Half-canonical divisors on variable Riemann surfaces
The main result of this paper is the following theorem: Let \(\pi_{p-1}: S_ T^{p-1}(V_ p)\to T_ p\), \(p\geq 2\), be the fiber space whose fiber over a point t of the Teichmüller space \(T_ p\) is \(\pi^{- 1}_{p-1}(t)=S^{p-1}(X_ t)\)- the (p-1)-fold symmetric product of the Riemann surface \(X_ t=\pi^{-1}(t)\) represented by t. Then \(\pi_{p- 1}\) has a half-canonical holomorphic section s (i.e. for all \(t\in T_ p\) a divisor 2s(t) is the divisor of a nontrivial Abelian differential of the first kind) if and only if \(p=2\), 3, or 4. The number of such sections is precisely 6 for \(p=2\), 28 for \(p=3\), and 120 for \(p=4\); that is, precisely the number of odd half periods in the Jacoby variety. The general problem of determining all sections of \(\pi_ n\) for arbitrary n is open; \textit{L. Bers} [Bull. Am. Math. Soc. 67, 206-210 (1961; Zbl 0102.067)] has constructed holomorhic sections of \(\pi_{2p- 2}\). Also it would be of interest to determinate the lowest value of n for which \(\pi_ n\) has holomorphic section. (The authors note that the second author's claim [Bull. Am. Math. Soc., New Ser., 4, 143-179 (1981; Zbl 0457.32011)] that \(\pi_{p-1}\) always has holomorphic sections was too optimistic. Such a result remains unknown). See also \textit{J. H. Hubbard}, Mem. Am. Math. Soc. 166, 1-137 (1976; Zbl 0318.32020); the authors, Contribut. Anal. 107-124 (1974; Zbl 0307.32016) and Acta Math. 137, 49-79 (1976; Zbl 0331.32018); the first author, Ann. Math., II. Ser. 107, 255-286 (1978; Zbl 0353.32027).
scientific article; zbMATH DE number 3969064