In this well-written research monograph the author studies Moishezon twistor spaces on connected sums of complex projective planes.
Let us recall the notation and basic definitions before formulating the main result. Let \(M\) be a \(4\)-manifold equipped with a self-dual conformal structure and let \(Z\) be the associated twistor space. Write \(\pi : Z \rightarrow M\) for the projection. The fibers of \(\pi\) are complex submanifolds of \(Z\) and are isomorphic to \(\mathbb{P}^{1}_{\mathbb{C}}\). These lines are called the twistor lines of the twistor space \(Z\). Recall that the anti-canonical bundle \(-K_{Z}\) of \(Z\) is of degree \(4\) over a twistor line. In particular, \(\kappa(Z) = -\infty\) for the Kodaira dimension when \(Z\) is compact. In a neighborhood of a twistor line, there exists a \(4\)-th square root of \(-K_{Z}\) and it exists globally when the base \(4\)-manifold \(M\) admits a spin structure. However, even if \(M\) is not spin, from the construction of a twistor space, there always exists a global and natural square root of \(-K_{Z}\) as a holomorphic line bundle. This is called the vertical line bundle or the fundamental line bundle, and is of degree two over a twistor line. We denote it by \(F(=K_{Z}^{-1/2})\) and we call the complete linear system \(|F|\) the fundamental system. An element of the fundamental system is called a fundamental divisor. The main result is a classification and a description of Moishezon twistor spaces on \(n\mathbb{P}^{2}_{\mathbb{C}}\) with \(n\geq 4\), where \(n\mathbb{P}^{2}_{\mathbb{C}}\) denotes the connected sum of \(n\) copies of \(\mathbb{P}^{2}_{\mathbb{C}}\). It can be formulated as follows.
Theorem. Let \(n\geq 4\) and \(Z\) be a Moishezon twistor space on \(n\mathbb{P}^{2}_{\mathbb{C}}\). Suppose that \(\dim |F| = 1\) and \(Z\) is not a generalized LeBrun space. Then there exists \(m\geq 2\) such that the pluri-system \(|mF|\) includes an \((m+2)\)-dimensional sub-system whose meromorphic map \(\Phi : Z \rightarrow \mathbb{P}^{m+2}_{\mathbb{C}}\) satisfies the following properties:
(i) The image \(\Phi(Z)\) is a scroll of planes over a rational normal curve in \(\mathbb{P}^{m}_{\mathbb{C}}\);
(ii) The meromorphic map \(\Phi: Z \rightarrow \Phi(Z)\) is two-to-one and the branch divisor is a cut of the scroll by a quartic hypersurface in \(\mathbb{P}^{m+2}_{\mathbb{C}}\);
(iii) The quartic hypersurface is defined by an equation of the form \[ h_{1}h_{2}h_{3}h_{4} = Q^{2}, \] where \(h_{i}\) are linear and \(Q\) is quadratic.
Furthermore, the integer \(m\) necessarily satisfies \(m \geq n-2\).