Bounded geometry for PCF-special subvarieties

2024/05/27 by DeMarco, Laura, Mavraki, Niki Myrto, Ye, Hexi
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.17343

Abstract

For each integer d≥ 2, let M_d denote the moduli space of maps f: ℙ¹→ ℙ¹ of degree d. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in M_d. A complex-algebraic subvariety Y ⊂ M_d is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in M_d with degree ≤ D. In addition, there exist constants N = N(D,d) and B = B(D,d) so that for any complex algebraic subvariety X ⊂ M_d of degree ≤ D, the Zariski closure X\capPCF~ has at most N irreducible components, each with degree ≤ B. We also prove generalizations of these results for points with small critical height in M_d(ℚ).

Related

• Special curves and postcritically-finite polynomials
• Sparsity of postcritically finite maps of ℙ^k and beyond: A complex analytic approach
• Geometry of PCF parameters in spaces of quadratic polynomials
• Post-Critically Finite Maps on ℙⁿ for n≥2 are Sparse
• Moduli spaces for families of rational maps on P¹