2020/10/05 by Yu Yang, Yang, Yu
Mathematics · #14G32 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Primary 14H30 #Secondary 14F35
paper · pdf · doi:10.48550/arxiv.2010.01806
openalex publication_date 2020/10/05 · openalex created_date 2020/10/08 · openalex updated_date 2026/07/28
In this series of papers, we investigate a new anabelian phenomenon of curves over algebraically closed fields of positive characteristic. Let Mg, n be the moduli space of curves of type (g, n) over \mathbbFp. We introduce a topological space Πg, n which can be determined group-theoretically from admissible fundamental groups of pointed stable curves of type (g, n). By introducing a certain equivalence relation ∼fe on the underlying topological space | Mg, n| of Mg, n, we obtain a topological space \mathfrakMg, n:= | Mg, n|/∼fe. Moreover, there is a natural continuous map πg,n\rm adm: \mathfrakMg, n → Πg, n. Furthermore, we pose a conjecture (=the Homeomorphism Conjecture) which says that πg,n\rm adm is a homeomorphism. The Homeomorphism Conjecture generalizes all the conjectures in the theory of anableian geometry of curves over algebraically closed fields of characteristic p. One of main results of the present series of papers says that the Homeomorphism Conjecture holds when dim( Mg, n)=1 (i.e., (g, n)=(0,4) or (g, n)=(1,1)). In the present paper, we establish two fundamental tools to analyze the geometric behavior of curves from open continuous homomorphisms of admissible fundamental groups, which play central roles in the theory developed in the series of papers. Moreover, we prove that π0,n\rm adm([q]) is a closed point of Π0,n when [q] is a closed point of \mathfrakM0, n. In particular, we obtain that the Homeomorphism Conjecture holds when (g, n)=(0, 4).