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In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.

atti della accademia nazionale dei lincei. rendiconti lincei. matematica e applicazioni

Some (big) irreducible components of the moduli space of minimal surfaces of general type with &lt;em&gt;p&lt;sub&gt;g&lt;/sub&gt;&lt;/em&gt; = &lt;em&gt;q&lt;/em&gt; = 1 and &lt;em&gt;K&lt;sup&gt;2&lt;/sup&gt;&lt;/em&gt; = 4