Brill-Noether theory for rank two vector bundles generated by their sections
The paper under review is a contribution to the Brill-Noether theory of vector bundles on algebraic curves [see \textit{I. Grzegorczyk and M. Teixidor I. Bigas}, \url{arXiv:0801.4740}] for an overview of the main known results about stable vector bundles. Let \(C\) be a reduced irreducible smooth non-hyperelliptic complex projective algebraic curve of genus \(g \geq 3\). A rank two vector bundle \(E\) on \(C\) is said to be \textit{generated by its sections} if \(E\) is an extension \(0 \to L_1 \to E \to L_2 \to 0\) of a line bundle \(L_2\) by a line bundle \(L_1\) such that \(H^0 (C,L_1) \neq 0\) and \(Im \{ H^0 (C,E) \to H^0 (C,L_2) \} \neq 0\). Recall also that a bundle \(E\) is called \textit{special} if both \(H^0 (C, E)\) and \(H^1 (C, E)\) are nonzero. The authors first construct a holomorphic vector bundle \(H_d\) of dimension \(d\) on \(C_d\) such that each point \((D, f) \in H_d\), \(D \in C_d\) and \(f \in H^0 (C, {O_C}_{|D})\), uniquely corresponds to a base point free rank two vector bundle \(E\) generated by its sections. Thus \(H_d\) can be regarded as a parameter space of all base point free rank two vector bundles of degree \(d\) on \(C\) generated by their sections. Then they obtain a condition for splitting of \(E\) in terms of the decomposition of \(D\) as a sum of effective divisors. Further, they prove that if \(E\) is indecomposable, generated by its sections, and \(d > 6g - 6\) then \(H^1 (C, E) = 0\), while in case \(d = 6g - 6\) there exists a unique \(E\) with these properties and such that \(H^1 (C, E) \neq 0\). Next, they produce two Clifford-like results according to whether the bundle splits, i.e. they find upper bounds for \(\dim H^0 (C, E)\) in terms of its degree and the genus \(g\). Finally, the authors prove that the existence of rank two vector bundles on a general curve depends on the non-negativity of the Brill-Noether number \(\rho (g,r,d) = g - (r+1)(g-d+r)\). Precisely, denote by \(H^r_d = \{ E \in H_d | \dim H^0 (C,E) \geq r + 1 \}\), and let \(C\) be a general curve. Then: (i) if \(r \geq 0\) and \(2 \leq d \leq 2g - 2\) there exists \(E \in H^r_d\) such that \(E / O_C\) is base point free and \(E \neq O_C \oplus E / O_C\) if and only if \(\rho (g, r, d) \geq 0\); (ii) if \(r \geq 0\) and \(1 \leq d \leq 2g - 2\) there exists \(E \in H^r_d\) such that \(E/{O_C}\) has a base point if and only if \(\rho (g, r - 1, d - 1) \geq 0\); (iii) if \(r +1 \geq d-2g +2 > 0\) and \(d\) is an even number there exists \(E \in H^r_d\) such that \(E\) is indecomposable if and only if \(\rho (g, r - {d \over 2} , {d \over 2} - g + 1) \geq 0\); (iv) if \(r + 1 \geq d - 2g + 2 > 0\) and \(d\) is an odd number there exists \(E \in H^r_d\) such that \(E\) is indecomposable if and only if \(\rho (g, r - {d-1 \over 2} , {d+1 \over 2} - g + 1) \geq 0\).
scientific article; zbMATH DE number 5485344