Bounding prime torsion by the genus

Let T be a numerical type of genus g ≥ 2 and ℓ a prime with ℓ > 768g. Then

dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g,

and if T is minimal, even dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top ≤ g, where g_top is the first Betti number of the intersection graph (Stacks, Proposition 55.7.4). In the proof of semistable reduction this is the numerical input which, confronted with the 2g-dimensional ℓ-torsion of the Jacobian, forces the special fibre of a minimal regular model to be reduced with nodal singularities.

The statements below only require ℓ > 768g - 768, which the bound on minimal types permits; this contains the hypothesis ℓ > 768g of the Stacks Project.

Main results

• TauCeti.NumericalType.IsMinimal.finrank_torsion_le_topologicalGenus_of_lt: dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top for a minimal type.
• TauCeti.NumericalType.finrank_torsion_le_arithmeticGenus: dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g.

theorem TauCeti.NumericalType.IsMinimal.finrank_torsion_le_topologicalGenus_of_lt {T : NumericalType} (hT : T.IsMinimal) (hg : 2 ≤ T.arithmeticGenus) (ℓ : ℕ) [Fact (Nat.Prime ℓ)] (hℓ : 768 * T.arithmeticGenus - 768 < ↑ℓ) : ↑(Module.finrank (ZMod ℓ) ↥(T.torsion ℓ)) ≤ T.topologicalGenus

Prime torsion of a minimal numerical type. In a minimal numerical type of genus g ≥ 2, every prime ℓ > 768g - 768 satisfies dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g_top, the first Betti number of the intersection graph (Stacks, Proposition 55.7.4).

theorem TauCeti.NumericalType.finrank_torsion_le_arithmeticGenus {T : NumericalType} (hg : 2 ≤ T.arithmeticGenus) (ℓ : ℕ) [Fact (Nat.Prime ℓ)] (hℓ : 768 * T.arithmeticGenus - 768 < ↑ℓ) : ↑(Module.finrank (ZMod ℓ) ↥(T.torsion ℓ)) ≤ T.arithmeticGenus

Prime torsion of a numerical type. In a numerical type of genus g ≥ 2, every prime ℓ > 768g - 768 satisfies dim_{𝔽_ℓ} Pic(T)[ℓ] ≤ g. In particular this holds for every prime ℓ > 768g, which is Stacks, Proposition 55.7.4.