Standard isotrivial fibrations with [formula omitted], II

Abstract

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 ≔ ( C × F ) / G . Standard isotrivial fibrations of general type with p g = q = 1 have been classified in [F. Polizzi, Standard isotrivial fibrations with p g = 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 p g = q = 1 , K S 2 = 5 and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where S is not minimal actually occurs.