The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves (Q5947471)

The author proves the following main result: Let \(m\geq 1\), \(n\geq 1\), \(g\geq 0\) be integers. Then, for a general smooth projective curve \(X\) of genus \(g\), and a general pair \((L,M)\in\text{Pic}^{g+m}X\times \text{Pic}^{g_n} X\), the multiplication map \(H^0(L)\otimes H^0(M)\to H^0(L\otimes M)\) is injective if \(g\geq mn\) and surjective if \(g\leq mn\). The idea of the proof is to construct inductively smooth curves \(X(m,n)\) of genus \(mn\), embedded in \(\mathbb{P}^m \times\mathbb{P}^n\) by a pair of line bundles \((L,M)\), of bidegree \((mn+m, mn+n)\) such that \(H^1(L)=0\), \(H^1(M)=0\) and such that the multiplication map \(H^0 (L) \otimes H^0(M)\to H^0(L\otimes M)\) is bijective. -- The author further uses this construction to study the multiplication map for line bundles on nodal or cuspidal curves and for rank 1 torsion free sheaves on nodal curves.