A model of Brill-Noether theory for rank two vector bundles and its Petri map
Let \(C\) be a smooth irreducible complex projective curve. The author concentrates on rank 2 vector bundles \(E\) on \(C\), of fixed degree \(d>0\), and such that the evaluation map \(\text{H}^0(E)\otimes {\mathcal O}_C \rightarrow E\) is surjective except at finitely many points where it has rank 1. In this case, a general global section \(s\) of \(E\) vanishes nowhere. If \(s^{\prime}\in \text{H}^0(E)\) is linearly independent of \(s\) then the zero locus of the global section \(s\wedge s^{\prime}\) of \(\text{det}E\) is an effective divisor \(D\) on \(C\), of degree \(d\), and \(s^{\prime}| D = f(s| D)\) for some \(f\in \text{H}^0({\mathcal O}_D)\). \(E\) can be described as an extension : \[ 0\rightarrow 2{\mathcal O}_C\overset {u} \longrightarrow E \overset {v} \longrightarrow {\mathcal O}_D[D]\rightarrow 0 \] where \(u = (s,s^{\prime})\) and \(v(t) = (s\wedge t)| D\), or by an exact sequence : \[ 0\rightarrow E[-D]\overset {a} \longrightarrow 2{\mathcal O}_C \overset {b} \longrightarrow {\mathcal O}_D\rightarrow 0 \] where \(a(t) = (s^{\prime}\wedge t,s\wedge t)\) and \(b\) is defined by 1 and \(-f \in \text{H}^0({\mathcal O}_D)\). It follows that these bundles \(E\) can be (over-)parametrized by the geometric vector bundle \(H_d\) over the \(d\)-fold symmetric product \(C_d\), whose fibre over \(D\in C_d\) is \(\text{H}^0({\mathcal O}_D)\). The author shows that the subset \(C^r_{2,d}\) of \(H_d\) corresponding to those \(E\) for which \(\text{h}^0(E)\geq r+1\) has a natural structure of a determinantal subscheme of \(H_d\) and that \(C^r_{2,d}\) is smooth, of the expected codimension, at a point \(E\in C^r_{2,d}\setminus C^{r+1}_{2,d}\) iff the multiplication map \(\text{H}^0(E^*\otimes {\omega}_X)\otimes W\rightarrow \text{H}^0(E^*\otimes {\omega}_X[D])\) (where \(W\) is the image of \(s\wedge - : \text{H}^0(E)\rightarrow \text{H}^0({\mathcal O}_C[D])\)) is injective. This can be proved using the deformation theory of cohomology groups, but the author uses a more concrete method, based on local computations, as in the first paragraph of Chapter IV of [\textit{E. Arbarello, M. Cornalba, P. A. Griffiths} and \textit{J. Harris}, Geometry of Algebraic Curves, I (Grundlehren der mathematischen Wissenschaften 267, Springer, New York) (1985; Zbl 0559.14017)].
scientific article; zbMATH DE number 2113427