Some bounds for the number of coincidences of morphisms between closed Riemann surfaces
type: Item
instance of: scholarly article
title: Some bounds for the number of coincidences of morphisms between closed Riemann surfaces
zbMATH Open document ID: 0928.30024
DOI: 10.1007/BF02775022
author: Yolanda Fuertes
published in: Israel Journal of Mathematics
publication date: 2000-01-03T00:00:00Z
review text: Let \(M\) and \(M'\) be compact Riemann surfaces of genera \(g\) and \(g'\), respectively. Assume there exist two morphisms \(f_i : M\rightarrow M'\) \((i=1,2)\) of degree \(d_i\). By definition we call \(P\in M\) is a coincidence of \(f_1\) and \(f_2\) provided if \(f_1(P)=f_2(P)\) and \(k:=m_P(f_1,f_2)\) is the multiplicity of \(f_1, f_2\) at \(P\) provided if \(f_1(z)-f_2(z)=c_kz^k+c_{k+1}z^{k+1}+\cdots, c_k\neq 0\) for a parametric coordinate \(z\) at \(P\). We then denote by \(L(f_1,f_2)\), the number of coincidences of \(f_1\) and \(f_2\) counted with multiplicities. In this paper, the author proves: Theorem 1.5. Let \(f_i: M\rightarrow M'\) be two different morphisms of degrees \(d_i\) (\(i=1,2\)) between compact Riemann surfaces of genera \(g\) and \(g'\geq 2\). Assume that \(L(f_1,f_2)>g'(d_1+d_2)\), then the following statements must hold: (i) \(L(f_1,f_2)=(g'+1)(d_1+d_2)=(2g'+2)d_i\). (ii) \(M'\) is a hyperelliptic surface. (iii) \(f_1=J'\circ f_2\), where \(J'\) is the hyperelliptic involution of \(M'\). In case \(M'\) has an automorphism, one obtains finer results. A typical one is: Theorem 2.1. Let \(f_i:M\rightarrow M'\) be two different morphisms of degrees \(d_i\) \((i = 1,2)\) between compact Riemann surfaces of genera \(g\) and \(g'\), respectively. Let \(\tau :M'\rightarrow M'\) be an automorphism on \(M'\), and \(\gamma\) the genus of the quotient surface \(X=M'/\langle \tau \rangle\). Then, we have two cases: (i) If \(X\) is hyperelliptic and \(L(f_1,f_2)> \text{ord}(\tau)(\gamma+1)(d_1+d_2)\), then \(f_1=\tau^k \circ f_2\) and \(L(f_1,f_2)=d_i\nu(\tau^k)\). (ii) If \(X\) is not hyperelliptic and \(L(f_1,f_2)> \text{ord}(\tau)\gamma(d_1+d_2)\), then \(f_1=\tau^k \circ f_2\) and \(L(f_1,f_2)=d_i\nu(\tau^k)\). Here, \(\nu(\tau^k)\) denotes the number of fixed points of \(\tau^k\).
reviewed by: Takao Kato
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 1286692
zbMATH Keywords: compact Riemann surfaces
zbMATH Keywords: conformal automorphism
zbMATH Keywords: Lefschetz number
MaRDI profile type: Publication
cites work: On the number of coincidences of morphisms between closed Riemann surfaces
cites work: Q4002564
cites work: Automorphisms of Compact Riemann Surfaces
cites work: Observations on Morphisms of Closed Riemann Surfaces II
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description: scientific article; zbMATH DE number 1286692