Semibreak divisors on banana graphs #
A semibreak divisor is represented by an optional interior chip on each strand. This representation is definitionally effective, puts no chips at the two core vertices, and makes both its degree and its Dhar burn explicit.
The main result is rank_semibreak_sub_vertex_eq_neg_one: if E has degree
at most the genus and w is outside its support, then E - w is w-reduced
with debt, hence has rank -1. This is the support lemma needed by the
length-two cross-exception argument.
The last sections prove the reducedness and rank formula for a supplied endpoint/semibreak normal form, extract that form from every left-reduced divisor, and hence construct one in every linear-equivalence class.
A divisor represented by at most one selected interior chip per strand and none at the core vertices.
Equations
- Bananas.IsSemibreak B E = ∃ (chips : (γ : Fin (g + 1)) → Option (Fin (B.length γ - 1))), E = Bananas.semibreakDivisor B chips
Instances For
Restricting a semibreak divisor to a set of vertices cannot increase its degree.
A semibreak divisor of degree at most the genus has rank exactly zero.
The endpoint/semibreak normal-form interface #
The divisor a·L + b·R + E occurring in the banana reduced-divisor
normal form.
Equations
- Bananas.bananaNormalForm B a b E = a • oneChip (Bananas.leftEndpoint B) + b • oneChip (Bananas.rightEndpoint B) + E
Instances For
The numerical side condition in the paper's banana normal form.
Equations
- Bananas.IsBananaNormalForm B b E = (Bananas.IsSemibreak B E ∧ 0 ≤ b ∧ b + CFDiv.degree E ≤ ↑g)
Instances For
Every divisor in the paper's endpoint/semibreak normal-form range is reduced at the left endpoint. The coefficient at the reducing endpoint is unrestricted, as reducedness only asks for effectivity away from it.
The endpoint coefficients and semibreak divisor of two linearly equivalent normal forms in the numerical range agree. This is the uniqueness half of the normal-form statement.
The endpoint-swapped effectivity statement used for reduction at the right endpoint.
Symmetrically, bounding the left-endpoint coefficient makes the same normal-form divisor reduced at the right endpoint.
Once reducedness is established, the sign criterion in banana normal form follows from the generic reduced-divisor API.
In the numerical normal-form range, the rank is negative exactly when the unrestricted left-endpoint coefficient is negative.
Rank of a supplied banana normal form. The semibreak inequalities bound
the right-endpoint coefficient; the left coefficient need only be at least
-1.
Extraction of normal-form parameters from a reduced divisor #
The coefficient at the right endpoint of a left-reduced divisor is nonnegative.
Every interior coefficient of a left-reduced divisor is nonnegative.
The singleton Dhar cut at a two-valent interior vertex bounds its coefficient by one.
On a fixed strand, two positive interior coefficients of a reduced divisor must occur at the same offset.
The optional interior chip selected on each strand of a left-reduced divisor. If the strand has a positive interior coefficient, its unique such offset is chosen; otherwise the strand is empty.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The semibreak divisor extracted from a left-reduced divisor.
Equations
Instances For
The selected chip family reproduces every interior coefficient of the left-reduced divisor.
The divisor extracted from a left-reduced divisor is semibreak.
A left-reduced divisor is exactly its two endpoint coefficients plus the semibreak divisor extracted from its interior coefficients.
Existential form of semibreak extraction from a left-reduced divisor.
The endpoint coefficient and extracted semibreak degree of a left-reduced divisor satisfy the numerical normal-form bound.
Full converse normal-form statement for a left-reduced divisor.
Every divisor class on a banana graph has a representative in banana normal form with the required semibreak and numerical conditions.
TeX label: Lemma 2.23 (unlabeled), final clause: r(D) ≥ 0 iff a ≥ 0.