Let \(X\) be a smooth projective curve of genus \(g\geq3\) defined over the complex numbers and let \({\mathcal N}\) denote the moduli space of stable bundles of rank \(2\) and determinant \({\mathcal O}_X(-x)\) for some fixed point \(x\in X\). The author proves that the moduli space \(\overline{\mathbf M}_{0,0}({\mathcal N},2)\) of stable maps of degree \(2\) from \({\mathbb P}^1\) to \({\mathcal N}\) has two irreducible components intersecting transversely. The first component, which he calls the Hecke component, can be identified with Kirwan’s partial desingularisation \(\widetilde{\mathcal M}_X\) of the moduli space \({\mathcal M}_X\) of semistable bundles of rank \(2\) with determinant isomorphic to \({\mathcal O}_X(y-x)\) for some \(y\in X\). The generic point of \(\widetilde{\mathcal M}_X\) corresponds to a Hecke curve; these were introduced by M. S. Narasimhan and S. Ramanan [in: C. P. Ramanujam. – A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 291–345 (1978; Zbl 0427.14002)] in connection with a desingularisation of this moduli space. A Hecke curve is obtained by fixing a bundle \(E\) in the stable part \({\mathcal M}_X^s\) of \({\mathcal M}_X\). Then, for any \(\nu\in{\mathbb P}E_y^\vee\cong{\mathbb P}^1\), let \(E^\nu\) denote the kernel of the composition \(E\to E_y\smash{\mathop{\rightarrow}\limits^{\tilde{\nu}}}\mathbb{C}\), where \(\tilde\nu\) is any lift of \(\nu\) to \(E^\vee_y\). As \(\nu\) varies, the \(E^\nu\) form a family of stable bundles of determinant \({\mathcal O}_X(-x)\) parametrised by \({\mathbb P}^1\) and hence a morphism \({\mathbb P}^1\to {\mathcal N}\); this morphism is an embedding and has degree 2.
The second component is called the extension component and a generic point of this corresponds to a curve which is obtained as follows. Let \(\xi\in\text{Pic}^0X\); then every non-trivial extension \(0\to\xi^{-1}(-x)\to E\to\xi\to0\) defines a bundle \(E\in{\mathcal N}\). Thus we have a morphism \({\mathbb P}\text{Ext}^1(\xi,\xi^{-1}(-x))\to{\mathcal N}\) which is an embedding of degree \(1\). For any conic in \({\mathbb P}\text{Ext}^1(\xi,\xi^{-1}(-x))\), we thus obtain a morphism \({\mathbb P}^1\to{\mathcal N}\) of degree \(2\). It turns out that the extension component \(\widetilde{Q}_J\) can also be identified with a partial desingularisation of a GIT quotient. In fact \(\widetilde{Q}_J\) is itself a moduli space, namely \(\overline{M}_{0,0}({\mathbb P}{\mathcal W}/J,2)\), where \(J=\text{Pic}^0X\) and \({\mathcal W}=R^1\pi_{J*}{\mathcal L}^{-2}(-x)\), with \({\mathcal L}\) being a Poincaré bundle on \(J\times X\). The intersection \(\widetilde{\mathcal M}_X\cap\widetilde{Q}_J=\widetilde{Q}_{\widetilde{X}}\) is yet again a moduli space, this time \(\widetilde{Q}_{\widetilde{X}}=\overline{M}_{0,0}({\mathbb P}{\mathcal W}_0/\widetilde{X},2)\) where \(\widetilde{X}=\{\xi\in J| \xi^2\cong{\mathcal O}_X(y-x)\text{ for some }y\in X\}\) and \({\mathcal W}_0\) is the restriction of \({\mathcal W}\) to \(\widetilde{X}\).
After an introduction which describes in a very clear way the objectives of the paper, the author gives a description of Hecke curves and extension curves and generalises these to higher degree. This is followed by a classification of rational curves in \({\mathcal N}\) (of any degree) which depends on a result of J. E. Brosius [Math. Ann. 265, 155–168 (1983; Zbl 0503.55012); Theorem 1]. The classification is spelt out in detail for degree \(\leq4\). Section 4 discusses stable maps to projective space and gives the identification of \(\widetilde{Q}_J\) with \(\overline{M}_{0,0}({\mathbb P}{\mathcal W}/J,2)\). The author then turns to the Hecke curves and partial desingularisations in order to identify \(\widetilde{\mathcal M}_X\) with the Hecke component of \(\overline{M}_{0,0}({\mathcal N},2)\); this involves a modification of a construction of I. Choe, J. Choy and the author [Topology 44, No. 3, 585–608 (2005; Zbl 1081.14045); §§5, 6]. The proof of the theorem is completed in section 6. In the final section, the author shows how the Hilbert scheme \({\mathbf H}\) and the Chow scheme \({\mathbf C}\) of conics in \({\mathcal N}\) are related to \(\overline{M}_{0,0}({\mathcal N},2)\). In particular \({\mathbf H}\) has two components, both smooth, one of which (the Hecke component) is the desingularisation of \({\mathcal M}_X\) constructed by Narasimhan and Ramanan [loc. cit.].
Some related results have been obtained by A.-M. Castravet [Int. J. Math. 15, No. 1, 13–45 (2004; Zbl 1092.14041)]. The two papers are independent and the only major overlap is the use of the results of Brosius [loc. cit.].