Polytope ↔ fan dictionary; the line bundle $L_{P}$ . Given a lattice polytope $P$ , define the normal fan $Σ_{P}$ and the polarisation line bundle $L_{P}$ on $X_{Σ_{P}}$ . Then $H^0(X_{\Sigma_P}, L_P) = \bigoplus_{m \in P \cap M} \mathbb{C} \cdot \chi^m $— t h e l a tt i ce p o in t so f$ P$ are an explicit basis of global sections. Ample iff $P$ has full $n$ -dimensional interior. Fulton §3.4 anchor; CLS §4.3, §6.1 anchors. Three-tier, ~2500 words. Distinctive content; bridge to representation theory and physics.
04.11.11 Algebraic moment map and the polytope. For a projective toric variety $X_{P}$ with the compact torus $T_{R}^{n}$ acting, the moment map sends $X_{P}$ onto $P$ . Explicit formula via $L_{P}$ and the Fubini-Study form pulled back from the projective embedding. Fulton §4.2 anchor; Audin Ch. VII–VIII parallel. Intermediate + Master. ~1500 words. Explicit cross-pointer to 05.04.04 Delzant theorem.

Priority 3 — cohomology and intersection theory:

04.11.12 Cohomology of a smooth complete toric variety. $H^{*} (X_{Σ}; Q)$ is the Stanley-Reisner ring of $Σ$ modulo the linear relations from $M$ . Includes the proof for the projective case (Jurkiewicz / Danilov). Fulton §5.2 anchor; CLS §12.4 anchor. Master-tier; Intermediate gives the statement and the computation for $P^{n}$ and $P^{1} \times P^{1}$ . ~2500 words.
04.11.13 Toric intersection theory and mixed volumes. For a smooth complete toric variety with ample divisors $L_{P_1}, \ldots, L_{P_n} $f r o m p o l y t o p es$ P_1, \ldots, P_n$, the top intersection number is $n! \cdot MV (P_{1}, \dots, P_{n})$ , the mixed volume. Fulton §5.3–5.4 anchor; CLS §13.4 anchor. Master-tier. ~2000 words.
04.11.14 Bernstein-Kushnirenko theorem. The number of solutions in $(C^{*})^{n}$ to a system of $n$ Laurent polynomial equations with generic coefficients and Newton polytopes $P_{1}, \dots, P_{n}$ is $n! \cdot MV (P_{1}, \dots, P_{n})$ . Toric Bezout. Originator citations: Kushnirenko 1976, Bernstein 1975. Fulton §5.5 anchor (brief); CLS §13.5 anchor. Master-tier. ~1500 words. Distinctive content; the classical Bezout reduces to this for $P_{i}$ = scaled simplex.

Priority 4 — Cox-Demazure construction; reflexive polytopes; pointers:

04.11.15 Cox homogeneous coordinate ring; Demazure quotient construction. $X_\Sigma = (\mathbb{C}^{|\Sigma(1)|} \setminus Z(\Sigma)) /!/ G $, w h er e$ G $i s t h e k er n e l o f t h e t or u s ma p$ (\mathbb{C}^*)^{|\Sigma(1)|} \to T $d u a l t o t h e l in e a r ma p se n d in g t h e$ i$-th standard generator to $v_{ρ_{i}}$ . The total coordinate ring is $\mathbb{C}[x_\rho : \rho \in \Sigma(1)]$. Cox 1995 originator citation (the construction sharpens Demazure 1970). Fulton's preface mentions this; CLS §5 is devoted to it. Master-tier; ~2500 words. Explicit cross-pointer to 05.04.04 Delzant: the symplectic-quotient $C^{d} / / K$ in the Delzant construction is the topologists' shadow of the Cox construction.
04.11.16 Reflexive polytope and Batyrev mirror duality (pointer). Lattice polytope $P$ with $0 \in int (P)$ is reflexive iff its polar dual $P^{\circ}$ is also a lattice polytope. Batyrev 1994: reflexive polytopes come in mirror pairs whose anti-canonical Calabi- Yau hypersurfaces in the associated Gorenstein toric Fano varieties are topologically mirror-symmetric. Pointer unit at FT-equivalence: statement + one worked pair (the quintic / mirror quintic via the standard 4-simplex). Master-only, ~1500 words. Deferred deep content on mirror symmetry — see §5.

§4 Implementation sketch (P3 → P4)

For full ITV coverage, items 1–11 are the minimum equivalence set (priority 1+2). Realistic production estimate (mirroring earlier algebraic-geometry batches like Hartshorne / Griffiths-Harris and the existing 05.04.04 Delzant unit):

~3–4 hours per unit. Toric units skew slightly higher than corpus average because of the bidirectional polytope-and-fan picture (every unit needs cone-side, polytope-side, and variety-side examples).
11 priority-1+2 units × ~3.5 hours = ~38–40 hours. Plus 3 priority-3 units (~12 hours) and 2 priority-4 units (~8 hours) to reach ≥95% equivalence: total ~58–60 hours, a focused 7–8 day window.
7 priority-1 units alone (~25 hours) lift coverage from ~5% to ~50%; adding priority-2 (~13 hours) reaches ~80%; priority-3 reaches ~92%; priority-4 closes the residual.

Originator-prose target. ITV's content was originated by a small set of authors with clear attributions per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10:

Demazure 1970 (Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. ENS 4) — originator of toric varieties as smooth completions of $(C^{*})^{n}$ via fan-and-cone data, motivated by classifying maximal-rank subgroups of $Cr_{n}$ .
Mumford 1973 (Kempf-Knudsen-Mumford-Saint-Donat, Toroidal Embeddings I, Lecture Notes in Math 339, Springer) — generalised the toric construction to toroidal embeddings of arbitrary varieties along divisors; this is the algebraic-geometry-of-singularities incarnation of the toric story.
Danilov 1978 (The geometry of toric varieties, Russian Math. Surveys 33, pp. 97–154) — the canonical pre-Fulton survey; introduced the cohomology computation now associated with the Stanley-Reisner ring.
Fulton 1993 — the book itself; consolidated the working combinatorial-algebraic treatment used in the field today.
Cox 1995 (The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4, 17–50) — the explicit quotient construction unifying the Demazure / GIT pictures.
Batyrev 1994 (Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3, 493–535) — reflexive-polytope mirror duality; entry point to the toric side of mirror symmetry.

Each anchor unit's Master Historical section should carry at least one of these originator citations; 04.11.03–04 (affine and gluing) should cite Demazure 1970 explicitly; 04.11.12 (cohomology) should cite Danilov 1978; 04.11.15 (Cox / Demazure quotient) should cite Cox 1995 and Demazure 1970 together; 04.11.16 (reflexive polytopes) should cite Batyrev 1994.

Notation crosswalk. ITV uses $N$ for the cocharacter lattice and $M$ for the character lattice, with $N_{R} = N \otimes R$ and similarly for $M_{R}$ ; cones live in $N_{R}$ and polytopes in $M_{R}$ . CLS adopts the same convention. The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should adopt $(N, M, N_{R}, M_{R})$ throughout the 04.11.* series, with $σ$ for cones, $Σ$ for fans, $X_{Σ}$ for the toric variety, $U_{σ}$ for affine pieces. Note the Delzant unit 05.04.04 uses $t ≅ R^{n}$ for the torus Lie algebra and $\Lambda = \mathbb{Z}^n \subset \mathfrak{t}$ for the integer lattice — these identify with $N_{R}$ and $N$ respectively, and the cross-pointer notes in 04.11.11 and 04.11.15 should make this identification explicit.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in ITV. Deferred until a working copy of the book is sourced (see §7) — current sourcing status is reduced; the plan was built from the universally- documented ToC plus the CLS / Oda / Audin / Danilov cross-references.
The deep content of Batyrev mirror symmetry. 04.11.16 is a single pointer unit at FT-equivalence; full mirror-symmetric content (Hodge number duality of CY hypersurfaces, the Strominger-Yau-Zaslow picture, Givental's J-function, mirror theorems) is deferred to FT 3.33 (Gross, Tropical Geometry and Mirror Symmetry) per the standing scope note.
The toric MMP beyond the resolution-of-singularities statement. Reid-Witten-style flips and the structure of the toric movable cone are out of scope for this audit; can be added as Master-tier deepenings to 04.11.07 if needed.
The toroidal-embeddings generalisation (Mumford 1973) beyond a single historical-context pointer in the Master section of 04.11.04.
Exercise-pack production. ITV exercises are conceptually rich and follow the chapter material directly; the exercise pack is a P3-priority-3 follow-up after the priority-1+2 units ship.
Toric stacks. Out of scope; pointer-only if at all.
The Cox construction in the orbifold (simplicial fan) generality. 04.11.15 covers the smooth case; the simplicial-orbifold extension is a Master-tier extension and is a candidate for a separate unit if the corpus later acquires orbifold infrastructure.

§6 Acceptance criteria for FT equivalence (ITV)

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

≥95% of ITV's named theorems in chapters 1–5 map to Codex units. Currently ~5% (Delzant + general Picard partial coverage). After priority-1 ships: ~50%. After priority-1+2: ~80%. After priority-1+2+3: ~92%. After priority-1+2+3+4: ≥95%.
≥90% of ITV's worked computations have a direct unit or are referenced from a unit that covers them. The standard examples (affine toric varieties $A^{n}$ , the quadric cone, $P^{n}$ , $\mathbb{P}^1 \times \mathbb{P}^1 $, t h eH i r z e b r u c h s u r f a ces$ \mathbb{F}_n$, blow-up of $A^{2}$ , the toric resolution of $A_{n}$ singularities) are distributed across priority-1 and priority-2 units' worked-example sections.
Notation decisions are recorded (see §4 above).
Cross-pointers to 05.04.04 Delzant theorem are explicit in 04.11.04, 04.11.10, 04.11.11, and 04.11.15. Reciprocal back- pointer in 05.04.04 Master section's "Algebraic-geometric counterpart" paragraph is updated to cite the new units.
Pass-W weaving connects the new 04-algebraic-geometry/11-toric/ chapter to 05-symplectic/moment-reduction/, to 04-algebraic-geometry/05-divisors/, to 04-algebraic-geometry/07-projective/, and (for the cohomology units) to 04-algebraic-geometry/03-cohomology/ and to 07-representation-theory/.

The 7 priority-1 units close the foundational gap. Priority-2 closes the divisor / polytope-bridge gap and supplies the explicit reciprocal pointer to Delzant. Priority-3 closes the cohomology / intersection- theory gap. Priority-4 is the Cox construction and the reflexive-polytope pointer to mirror symmetry. Once all four priority bands are shipped, ITV is at the FT-equivalence threshold.

§7 Sourcing

Status: REDUCED. No local PDF was found in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/. The book is in active copyright (Princeton University Press, Annals of Mathematics Studies 131, 1993) and is not on Princeton's open backlist, AMS open archive, or any author-hosted page.
Action required to upgrade to full P1 audit. Acquire a copy. Listed as BUY in docs/catalogs/FASTTRACK_BOOKLIST.md line for 3.32 ("Introduction to Toric Varieties — William Fulton — Toric geometry, momentum maps — BUY") and similarly in docs/catalogs/NEED_TO_SOURCE.md. Estimated cost ~$45–55 paperback, ~$70 hardcover; library access via institutional ILL is the fastest path. Once acquired, the line-number P1 inventory in §2 above can be upgraded from ToC-level to theorem-level.
Local copy target. Add to reference/fasttrack-texts/03-modern- geometry/ as Fulton-IntroductionToToricVarieties.pdf to mirror the pattern of other paid FT texts.
Substitute exposition (used in this audit). Cox-Little-Schenck Toric Varieties (AMS GSM 124, 2011) is the open-readable modern successor and follows Fulton's structure closely. CLS chapters 1–4 cover Fulton chapters 1–3; CLS chapters 8–9 cover Fulton chapter 4; CLS chapters 12–13 cover Fulton chapter 5. Worked-example coverage in CLS is strictly broader than in Fulton; ITV-anchored units can use CLS as the working day-to-day reference and cite Fulton 1993 for the canonical short statements.
Audin reference. Already in reference/textbooks-extra/ via the 05.04.04 Delzant unit's master anchor; Audin Ch. VII–VIII parallel ITV Ch. 4 directly.
Danilov 1978. Russian Math. Surveys 33, pp. 97–154; available open-access through the journal's archive. Cited in 05.04.04 Historical section already; ITV units can cite the same source.

Unusual finding. The existing 05.04.04-delzant-theorem.md already carries substantial toric-geometry cross-pointers in its Master sections: the "Algebraic-geometric counterpart" paragraph in Advanced Results, the explicit Demazure / Mumford / Danilov citations in Historical Context, and the Bibliography entry @article{Danilov1978Toric, ...}. This is the natural Codex-side anchor to which ITV-derived units should back-link, and the Delzant unit's existing master paragraph can be extended with one-line cross-pointers to 04.11.04, 04.11.10, 04.11.11, and 04.11.15 once those ship. No other Codex unit currently mentions toric / fan / Demazure / Batyrev material.

Fulton — *Introduction to Toric Varieties* (Fast Track 3.32) — Audit + Gap Plan