SURFACES OF GENERAL TYPE WITH pg = q = 0 HAVING A PENCIL OF HYPERELLIPTIC CURVES OF GENUS 3

We prove that the bicanonical map of a surfaces of general type S with pg = q = 0 is non birational if there exists a pencil |F | on S whose general member is an hyperelliptic curve of genus 3. Let S be a minimal surface of general type and let fg : S 99K B be a rational map onto a smooth curve such that the normalization of the general fibre is an hyperelliptic curve of genus g. Then the hyperelliptic involution of the general fibre induces a (biregular) involution σ on S and one has a map of degree two ρ : S 99K Σ onto a smooth ruled surface with ruling induced by fg. It is known (e.g [2],[4]) that if g = 2 than the bicanonical map φ2K of S factors through ρ and moreover, if K S ≥ 10 then the non birationality of φ2K implies the existence of fg with g = 2 ([7]). On the other hand, we proved in [3] that if φ2K factors through a rational map ρ (generically) of degree two onto a ruled surface, then there exists a map fg where g ≤ 4 and, more precisely, if g 6= 2 then we have g = 3 unless KS is ample and q(S) = 0, pg(S) = 1 2 (d − 3)d + 1, K S = 2(d − 3) , d = 4, 5. Finally, we recall that if pg(S) = 0 and K 2 S ≥ 3, then there not exists fg with g = 2 ([8]). In this note we prove that if pg = 0 then the existence of fg with g = 3 implies φ2K non birational. In particular, it follows that if pg = 0 and K 2 S ≥ 3 then φ2K factors through a map of degree two onto a ruled surface if and only if there exists fg with g = 3. To motivate this work we notice that, as far as we know, all the examples of surfaces of general type with pg = 0 and non birational bicanonical map have an fg with g = 3, except one case when K S = 3 and S is a double cover of an Enriques surface. Notation and conventions. We work over the complex numbers. We denote by KS a canonical divisor, by pg = h (S,OS(KS)) = 0 the geometric genus and by q = h(S,OS(KS)) = 0 the irregularity of a smooth (projective algebraic) surface S. The symbol ≡ (resp. ∼) will denote the linear (resp. numerical) equivalence of divisors. A curve on a surface has an [r, r]-point at p if it has a point of multiplicity r at p which resolves to a point of multiplicity r after one blow up. 1. surfaces with a pencil of curves of genus 3 Assumption 1.1. Throughout the end we assume that a) S is a minimal surface of general type with pg = q = 0 and b) f : S 99K P is a rational map with connected fibres such that the normalization of general fibre F is a curve of genus g = 3. This work was supported by bolsa DTI Instituto do Milenio/CNPq.