Surfaces of general type with 𝑝_{𝑔}=𝑞=1,𝐾²=8 and bicanonical map of degree 2

Abstract

Abstract. We classify the minimal algebraic surfaces of general type with p g = q = 1, K 2 = 8 and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, i.e. if S is such a surface, then there exist two smooth curves C, F and a finite group G acting freely on C × F such that S = (C × F )/G. We describe the C, F and G that occur. In particular the curve C is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map φ of S is composed with the involution σ induced on S by τ × id : C × F −→ C × F , where τ is the hyperelliptic involution of C. In this way we obtain three families of surfaces with p g = q = 1, K 2 = 8 which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space M of surfaces with p g = q = 1, K 2 = 8. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val. IntroductionIn [Par03] R. Pardini classified the minimal surfaces S of general type with p g = q = 0, K 2 S = 8 and a rational involution, i.e. an involution σ : S −→ S such that the quotient T := S/σ is a rational surface. All the examples constructed by Pardini are isogenous to a product, i.e. there exist two smooth curves C, F and a finite group G acting faithfully on C, F and whose diagonal action is free on the product C × F , in such a way that S = (C × F )/G. Pardini's classification contains five families of such surfaces; in particular, four of them are irreducible components of the moduli space of surfaces with p g = q = 0, K 2 S = 8, and represent the surfaces with the above invariants and non-birational bicanonical map.In this paper we deal with the irregular case, in fact we study the case p g = q = 1, K 2 S = 8. Surfaces with p g = q = 1 are the minimal irregular surfaces of general type with the lowest geometric genus, therefore it would be very interesting to obtain their complete classification; for such a reason, they are currently an active topic of research. However, such surfaces are still quite mysterious, and only a few families have been hitherto discovered. If S is a surface with p g = q = 1, then 2 ≤ K 2 S ≤ 9; the case K

Surfaces with p g = q = 1, K 2 = 8 and nonbirational bicanonical map

“…The case when ϕ 2 factors through an involution. The following proposition fix the gap of [18] (cf. the introduction).…”

“…Surfaces with K 2 S = 8 have been studied by Polizzi in [18], who considered those whose bicanonical map factors through a double cover onto rational surface. He proved that in this case ϕ 2K has degree 2 and that these surfaces are quotient of the product of two curves by the action of a finite group.…”

“…In [18] it is also claimed that if the bicanonical map of a minimal surface with p g = q = 1 and K 2 S = 8 factors through a double cover onto a surface X, then X is rational. For this assertion the author refers to a theorem by Xiao ([22,Theorem 3]), which unfortunately has a gap, as pointed out by Rito (cf.…”

“…In Section 1 we recall some facts which will be used in the paper. In Section 2 we first fix the gap in [18], and the we study the case when the bicanonical map has degree 4. In the last section, first we prove Theorem 0.1 and then, as application, we provide the classification of surfaces with p g = q = 1, K 2 S = 8 and nonbirational bicanonical map.…”

Bicanonical map of surfaces with χ = 1 fibered by hyperelliptic curves of genus 3

“…Let r be the rotation. In [23] and [24] F. Polizzi shows that for any g ∈ {2, 3, 4, 5} there exist a curve F of genus 3 and a curve C of genus g such that • G acts on F and C, and hence acts diagonally on the product F × C, i.e. g(x, y) = (g(x), g(y)), for g ∈ G and (x, y) ∈ F × C, • F/G (respectively C/G) is a smooth elliptic (respectively rational) curve,…”