Even canonical surfaces with small \(K^ 2\). II
[For part I of this paper see Nagoya Math. J. 129, 115-146 (1993; Zbl 0780.14020).] A surface \(S\) is called canonical if it is minimal and its canonical divisor defines a birational map onto its image \(X\) in \(\mathbb{P}^{p_g-1}\) (where \(p_g\) is the geometric genus of \(S)\). Let \(Q(X)\) be the quadric hull of \(X\), that is, the intersection of all quadrics containing \(X\). It was conjectured by Reid that a (complex) canonical surface with \(K^2< 4p_g-12\) has a three-dimensional quadric hull \(Q(X)\). In part I (loc. cit.), \textit{K. Konno} gave a partial positive answer to Reid's conjecture. More precisely, if \(S\) is even (i.e. the canonical divisor can be written as \(K=2L)\) and \(q\) is the irregularity of \(S\), he proved that the conjecture is true when \(q=0\). He also proved that if \(S\) is even, \(K^2\leq 4p_g-12+q\) and \(S\) contains a pencil of trigonal or plane quintic curves, then \(Q(X)\) has dimension three. In the paper under review the author improves his previous work. In particular, he classifies even canonical surfaces with \(K^2=4p_g-12\) and \(Q(X) =X\), showing that they must be one of the following: -- Some explicit weighted complete intersection. -- A hyperquartic section of a normal Del Pezzo surface. -- A surface with a pencil of non-hyperelliptic curves of genus 7 whose general member has a \(g^2_6\). -- A double covering of a \(K3\) surface. -- A surface with a pencil of non-hyperelliptic, non-trigonal curves of genus 5. He also proves that, if \(q>0\), then \(S\) has a pencil of trigonal curves and the irreducible component of \(Q(X)\) containing \(X\) has dimension three. The above classification is done by studying the possible behavior of the map defined by the line bundle \(L\).
scientific article; zbMATH DE number 826767