On certain even canonical surfaces

TLDR: In this paper, the moduli space is non-reduced in many cases and the rational map associated with a semi-canonical bundle induces a linear pencill of nonhyperelliptic curves of genus three.

Abstract: We classify even canonical surfaces on the Castelnuovo lines, and show that the moduli space is non-reduced in many cases. We show that, in most cases, the rational map associated with a semi-canonical bundle induces a linear pencill of nonhyperelliptic curves of genus three, and that a nonsingular rational curve with self-intersection number —2 appears as a fixed component of the semi-canonical system. By the latter, we can apply a result of Burn and Wahl to show that they are obstructed surfaces. Introduction. According to [8], we call a minimal surface a canonical surface if the canonical map induces a birational map onto its image. Canonical surfaces with c\ = 3pg — l and 3pg — β were studied in our previous papers [1] and [10] (see also [4] and [8]). 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 [1] and [10]. Here, we call a compact complex manifold of dimension 2 an even surface if its second Steifel-Whitney class w2 vanishes ([8]). This topological condition implies the existence of a line bundle L with K=2L. In a recent paper [9], Horikawa classified all the even surfaces with/^ = 10, # = 0 and K = 24 (numerical sextic surfaces). Following [9], we consider the rational map ΦL associated with \L\ also in the remaining cases. Recall that most canonical surfaces with cj = 3pg — Ί, 3pg — 6 have a pencil | Z> | of nonhyperelliptic curves of genus 3. Therefore, it is naturally expected that ΦL should be composed of such a pencil. We show that this is the case, except for numerical sextic surfaces. Let/ : S-+P be the corresponding fibration. It turns out that the fact that S is an even surface forces f*Θ{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. When K = 3pg — 1, we have no difficulty in doing this, since the (relative) canonical image itself is the canonical model. On the other hand, when K = 3pg — 6, we need to study the bi-graded ring 0// o (αZ) + jSZ) as in [9]. The calculation after Lemma 2.3 is a verbatim translation of [9]. 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 Burns and Wahl [3] can be applied to show that the Kuranishi space is everywhere singular. As far as surfaces of general type are concerned, such pathological 1991 Mathematics Subject Classification. Primary 14J29; Secondary 14J15.