Brill-Noether theory of stable vector bundles on ruled surfaces (Q6552253)

In this paper, the authors study Brill-Noether theory for a ruled surface. Brill-Noether theory deals with the number of independent sections of stable vector bundles on smooth projective varieties.\N\NLet \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field \(K\) of characteristic \(0\), and let \(M_{X,H}(r; c_1, \ldots, c_s)\) be the moduli space of stable rank-\(r\) vector bundles \(E\) with respect to an ample divisor \(H\) on \(X\) and with fixed Chern classes \(c_i := c_i(E)\) for \(1 \le i \le s := \min \{r, n\}\).\N\NThe Brill-Noether locus \(W_H^k(r; c_1, \ldots, c_s)\) in \(M_{X,H}(r; c_1, \ldots, c_s)\) is defined as the set of stable vector bundles in \(M_{X,H}(r; c_1, \ldots, c_s)\) having at least \(k\) independent sections. Assume further that \(X\) is a ruled surface over a nonsingular curve \(C\) of genus \(g \ge 0\). Let \(C_0 \subset X\) be a section of the ruling such that \(C_0^2\) is the least, and let \(e = -C_0^2\).\N\NThe authors show that for \(m \in \{0, 1\}\), a divisor \(\mathfrak m\) on \(C\) with \(\deg(\mathfrak m) = m\), an integer \(c_2 \gg 0\) and an ample divisor \(H \equiv \alpha C_0 + \beta f\) on \(X\) with \(\alpha(e+m) < \beta\), if\N\[\N\max\{1, g\} \le k < \frac1{2\alpha}[\beta -\alpha(e-m+2g-2)],\N\]\Nthen the Brill-Noether locus \(W^k_H(2;C_0 + \mathfrak mf, c_2)\) is nonempty. In another direction, it is verified that for an integer \(c_2 \gg 0\) and an ample divisor \(H \equiv C_0 + \beta f\) on \(X\), \(W^1_H(2; f, c_2) \ne \emptyset\) and \(W^k_H(2; f, c_2) \ne \emptyset\) when \(k \ge 3\). Moreover, \(W^1_H(2; f, c_2) \backslash W^2_H(2; f, c_2)\) is smooth and has the expected dimension. The main idea of the proofs is to apply the theory of walls and chambers developed by \textit{Z. Qin} [J. Differ. Geom. 37, No. 2, 397--415 (1993; Zbl 0802.14005)].\N\NSection~2 fixes standard notations, proves some basic facts about the cohomology of line bundles on ruled surfaces, recalls the notion of stability of a vector bundle, and presents the general definition of the Brill-Noether locus. Section~3 applies Qin's theory of walls and chambers to describe how Brill-Noether loci change when the ample divisor crosses a wall between two chambers. The authors prove the main results in Section~4.