Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Moduli spaces of fundamental groups of curves in positive characteristic
  I   In this series of papers, we investigate a new anabelian phenomenon of curves
over algebraically closed fields of positive characteristic. Let $\overline
M_{g, n}$ be the moduli space of curves of type $(g, n)$ over $\overline
{\mathbb{F}}_p$. We introduce a topological space $\overline \Pi_{g, n}$ which
can be determined group-theoretically from admissible fundamental groups of
pointed stable curves of type $(g, n)$. By introducing a certain equivalence
relation $\sim_{fe}$ on the underlying topological space $|\overline M_{g, n}|$
of $\overline M_{g, n}$, we obtain a topological space $\overline
{\mathfrak{M}_{g, n}}:= |\overline {M_{g, n}}|/\sim_{fe}$. Moreover, there is a
natural continuous map $\pi_{g,n}^{\rm adm}: \overline {\mathfrak{M}}_{g, n}
\rightarrow \overline \Pi_{g, n}.$ Furthermore, we pose a conjecture (=the
Homeomorphism Conjecture) which says that $\pi_{g,n}^{\rm adm}$ 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 $\text{dim}(\overline M_{g, n})=1$
(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
$\pi_{0,n}^{\rm adm}([q])$ is a closed point of $\overline \Pi_{0,n}$ when
$[q]$ is a closed point of $\overline {\mathfrak{M}}_{0, n}$. In particular, we
obtain that the Homeomorphism Conjecture holds when $(g, n)=(0, 4)$.

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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin