Dyson's lemma for products of two curves of arbitrary genus (Q1116967)

Let \(C_1\), \(C_2\) be smooth irreducible complex curves of genus \(g_1\), \(g_2\), respectively. Let \({\mathcal L}_1\), \({\mathcal L}_2\) be invertible sheaves on \(C_1\), \(C_2\) respectively of degrees \(d_1\), \(d_2\); \(d_1\ge d_2\). Let \(\pi_i: C_1\times C_2\to C_i\) be the projection morphisms and let \(s\) be a non-identically vanishing section of \(\pi^*_i{\mathcal L}_1\otimes \pi^*_2{\mathcal L}_2\). Let \(\xi_1,\ldots,\xi_m\) be points on \(C_1\times C_2\) with \(\pi_i(\xi_j)\ne \pi_i(\xi_k)\); \(i=1,2\), \(1\le j<k\le m\). Then, using the index and volume notation of the classical Dyson lemma, we have \[ \sum^{m}_{h=1} \text{Vol}_ h\quad \le 1+\frac{d_2}{2d_1}\max (2g_1-2+m,0). \] The proof uses a modification of Viola's approach. [Cf. \textit{C. Viola}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 105--135 (1985; Zbl 0596.10032)].