Algebraic surfaces with pg = q = 1,K2 = 4 and nonhyperelliptic Albanese fibrations of genus 4

Abstract

In this article, we study minimal algebraic surfaces with and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type (2, 3) in a -bundle over an elliptic curve. For the surfaces we construct here, the direct image of the canonical sheaf under the Albanese map is decomposable (which is a topological invariant property). Moreover, we prove that all minimal surfaces with and nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of the canonical sheaf under the Albanese map is decomposable are contained in our family. As a consequence, we show that these surfaces constitute a four-dimensional irreducible subset of , the Gieseker moduli space of minimal surfaces with . Moreover, the closure of is an irreducible component of .