Elementary rows in the theta transmission case table #
This module starts the direct rank-difference portion of Proposition 4.5. The remaining task is the exhaustive classification of the default case; the exceptional rows below are graph-independent once their stated divisor-class conditions hold.
The marked second rank difference of the zero divisor is one.
The first row in Proposition 4.5: if the degree-two twist at t is
2u, then the transmission permutation takes t to t - 2.
A one-chip divisor has marked second difference one when its class is different from both marked one-chip classes.
The second row in Proposition 4.5: if the degree-two twist at t is
u+w with w in neither marked one-chip class, then the transmission value
is t - 1.
An effective degree-two pair not in the canonical class has rank zero on a theta graph.
The third row in Proposition 4.5: if the degree-two twist at t is
v+w, and neither pair involving w is canonical, then the transmission
value is t + 1.
The fourth row in Proposition 4.5: writing the reflected class of u as
K-u, a twist equivalent to v + (K-u) forces transmission value t + 2.
The exhaustive default row #
Proposition 4.5's t - 2 exceptional class, stated independently of a
chosen transmission permutation.
Equations
Instances For
Proposition 4.5's t - 1 exceptional class. The two inequalities in
the paper mean that the residual chip belongs to neither marked degree-one
class.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Proposition 4.5's t + 1 exceptional class. Saying that w is
neither reflected marked point is precisely saying that neither marked pair
with w is canonical.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Proposition 4.5's t + 2 exceptional class, with the reflected class
bar u written coordinate-freely as K-u.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The fifth row of Proposition 4.5 at the rank-theoretic level. On a
rigid theta graph, a degree-two class outside the four exceptional classes
has rank pattern (0,-1,-1,-1) after deleting neither, either, or both
marked chips.
The exhaustive default row of Proposition 4.5: if the degree-two twist
at t belongs to none of the four exceptional divisor classes, its
transmission value is t.
TeX label: prop-thetaTransChar (Proposition 4.5).
The complete rowwise transmission table for a rigid theta marking. The four
exception predicates are the coordinate-free versions of the paper's classes
2u, u+w, v+w, and v+\bar u; the final implication is the exhaustive
default row. Their mutual exclusivity is not needed to state or use the
table, since each implication is proved directly from the corresponding rank
difference.