Surfaces of general type whose canonical map is composed of a pencil of genus 3 with small invariants (Q1290852)

For a surface of general type with canonical system composed with a pencil of genus \(b\), if the curves of the pencil have genus \(g\) at least \(3\), \(K^2 \geq 4 p_g + 4 (b-1)\) with few exceptions [cf. \textit{F. Catanese}, in: Algebraic Geometry, Proc. Summer Res. Inst., Brunswick 1985, part I, Proc. Symp. Pure Math. 46, 175-194 (1997; Zbl 0656.14021) and \textit{A. Beauville}, Invent. Math. 55, 121-140 (1979; Zbl 0403.14006)]. In this paper the authors prove that, if \(b=0\) and \(p_g \geq 3\) and the inequality does not hold, then \(K^2=7\) and \(p_g=3\), excluding the other unknown cases \(p_g=3\), \(K^2=6\) and \(p_g=4, K^2=9\). Interesting is the example with \(K^2=7\) and \(p_g=3\), which is given in three different ways: as hypersurface in a suitable \({\mathbb P}^2\)-bundle over \({\mathbb P}^1\); as Galois triple cover of a suitable blow up of the Hirzebruch surface \(F_3\); as a minimal resolution of a sextic with three singular points, one with local equation \(x^2+y^3+z^6=0\), two with local equation \(x^2+y^6+z^6=0\). All the constructions are explicit. The non-existence theorems are obtained carrying out a very detailed analysis of the possible behaviour of the relative canonical map.