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

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. 0. Introduction In [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 2 S = 2 is studied in [Ca81], whereas [CaCi91] and [CaCi93] deal with the case K 2 S = 3. For higher values of K 2 S only some sporadic examples

On the Tate and Mumford–Tate conjectures in codimension 1 for varieties with h2,0=1

“…For the surfaces in (g), which make up three irreducible components of M 8,1,1 , we use the results in [29] and [7]. The surfaces in question are of the form S = (C × F )/G, where C and F are curves with a faithful action of a finite group G such that the diagonal action on C × F is free.…”

Surfaces with $p_g = q= 1$, $K^2 = 7$ and non-birational bicanonical mpas

“…Abusing notations, we also denote the composed map X → Y → S by π. By use of formula 2.1, X has the following invariants χ(X) = 2, p g (X) = h 0 (S, 2K S − L) + p g (S) = 5, q(X) = 4, K 2 X = 16 Then arguing as in the proof of Lemma 1.6 in [Bor2], we get a fibration f : X → B where B is a curve of genus ≥ 2, a fibration g : S → C and a double cover π ′ : B → C such that g • π = π ′ • f . Note that the two fibrations f and g have the same genus.…”

“…• the image of ψ : P → P 7 , where P is the blow-up of P 2 at two points P 1 , P 2 and ψ is given by the linear system | − K P |. First remark that the map f in the proof of Lemma 1.5 of [Bor2] factors through a morphism f : S → C since g(C) ≥ 1. The argument applied to f : S → C shows that the first case does not occur.…”

“…When we studied the degree of φ, we notice that since p g (S) = 1, the unique effective canonical divisor is nearly "determined", then we got much information using double cover trick and ramification formula. This method also applies when we consider the case K 2 = 8, so we give an alternative simpler proof of the results in [Bor2] (cf. Remark 3.7).…”

“…[Pol]); and Borreli proved it is the case as in [Pol] indeed, hence the bicanonical map is of degree 2 and factors through a double cover over a rational surface (cf. [Bor2]). The results are analogous to the cases when p g = q = 0, K 2 = 9, 8.…”