Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

Abstract

Abstract. This paper is devoted to the irregular surfaces of general type having the smallest invariants, p g = q = 1. We consider the still unexplored case K 2 = 4, classifying those whose Albanese morphism has general fibre of genus 2 and such that the direct image of the bicanonical sheaf under the Albanese morphism is a direct sum of line bundles.We find 8 unirational families, and we prove that all are irreducible components of the moduli space of minimal surfaces of general type. This is unexpected because the assumption on the direct image bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension strictly bigger than the expected one.

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

“…Over a Zariski-open subset S • ⊂ S containing ξ this D is relatively ample, and the proposition follows by applying Theorem 6.2 to the restriction of the family X to S • . (f) Surfaces with q = 1 and K 2 = 4 in any of the eight moduli components described by Pignatelli in [26]. (g) Surfaces with q = 1 and K 2 = 8 whose bicanonical map is not birational.…”

Involutions on surfaces withp g =q = 1

“…Several authors have studied surfaces of general type with p g = q = 1 ( [6,7,8,9,10,17,18,16,15]), but these surfaces are still not completely understood.…”

Standard isotrivial fibrations with pg=q=1, II

“…• Case (2i). G = GL 2 (F 3 ), m = (2, 3,8), g(C ) = 21, Sing(T ) = 2 × 1 2 (1, 1) + 1 3 (1, 1) + 1 3 (1, 2).…”

“…We denote by g alb the genus of a general Albanese fibre of S. A classification of surfaces with K 2 S = 2, 3 has been obtained by Catanese, Ciliberto, Pignatelli in [1][2][3][4]. For higher values of K 2 S some families are known, see [5][6][7][8][9][10][11]. As the title suggest, this paper considers surfaces with p g = q = 1 which are standard isotrivial fibrations.…”