Surfaces of general type with \(\chi({\mathcal O}_ S)=1\) and fibrations of genus two. III (Q803222)

This paper is the third part of my thesis [at the Mathematics Institute of Fudan Univ., P.R. China (1987)]. Roughly speaking, the topics discussed in my thesis are so-called surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibrations of genus two. For this kind of surfaces, we may find a double covering over a ruled surface, say P. Therefore, one can classify them (as we did in part I and II) by giving the invariants of P and classify the branch locus of the corresponding double covering. Once we know this basic method, we may use certain results of \textit{Xiao}, \textit{Persson} and \textit{Beauville} for double covering. For example, one can easily know that basically, we only need to classify the surfaces with \(q(S)=p_ g(S)=0, 1\) and 2. For example for surfaces with \(q=0\) in our case, P should be a product of two projective lines. In this situation, using a result of \textit{Horikawa}, one can show that there are only a few choices for the branch curves. - Now we may study the singular points on the branch locus. Suppose its non-fiber part are the summation of curves \(C_ j\). We at first determine the types of \(C_ j's\). Then we give the singularities on \(C_ j's\). Finally, we also show the situation for the intersection among \(C_ i\) and \(C_ j\) for \(i\neq j\). In this sense, we classify all the surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibration of genus 2. As a by-product, we also can give the 2-torsion of the algebraic fundamental group of our surface. Having classified those surfaces, we try to construct them. We pay our most attention to the surfaces with \(p_ g=0\), as they are very interesting. - By our classification, now the self-intersection of the classical sheaf is 1 or 2. Such surfaces do exist by the constructions of \textit{Oort-Peters} and \textit{Xiao} respectively. Now in this part III (under review), we construct another one. We give the exact bipolynomials, which define the curves in the branch locus. By the way, in a paper of \textit{Reid}, there is also constructed another example by our classification. Now by additional work of mine, there are only four types which we do not know if they exist. For this, please see a forthcoming book written by \textit{Xiao} about surfaces with fibrations.