Involutions on surfaces with pg=q=1p_g = q = 1

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g = q = 1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i$. A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g = q = 1$ and $K^2 = 7$ having birational bicanonical map is obtained

Standard isotrivial fibrations with pg = q = 1, II

A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G which acts faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T {colon equals} (C × F) / G. Standard isotrivial fibrations of general type with pg = q = 1 have been classified in [F. Polizzi, Standard isotrivial fibrations with pg = q = 1, J. Algebra 321 (2009),1600-1631] under the assumption that T has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where S is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with pg = q = 1, KS2 = 5 and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where S is not minimal actually occurs. © 2009 Elsevier B.V. All rights reserved