Surfaces with \(K^2=7\) and \(p_g=4\) (Q2734764)

In this monograph, the author gives a very precise description of smooth projective minimal surfaces over the complex number field with the invariants \(K^2=7\) and \(p_g=4\). The minimal surface with \(K^2=7\) and \(p_g=4\) is of general type, i.e., of Kodaira dimension 2. By Noether's inequality, we have \(K^2\geq 2p_g-4\). On the other hand, we have \(K^2\geq 2p_g\) for irregular surfaces. Hence the minimal surfaces of general type with \(p_g=4\) and \(K^2<8\) must have \(q=0\) and \(K^2\geq 4\). The cases \(p_g=4\), \(K^2=4,5,6\) have been studied by several authors (e.g. M. Noether, F. Enriques, E. Horikawa). Even the case \(K^2=6\) has become rather complicated and it is unknown until now whether the moduli space of these surfaces is connected or not. At first, these surfaces are divided into several families according to the behavior of the canonical system \(|K|\): whether \(|K|\) has fixed part or has 0, 1, 3 base points. Among these families, two of them has already been described by F. Catanese and F. Zucconi. Then by analysing each family, it is proved that the moduli space \(\mathcal M_{K^2=7, p_g=4}\) has three irreducible components of respective dimension 36, 36 and 38. A very careful study of the deformations of these surfaces makes it possible to show that the two components of dimension 36 have nonempty intersection. But it is not yet possible to decide whether the component of dimension 38 intersects the other two or not. To sum up, the main result of this monograph is the following:NEWLINENEWLINENEWLINE(1) The moduli space \(\mathcal M_{K^2=7, p_g=4}\) has three irreducible components \(\mathcal M_{36}\), \(\mathcal M'_{36}\) and \(\mathcal M_{38}\) with dimension 36, 36, 38, respectively,NEWLINENEWLINENEWLINE(2) \(\mathcal M_{36}\cap\mathcal M'_{36}\neq\emptyset\), hence \(\mathcal M_{K^2=7, p_g=4}\) has at most two connected components.NEWLINENEWLINENEWLINE(3) \(\mathcal M'_{36}\cap\mathcal M_{38}=\emptyset\).