Rank characterizations of raw transmission permutations #
This file isolates the generic bridge used by the non-recurrence argument in
Section 4 of the twice-marked banana paper. The paper's raw transmission
permutation is the unique ASP permutation whose slipface is the marked rank
surface. Consequently its southeast and northwest quadrant cardinalities
are exactly the two Riemann--Roch rank terms (paper Lemma lem:tauChars).
Concrete degree-twist representatives #
Integer-indexed degree twists have the prescribed degree.
Varying the index translates a fixed-degree representative by the marked difference.
Indexing degree twists modulo a torsion witness preserves their linear equivalence class.
The paper's nonrecurrence condition, formulated on concrete torsion residues instead of Picard-group quotient classes.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A convenient finite-residue consequence of nonrecurrence: two effective twists at the same vertex either use the zero residue or coincide.
A no-wrap prefix calculation #
Before either marked position wraps around its strand, multiplying the marked difference simply advances both points by the same multiplier.
A multiple of one normalized strand prefix is its quotient number of endpoint differences plus its residue prefix.
If two marked strand multiples have the same quotient, their difference is represented by the two residue positions.
The residue representative for an arbitrary evenly-marked theta
multiple. This is the paper's eq:multDiffMarkedPts in normalized
coordinates.
The canonical complement of an evenly-marked multiple has the reflected first residue point and the second residue point as a representative.
Rank support is invariant under linear equivalence of the ambient divisor.
In the no-wrap range, the canonical complement of a multiple of the marked difference is represented by the reflection of the first advanced point together with the second advanced point.
The indicator characterization makes a raw transmission permutation unique.
Any raw transmission permutation is the function underlying the ASP permutation whose slipface is the marked rank surface.
Paper Lemma lem:tauChars, southeast/rank form.
Paper Lemma lem:tauChars, northwest/canonical-complement form.
The geometric support input for evenly marked theta graphs #
A pair of chips on distinct interior strands of a theta has rank support
exactly at the two chip locations. This is paper Corollary cor:suppUV in
the form needed for the non-recurrence argument.
The preceding canonical representative has exactly its two visible interior chips as rank support.
In the nonzero part of its exact period, an evenly-marked theta multiple has a canonical complement with rank support exactly at its two residue points (with the first one reflected).
Two cross-strand two-chip supports are disjoint when the positions on both strands are distinct.
Distinct nonzero residues in one evenly-marked theta period have disjoint canonical rank supports. This is the geometric nonrecurrence input of Lemma 4.15.
Evenly-marked theta marks satisfy the paper's nonrecurrence condition: no vertex is effective in two distinct nonzero torsion-residue twists.
The two evenly-marked theta chips form a rank-zero semibreak divisor. This is the rigidity condition that eliminates the exceptional correction in the genus-two inversion formula.
An evenly-marked pair is not linearly equivalent to the canonical divisor.