On certain even canonical surfaces
According to \textit{E. Horikawa} [Tôhoku Math. J., II. Ser. 43, No. 1, 141-148 (1991; Zbl 0748.14014)], we call a minimal surface a canonical surface if the canonical map induces a birational map onto its image. Canonical surfaces with \(c^ 2_ 1 =3p_ g-7\) and \(3p_ g-6\) were studied in our previous papers [cf. \textit{T. Ashikaga} and the author, ibid. 42, No. 4, 517-536 (1990; Zbl 0735.14026) and the author, Math. Ann. 290, No. 1, 77-107 (1991; Zbl 0711.14021)]. These are regular surfaces whose canonical linear system \(| K |\) has neither fixed components nor base points. In this article, we list up those which are even surfaces in order to supplement our cited papers. Here, we call a compact manifold of dimension 2 an even surface if its second Stiefel-Whitney class \(w_ 2\) vanishes. This topological condition implies the existence of a line bundle \(L\) with \(K=2L\). In a recent paper, \textit{E. Horikawa} [cf. Topology 32, No. 4, 757-772 (1993)] classified all the even surfaces with \(p_ g=10\), \(q=0\) and \(K^ 2=24\) (numerical sextic surfaces). Following Horikawa (loc. cit.), we consider the rational map \(\Phi_ L\) associated with \(| L |\) also in the remaining cases. Recall that most canonical surfaces with \(c^ 2_ 1=3p_ g-7\), \(3p_ g-6\) have a pencil \(| D |\) of non-hyperelliptic curves of genus 3. Therefore, it is naturally expected that \(\Phi_ L\) should be composed of such a pencil. We show that this is the case, except for numerical sextic surfaces. Let \(f:S \to \mathbb{P}^ 1\) be the corresponding fibration. It turns out that the fact that \(S\) is an even surface forces \(f_ *{\mathcal O} (K)\) to be very special (lemmas 1.2 and 2.2). Using this, we can determine the fixed part \(Z\) of \(| L |\). The remaining problem is to write down the equation of the canonical model. As a by-product, we find that the moduli space is non-reduced in many cases (theorems 1.5 and 2.5). The point is the presence of a (--2)-curve contained in \(Z\). Then a general result of \textit{D. M. Burns} jun. and \textit{J. M. Wahl} [Invent. Math. 26, 67-88 (1974; Zbl 0288.14010)] can be applied to show that the Kuranishi space is everywhere singular.
scientific article; zbMATH DE number 570303