Unlikely intersections in families of polynomial skew products

Abstract

Motivated by the study of unlikely intersection in the moduli space of rational maps, we initiate our investigation on algebraic dynamics for families of regular polynomial skew products in this article. Our goals are threefold. (1) We classify special loci -- which contain a Zariski dense set of postcritically finite points -- in the moduli space of quadratic regular polynomial skew products. More precisely, special loci include families of homogeneous polynomial endomorphisms, families of split endomorphisms, and polynomial endomorphisms of the form $(x^2,y^2+bx)$ up to conjugacy. As a consequence, we verify a special case of a conjecture proposed by Zhong. (2) Let $F_t$ be a family of regular polynomial skew products defined over a number field $K$ and let $P_t, Q_t\in K[t]\times K[t]$ be two initial marked points. We introduce a good height $h_{P_t}(t)$ which is built from the theory of adelic line bundles for quasi projective varieties. We show that the set of parameters $t_0\in \overline{K}$ for which $P_{t_0}$ and $Q_{t_0}$ are simultaneously $F_{t_0}$-preperiodic is infinite if and only if $h_{P_t}=h_{Q_t}$. (3) As an application of $h_{P_t}$, we show that, under some degree conditions of $P_t$, if there is an infinite set of parameters $t_0$ for which the marked point $P_{t_0}$ is preperiodic under $F_{t_0}$, then the Zariski closure of the forward orbit of $P_t$ lives in a proper subvariety of $\mathbb{P}^2$. As a by-product, we conditionally verify a special case of a conjecture of DeMarco--Mavraki which is a relative version of the Dynamical Manin--Mumford Conjecture.