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Abstract

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants $p_g=q=1$ and $K^2=3$, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of $\Bbb{Q}$, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle $\ell$-adic cohomology of the surface.

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