On multiply of sections of line bundles on compact Riemann surfaces (Q943421)

The author gives some necessary and sufficiency conditions on the surjective of multiply maps \(H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K),\) \((s, t) \rightarrow s \cdot t.\) Here \(R\) is a compact Riemann surface, \(L\) a holomorphic line bundle and \(K\) is the canonical line bundle on \(R.\) A line bundle \(L\) on \(R\) is called base point free, if sections in \(H^{0}(R, L)\) do not have a common zero point. Theorem 1.1. For any compact Riemann surface \(R,\) the multiply maps \(H^{0}(R, L) \times H^{0}(R, K)\rightarrow H^{0}(R, L \otimes K)\) is surjective if and only if \(L\) is not a composite line bundle. Theorem 1.2. For a generic compact Riemann surface \(R,\) if the genus \(g\) of \(R\) is odd, then the map \(H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K)\) is surjective if and only if \(L\) is base point free; if the genus \(g\) of \(R\) is even, and \(L\) is not the form \(L_{1} \otimes L_{2}\) with \(\deg(L_{1}) = \deg(L_{2}) = \frac{g}{2} + 1,\) and \(\dim H^{0}(R, L_{1}) = \dim H^{0}(R, L_{2}) = 2;\) \(\dim H^{0}(R, L_{1} \otimes L_{2}) = 3,\) then the map \[ H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K) \] is surjective if and only if \(L\) is base point free.