Chapter 6 · Part II — Counting curves by taking them apart

Genus Is Cycles

Allow one doubled elevator, and genus stops being a hypothesis about the classical curve — it becomes something you can count in the picture.

8 min read

Chapters 4 and 5 lived under a quiet restriction: genus 0. A genus-0 floor diagram is a Intro ¶1tree on its floors — d − 1 elevators, no cycles, and, quietly doing work the whole time, no parallel edges: two elevators between the same pair of floors would already be a cycle.

This chapter breaks the restriction with exactly one new ingredient. Allow parallel edges, and the first Betti number of the diagram,Intro ¶2

b₁ = |E_fin| − d + 1,

can be positive — and Intro ¶3b₁ is the genus g of the curves being counted. A doubled elevator is the smallest possible cycle. Genus stops being a hypothesis about the classical curve and becomes something you can count in the picture.

Prediction, before we compute: everything else should survive — Intro ¶4d floors, divergence in − out = 1 at every floor, no directed cycles, markings as linear extensions up to automorphism, and the same multiplicity μ^C = I^β · Π w² (Dedieu (1.1), whose correspondence theorem (1.2) is stated for general g). The absolute counts are again the relative type (0, (1^d)), and the point count should grow with the genus: curves of degree d and genus g are counted through 3d − 1 + g points. And each tropical count should have a classical witness — a nodal-curve formula it has to match.

6.1 The one real correction: multisets, not sets

Honest Box — the enumeration refused to draw the picture. The study’s first attempt reused Chapter 5’s enumeration, which chose the elevators as a set of ordered floor pairs — and found zero genus-1 diagrams. That is not a theorem; it is the enumeration refusing to draw the one picture that matters. On 3 floors the only 3-edge acyclic set is the transitive tournament, whose sink always violates divergence 1. The fix is one word: enumerate multisets of pairs (combinations_with_replacement), and the genus-1 cubic appears immediately — a doubled edge 2→0 plus 0→1. The automorphism count needed the matching fix: floor permutations must preserve per-pair weight multisets, and parallel edges of equal weight contribute swap factors Π m_{i,j,w}!. The action on labellings stays free, so markings = (linear extensions)/|Aut| still holds exactly. The chapter program contains both fixes; its assertions check b₁ = g on every shape it enumerates.

With that, a genus-§6.1 ¶1g diagram is a Chapter 4 diagram with |E_fin| = d − 1 + g finite edges, parallel edges allowed. Nothing else in the definition moves.

6.2 Four counts, each with a classical witness

§6.2 ¶1N_{3,1} = 1 (1 shape, through 9 points). The unique diagram is the one the Honest Box promised: doubled 2→0 plus 0→1, with μ = 1 and exactly 1 marking. Classically this had to be 1: there is exactly one cubic through 9 general points, and it is smooth — genus 1.

§6.2 ¶2N_{4,1} = 225 (11 shapes, through 12 points). Per-shape contributions, in the order the chapter program prints them sorted:

9 + 15 + 21 + 21 + 26 + 16 + 24 + 24 + 28 + 9 + 32 = 225

Classical check: genus-1 quartics are 2-nodal quartics, and the node polynomial §6.2 ¶3N₂(d) = (3/2)(d−1)(d−2)(3d²−3d−11) (Vainsencher; Kleiman–Piene for higher δ) gives 225 at d = 4. The formula itself cross-checks at d = 3, where it predicts 21 = C(7,2) — the reducible line+conic count. The program asserts both evaluations.

§6.2 ¶4N_{4,2} = 27 (5 shapes, contributions 3 + 5 + 7 + 4 + 8, through 13 points). Classical: 1-nodal quartics, and Steiner’s N₁(d) = 3(d−1)² gives 27 at d = 4.

§6.2 ¶5N_{4,3} = 1 (1 shape, through 14 points). The unique diagram — doubled 0→1, single 1→2, tripled 3→0: three independent cycles, μ = 1, 1 marking — is the unique smooth quartic through 14 points.

(d, g)	Points (3d−1+g)	Shapes	N_{d,g}	Classical witness
(3, 1)	9	1	1	the unique cubic is smooth
(4, 1)	12	11	225	node polynomial N₂(4)
(4, 2)	13	5	27	Steiner’s 3(d−1)²
(4, 3)	14	1	1	the unique quartic is smooth

§6.2 ¶6Figure 6.1 — Three panels: the unique (3,1) diagram (doubled edge = the first cycle), a (4,1) diagram whose cycle is a genuine triangle on floors {0,2,3} (μ = 1, 26 markings — the largest single marking count in the chapter), and a (4,2) diagram with two doubled edges = two independent cycles. Cycle edges in red. Every panel was checked by the chapter program — axioms, b₁ = g, multiplicity, markings — before it was rendered.

§6.2 ¶7The table, drawn. Left: the four counts — note that genus 3 costs two more points than genus 1 and collapses back to a single shape. Right: 225 taken apart; the biggest bar (32) is the shape with two weight-2 elevators (μ^C = 2²·2² = 16) and only 2 markings, while the 26-marking triangle shape contributes 26. Markings and multiplicity trade off all the way down the list.

Honest Box — what the witnesses are. The node-polynomial formulas are checks, quoted from Vainsencher and Steiner via the Day 5 log — this chapter does not derive them. The chapter’s claims are the tropical counts; the classical formulas are the independent witnesses those counts have to answer to, and the program asserts the formulas’ values (225, 21 = C(7,2), 27) alongside the enumerations.

6.3 The whole chapter, as a program

Everything in this chapter is one runnable program: §6.3 ¶1ch06_genus_floor_diagrams.py — self-contained: Chapter 5’s relative machinery is inlined, then the genus extension (multiset enumeration, multiset-preserving automorphisms). The assertions are the chapter’s claims — all four totals, every per-shape contribution above, b₁ = g on every shape, the classical witnesses as formulas, and a robustness re-run with the finite-edge weight bound raised from d to d+1 for (3,1), (4,1) and (4,2) (no new diagrams; the bound is exhaustive). One honest limitation, stated in the program’s header: N_{4,3} is not re-run at bound d+1 — its bound-d enumeration already takes about a minute, and the exhaustive-bound argument is the same upward-closed one as Chapters 4–5. Run it, and if it prints its final line, the counts are what we said they are. (This section used to print the listing inline; a screenful of code you can download is just a download with extra scrolling.)

6.4 Try It

1. A diagram on 4 floors has 6 finite edges. What is its genus?
2. Why can’t a doubled edge appear in a genus-0 diagram? Answer in one sentence using b₁.
3. How many points does N_{5,1} run through, by the formula 3d − 1 + g?

Answers

• 1. g = b₁ = 6 − 4 + 1 = 3.
• 2. A doubled edge already makes b₁ ≥ 1, but genus 0 means b₁ = 0 — the finite edges must form a tree, and a tree has no parallel edges.
• 3. 3·5 − 1 + 1 = 15 points.

What we learned

• Genus is a cycle count. One formula, b₁ = |E_fin| − d + 1, turns the genus from a property of an unseen classical curve into something visible in the diagram — and the smallest cycle is just a doubled elevator.
• The machinery didn’t care. Divergence, markings, multiplicity and the correspondence theorem were already stated for general g; the only real work was combinatorial honesty — multisets in the enumeration, multisets in the automorphisms.
• Every count has a witness. 1, 225, 27, 1 each meet an independent classical formula or uniqueness fact. Part II’s counting is now complete in both directions Chapter 4 opened: tangency (Chapter 5) and genus (this chapter).

Next

An interlude for programmers. Chapter 7 leaves curves behind and looks at the semiring itself as linear algebra: min-plus matrix powers are shortest walks, the Kleene star is Floyd–Warshall, and the tropical eigenvalue is a minimum cycle mean — Karp’s algorithm and the assignment problem turn out to be the same subject wearing different hats.Next ¶1

---

Next ¶2Source & provenance: Day 5 (2026-10-15) of the tropical geometry study — code hidden_files/code/day05_genus.py (run log hidden_files/day05-run.log, shapes in hidden_files/code/day05_shapes.json), figure files/figures/day05-genus-is-cycles.png, study log Day 5 entry. Definitions: Dedieu arXiv:2511.21123 (1.1)–(1.2) at general genus; classical witnesses: Vainsencher node polynomials, Steiner’s 3(d−1)². The chapter program was made self-contained and re-run end-to-end on 2026-11-03 while finishing this chapter: all assertions passed, including all four totals, the per-shape breakdowns, b₁ = g on every shape, and the weight-bound robustness re-runs noted in §6.3. Figures regenerated as SVG the same day: code hidden_files/code/ch06_extra_figures.py, which imports the chapter program and asserts each drawn shape’s axioms, genus, multiplicity and markings — and every bar in the counts figure against a fresh enumeration — before rendering.

Chapter code

The full program behind this chapter, as a download: Chapter code ¶1ch06_genus_floor_diagrams.py — the same file §6.3 points at. Its assertions are the chapter’s claims; run it.