Minimal numerical types of genus one

This file describes the minimal numerical types of genus one, the numerical shadows of the special fibres of minimal regular models of genus-one curves.

A numerical type with a single component i has genus 1 + mᵢwᵢ(gᵢ - 1), so it has genus one exactly when gᵢ = 1. With more than one component, the signed genus is 1 + ∑ᵢ Φᵢ with every contribution Φᵢ of a minimal type nonnegative and zero exactly at the (-2)-indices; hence a numerical type with more than one component is minimal of genus one exactly when every component is a (-2)-index.

For such a type the intersection graph is either a tree or a single cycle. More generally, suppose only that every component satisfies aᵢᵢ ≥ -2wᵢ and that the intersection graph is not a tree, that is, has positive topological genus. For every component i of a cycle C in the graph, the two neighbours of i along C each meet i with aᵢⱼ ≥ wᵢ, so the row sums ∑_{j ∈ C} aᵢⱼ of the all-ones vector on C are nonnegative. Since the intersection form is negative definite on proper subsets of the components and vanishes only on the multiples of the multiplicity vector, the cycle passes through every component, all multiplicities are equal, every component satisfies aᵢᵢ = -2wᵢ and meets exactly two others, each with aᵢⱼ = wᵢ, and consequently all weights are equal; the topological genus is then one. The intersection matrix is -w times the Cartan matrix of the affine Dynkin diagram Ã_{n-1}, and for a minimal type of genus one, where moreover every gᵢ vanishes, this is the numerical type I_n of a cycle of rational curves.

Main results

• TauCeti.NumericalType.arithmeticGenus_eq_one_iff_of_card_eq_one: a numerical type with one component i has genus one exactly when gᵢ = 1.
• TauCeti.NumericalType.isMinimal_and_arithmeticGenus_eq_one_iff: with more than one component, a numerical type is minimal of genus one exactly when every component is a (-2)-index.
• TauCeti.NumericalType.multiplicity_eq_of_topologicalGenus_pos, TauCeti.NumericalType.weight_eq_of_topologicalGenus_pos, TauCeti.NumericalType.intersection_self_eq_of_topologicalGenus_pos, TauCeti.NumericalType.intersection_eq_weight_of_topologicalGenus_pos and TauCeti.NumericalType.ncard_neighborSet_eq_two_of_topologicalGenus_pos: if aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then the intersection graph is a single cycle through all components, with constant multiplicities and weights, as described above.
• TauCeti.NumericalType.topologicalGenus_le_one: if aᵢᵢ ≥ -2wᵢ for every component, the topological genus is at most one.

References

The numerical types are those of the Stacks Project chapter Semistable Reduction, Section Numerical types; the semidefiniteness of the intersection form used here is Stacks, Tag 0C5X.

Genus one

theorem TauCeti.NumericalType.arithmeticGenus_eq_one_iff_of_card_eq_one {T : NumericalType} (h : Fintype.card T.Component = 1) (i : T.Component) : T.arithmeticGenus = 1 ↔ T.genus i = 1

A numerical type with a single component i has genus one exactly when gᵢ = 1.

theorem TauCeti.NumericalType.isMinimal_and_arithmeticGenus_eq_one_iff {T : NumericalType} (h : 1 < Fintype.card T.Component) : T.IsMinimal ∧ T.arithmeticGenus = 1 ↔ ∀ (i : T.Component), T.IsMinusTwoIndex i

With more than one component, a numerical type is minimal of genus one exactly when every component is a (-2)-index.

Numerical types whose intersection graph has a cycle

theorem TauCeti.NumericalType.multiplicity_eq_of_topologicalGenus_pos {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) (htop : 0 < T.topologicalGenus) (i j : T.Component) : T.multiplicity i = T.multiplicity j

If aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then all multiplicities are equal.

theorem TauCeti.NumericalType.intersection_self_eq_of_topologicalGenus_pos {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) (htop : 0 < T.topologicalGenus) (i : T.Component) : T.intersection i i = -(2 * ↑↑(T.weight i))

If aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then aᵢᵢ = -2wᵢ for every component.

theorem TauCeti.NumericalType.intersection_eq_weight_of_topologicalGenus_pos {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) (htop : 0 < T.topologicalGenus) {i j : T.Component} (h : T.Adj i j) : T.intersection i j = ↑↑(T.weight i)

If aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then any two components that meet do so with intersection number aᵢⱼ = wᵢ.

theorem TauCeti.NumericalType.ncard_neighborSet_eq_two_of_topologicalGenus_pos {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) (htop : 0 < T.topologicalGenus) (i : T.Component) : (T.intersectionGraph.neighborSet i).ncard = 2

If aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then every component meets exactly two others.

theorem TauCeti.NumericalType.weight_eq_of_topologicalGenus_pos {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) (htop : 0 < T.topologicalGenus) (i j : T.Component) : T.weight i = T.weight j

If aᵢᵢ ≥ -2wᵢ for every component and the intersection graph is not a tree, then all weights are equal.

theorem TauCeti.NumericalType.topologicalGenus_le_one {T : NumericalType} (hself : ∀ (i : T.Component), -(2 * ↑↑(T.weight i)) ≤ T.intersection i i) : T.topologicalGenus ≤ 1

If aᵢᵢ ≥ -2wᵢ for every component, then the intersection graph has topological genus at most one: it is either a tree or a single cycle.