2016/04/27 by John Lesieutre, Daniel Litt, Lesieutre, John +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1604.08216
openalex publication_date 2016/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that if ϕ: X → X is an automorphism of a smooth projective variety and D ⊂ X is an irreducible divisor for which the set of d in D with ϕn(d) in D for some nonzero n is not Zariski dense, then (X, ϕ) admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of \textrmAut(X), extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. These results follow from a non-reduced analogue of the dynamical Mordell-Lang conjecture. Namely, let ϕ: X → X be an étale endomorphism of a smooth projective variety X over a field k of characteristic zero. We show that if Y and Z are two closed subschemes of X, then the set Aϕ(Y,Z) = \n : ϕn(Y) ⊆ Z\ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of Y.