Positive divisors and Poincaré series on variable Riemann surfaces (Q1821230)

Let \(\pi:V_ p\to T_ p\) be the universal Teichmüller curve over the Teichmüller space \(T_ p\) of closed Riemann surfaces of genus \(p\geq 2\). The zero set of a suitable holomorphic section of a line bundle \(L\to V_ p\) defines a positive divisor on each closed Riemann surface \(\pi ^{-1}(t)\), \(t\in T_ p\). An explicit construction of such sections by Poincaré series produces divisors of any degree \(\geq 4p-4\), and linear algebra then produces divisors of any degree \(\geq 2p-2\), including divisors of Prym differentials. \textit{L. Bers} [Bull. Am. Math. Soc. 67, 206-210 (1961; Zbl 0102.067)] constructed canonical divisors by essentially the same methods.