Some bounds for the number of coincidences of morphisms between closed Riemann surfaces
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 \(\tau^k\).
scientific article; zbMATH DE number 1286692