Bounding prime torsion by the topological genus

Let T be a numerical type with n components, intersection matrix A and multiplicities mᵢ, and let e be the number of edges of its intersection graph, so that the topological genus g_top = 1 - n + e is the first Betti number of that graph. This file proves that for a prime ℓ dividing no multiplicity and no intersection number of two distinct meeting components,

dim_{𝔽_ℓ} Coker(A)[ℓ] ≤ g_top and hence dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top,

the second bound following from the first through the injection Pic(T) → Coker(A) (Stacks, Tag 0CE7). This is Stacks, Lemma 55.2.6, the combinatorial input for the bound dim Pic(T)[ℓ] ≤ g_top on minimal numerical types in Stacks, Proposition 55.7.4, which in turn is what forces a reduced nodal special fibre in the proof of semistable reduction.

Main results

• TauCeti.NumericalType.finrank_cokerTorsion_le_topologicalGenus: dim_{𝔽_ℓ} Coker(A)[ℓ] ≤ g_top.
• TauCeti.NumericalType.finrank_torsion_le_topologicalGenus: dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top.

Implementation notes

The Stacks Project argues with dual lattices. The proof here is instead linear algebra over 𝔽_ℓ, in two steps.

• Lifting kernel vectors of A modulo ℓ to ℤ and dividing their images under A by ℓ gives a surjection from the kernel of A over 𝔽_ℓ onto Coker(A)[ℓ]. It kills the multiplicity vector, which is nonzero modulo ℓ, so dim Coker(A)[ℓ] + 1 ≤ dim ker (A mod ℓ).
• Rescaling by the multiplicities, which are units modulo ℓ, turns A into the Laplacian Bᵀ W B of the intersection graph, where B is an oriented incidence matrix and W is the diagonal matrix of the edge weights -mᵢ aᵢⱼ mⱼ, again units modulo ℓ. Connectedness gives rank B ≥ n - 1, and Sylvester's rank inequality Matrix.rank_add_rank_le_rank_mul_add_card then gives rank (A mod ℓ) ≥ 2(n - 1) - e.

Together these give dim Coker(A)[ℓ] ≤ n - rank (A mod ℓ) - 1 ≤ 1 - n + e.

Torsion classes from the kernel of the intersection matrix modulo ℓ

The intersection matrix modulo ℓ as a weighted graph Laplacian

The bound

theorem TauCeti.NumericalType.finrank_cokerTorsion_le_topologicalGenus (T : NumericalType) (ℓ : ℕ) [Fact (Nat.Prime ℓ)] (hm : ∀ (i : T.Component), ¬ℓ ∣ ↑(T.multiplicity i)) (ha : ∀ (i j : T.Component), T.Adj i j → ¬↑ℓ ∣ T.intersection i j) : ↑(Module.finrank (ZMod ℓ) ↥(T.cokerTorsion ℓ)) ≤ T.topologicalGenus

The ℓ-torsion of Coker(A) is bounded by the topological genus. If the prime ℓ divides no multiplicity and no intersection number of two distinct meeting components, then dim_{𝔽_ℓ} Coker(A)[ℓ] ≤ 1 - n + e, the first Betti number of the intersection graph.

theorem TauCeti.NumericalType.finrank_torsion_le_topologicalGenus (T : NumericalType) (ℓ : ℕ) [Fact (Nat.Prime ℓ)] (hm : ∀ (i : T.Component), ¬ℓ ∣ ↑(T.multiplicity i)) (ha : ∀ (i j : T.Component), T.Adj i j → ¬↑ℓ ∣ T.intersection i j) : ↑(Module.finrank (ZMod ℓ) ↥(T.torsion ℓ)) ≤ T.topologicalGenus

The ℓ-torsion of Pic(T) is bounded by the topological genus. If the prime ℓ divides no multiplicity and no intersection number of two distinct meeting components, then dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top.