Proper subgraphs of (-2)-indices in a numerical type #
A (-2)-index of a numerical type is a component i with gᵢ = 0 and aᵢᵢ = -2wᵢ. The
connected configurations of (-2)-indices that can occur as proper subsets of the components
of a numerical type form a short explicit list, of Dynkin-diagram shape
(Stacks, Section 0C7L). This classification is
what bounds the multiplicities along chains of (-2)-indices in a minimal numerical type, and
hence its Picard group.
This file classifies the configurations on two, three, four, and five components. If a numerical
type has
more than two components and two (-2)-indices i and j meet, then up to swapping i and j,
(wᵢ, wⱼ, aᵢⱼ) = (w, w, w), (w, 2w, 2w) or (w, 3w, 3w)
for some positive integer w (Stacks, Tag 0C7M). If
a numerical type has more than three components and two distinct (-2)-indices i and k both
meet a (-2)-index j, then i and k do not meet, so that the configuration is a chain, and
up to reversing it
(wᵢ, wⱼ, wₖ, aᵢⱼ, aⱼₖ) = (w, w, w, w, w), (w, w, 2w, w, 2w) or (2w, 2w, w, 2w, 2w)
On four components, a chain has four possible unoriented weight patterns, and a component meeting three others has the simply-laced star pattern. In both cases all intersections not displayed in the graph vanish (Stacks, Tags 0C7V and 0C80).
On five components a chain has only three unoriented weight patterns: all five weights equal, or
four equal weights together with, at one end of the chain, their double or their half
(Stacks, Tag 0C82). In particular a double edge can
occur only at an end, so the two middle edges are simply laced, and again no intersection outside
the chain is nonzero: five (-2)-indices never form a pentagon. Likewise, a component meeting
three other (-2)-indices cannot meet a fourth one
(Stacks, Lemma 55.5.6).
The remaining five-component tree has a chain of length three ending in a fork. All five weights and all four displayed intersections are equal, and every other intersection vanishes (Stacks, Lemma 55.5.7).
On six components, a chain again has equal weights and simple edges except possibly at one end, where the endpoint may have twice or half the common interior weight. This is the base case for the arbitrary-length chain classification of Stacks, Lemma 55.5.8.
These arguments use only the self-intersections aᵢᵢ = -2wᵢ, not the genera. For a pair, negative
definiteness of the principal 2 × 2 submatrix gives aᵢⱼ² < 4wᵢwⱼ, and lcm(wᵢ, wⱼ) ∣ aᵢⱼ
leaves only the three solutions above. For a triple, negative definiteness of the principal
3 × 3 submatrix says that
p₁q₁ + p₂q₂ + p₃q₃ + q₁q₂p₃ < 4
where aᵢⱼ = wᵢp₁ = wⱼq₁, aⱼₖ = wⱼp₂ = wₖq₂ and aᵢₖ = wᵢp₃ = wₖq₃ are the factorisations
supplied by the divisibility axiom. All four summands are nonnegative integers and the first two
are positive, so the last two vanish; in particular aᵢₖ = 0 and p₁q₁ + p₂q₂ ≤ 3. The constraints
aᵢⱼ mⱼ ≤ 2wᵢ mᵢ and aᵢⱼ mᵢ ≤ 2wⱼ mⱼ on the multiplicities listed alongside the Stacks
statements are instances of TauCeti.NumericalType.multiplicity_mul_intersection_le.
Beyond four components the principal determinants become unwieldy, and the five-component classification instead evaluates the intersection form at an explicit positive integral vector. Each ratio pattern excluded there is the diagram of an affine generalized Cartan matrix, so the form vanishes at the vector spanning its kernel; this is what negative definiteness of the intersection form on the vectors supported on a proper subset of the components forbids.
Main results #
TauCeti.NumericalType.exists_weight_intersection_triple_mem: the classification of the weights and the intersection number of two meeting(-2)-indices.TauCeti.NumericalType.intersection_eq_max_weight: two meeting(-2)-indices have intersection number the larger of their two weights.TauCeti.NumericalType.intersection_eq_zero_of_intersection_pos_of_intersection_pos: two(-2)-indices meeting a common third one do not meet each other.TauCeti.NumericalType.exists_weight_intersection_quintuple_mem: the classification of the weights and the intersection numbers of a chain of three(-2)-indices.TauCeti.NumericalType.exists_intersection_ratio_chain_four_mem: the classification of a chain of four(-2)-indices by its normalized adjacent intersection ratios.TauCeti.NumericalType.exists_weight_intersection_star_four_eq: the classification of the four-component star.TauCeti.NumericalType.exists_intersection_ratio_chain_five_mem: the classification of a chain of five(-2)-indices by its normalized adjacent intersection ratios.TauCeti.NumericalType.intersection_eq_weight_of_chain_five: the two middle edges of a chain of five(-2)-indices are simply laced.TauCeti.NumericalType.intersection_eq_zero_of_chain_five: a chain of five(-2)-indices does not close up into a pentagon.TauCeti.NumericalType.intersection_eq_zero_of_star_five: a(-2)-index meeting three others meets no fourth one.TauCeti.NumericalType.exists_weight_intersection_fork_five_eq: the five-component fork is simply laced, with equal weights and no additional edges.TauCeti.NumericalType.intersection_eq_zero_of_chain_six: a chain of six(-2)-indices has no intersections outside its five displayed edges.TauCeti.NumericalType.exists_intersection_ratio_chain_six_mem: a six-component chain has at most one nonsimple edge, and only at an end.
Two meeting components i and j of self-intersections aᵢᵢ = -2wᵢ and aⱼⱼ = -2wⱼ, such
as two (-2)-indices, in a numerical type with more than two components have
(wᵢ, wⱼ, aᵢⱼ) equal to (w, w, w), (w, 2w, 2w), (2w, w, 2w), (w, 3w, 3w) or
(3w, w, 3w) for some positive integer w
(Stacks, Tag 0C7M).
Two meeting components i and j of self-intersections aᵢᵢ = -2wᵢ and aⱼⱼ = -2wⱼ in a
numerical type with more than two components have aᵢⱼ = max(wᵢ, wⱼ). This is the uniform
reading of the five weight patterns of
Stacks, Tag 0C7M.
Three components #
Two components i and k of a numerical type with more than three components whose
self-intersections are aᵢᵢ = -2wᵢ and aₖₖ = -2wₖ, such as two (-2)-indices, and which both
meet a third component j with aⱼⱼ = -2wⱼ, do not meet each other: such a configuration is a
chain, never a triangle (Stacks, Tag 0C7R).
Three components i, j, k of self-intersections aᵢᵢ = -2wᵢ, aⱼⱼ = -2wⱼ and
aₖₖ = -2wₖ, such as three (-2)-indices, in a numerical type with more than three components,
with i and k distinct and both meeting j, have (wᵢ, wⱼ, wₖ, aᵢⱼ, aⱼₖ) equal to
(w, w, w, w, w), (w, w, 2w, w, 2w), (2w, w, w, 2w, w), (2w, 2w, w, 2w, 2w) or
(w, 2w, 2w, 2w, 2w) for some positive integer w. The first, second and fourth of these are the
three chains of Stacks, Tag 0C7R, the remaining two
their reverses. The intersection number aᵢₖ of the two ends vanishes by
TauCeti.NumericalType.intersection_eq_zero_of_intersection_pos_of_intersection_pos.
Four components #
In a chain of four components of self-intersection -2w, every nonconsecutive intersection
vanishes. This is the graph-shape part of
Stacks, Lemma 55.5.3.
For four components in a chain, all of self-intersection -2w, the three normalized adjacent
intersection ratios are (1, 1, 1), (1, 1, 2), (1, 2, 1), or (2, 1, 1). The factors in
the statement express aᵢⱼ²/(wᵢwⱼ) without division and recover the four unoriented weight
patterns of Stacks, Lemma 55.5.3.
Four components of self-intersection -2w forming a three-legged star all have the same
weight, and every displayed intersection equals that weight
(Stacks, Lemma 55.5.4).
Five components #
For five components in a chain, all of self-intersection -2w, the four normalized adjacent
intersection ratios are (1, 1, 1, 1), (1, 1, 1, 2) or (2, 1, 1, 1): a double edge occurs
only at one of the two ends of the chain, and a triple edge not at all. The factors in the
statement express aᵢⱼ²/(wᵢwⱼ) without division, and together with the divisibility of each row
by its weight they give the weight patterns of
Stacks, Lemma 55.5.5: all five weights equal, or
four equal weights together with, at one end, their double or their half.
The two windows of four consecutive components leave three further ratio patterns, carrying a double edge in the middle of the chain or double edges at both of its ends. Each is ruled out by an explicit positive vector at which the intersection form vanishes, which negative definiteness on the vectors supported on a proper subset of the components forbids.
In a chain of five components of self-intersection -2w, the two middle edges are simply
laced: the three middle components share a weight, and it is the intersection number of both
middle pairs. This is the part of
Stacks, Lemma 55.5.5 common to its three cases, in
all of which a double edge can only occur at one of the two ends of the chain.
In a chain of five components of self-intersection -2w, every nonconsecutive intersection
vanishes: in particular the chain does not close up into a pentagon. This is the graph-shape part
of Stacks, Lemma 55.5.5.
A component of self-intersection -2w meeting three others of self-intersection -2w meets
no fourth such component, in a numerical type with more than five components. In particular, the
four-legged star does not occur as a proper subgraph of (-2)-indices
(Stacks, Lemma 55.5.6).
Five components of self-intersection -2w in a numerical type with more than five components
forming the fork
h - i - j - k, with a second leaf l at j,
all have the same weight, every displayed intersection equals that weight, and every other
intersection vanishes. This is the classification of
Stacks, Lemma 55.5.7. The corresponding inequalities
on the multiplicities follow from TauCeti.NumericalType.multiplicity_mul_intersection_le.
Six components #
Six components of self-intersection -2w forming a chain in a numerical type with more than
six components have no additional intersections. This is the graph-shape part of the
six-component base case for
Stacks, Lemma 55.5.8.
In a chain of six components of self-intersection -2w, all adjacent normalized intersection
ratios are one except possibly a ratio two at one end. Equivalently, their ratio pattern is
(1,1,1,1,1), (2,1,1,1,1), or (1,1,1,1,2). This is the numerical classification in the
six-component base case for
Stacks, Lemma 55.5.8.