Moduli spaces for dynamical systems with portraits

Source abstract

A portrait P on P N is a pair of finite point sets Y ⊆ X ⊂ P N , a map Y → X , and an assignment of weights to the points in Y . We construct a parameter space End d N [ P ] whose points correspond to degree d endomorphisms f : P N → P N such that f : Y → X is as specified by a portrait P , and prove the existence of the GIT quotient moduli space M d N [ P ] : = End d N / / SL N + 1 under the SL N + 1 -action ( f , Y , X ) ϕ = ( ϕ − 1 ∘ f ∘ ϕ , ϕ − 1 ( Y ) , ϕ − 1 ( X ) ) relative to an appropriately chosen line bundle. We also investigate the geometry of M d N [ P ] and give two arithmetic applications.