Thesis abstract
The main subject of this thesis is the study of surfaces of general type $S$ with $p_g=q=1$ having an involution $i.$ For such surfaces one has $2\leq K_S^2\leq 9$ and only few examples with $K^2=2,\ldots,5$ or 8 are known. The quotient surface $S/i$ is a surface with $p_g\leq 1$ and $q\leq 1$ and its Kodaira dimension, ${\rm Kod}(S/i)$, can be any. A list of possibilities for the case ${\rm Kod}(S/i)=-\infty$ and bicanonical map $\phi_2$ composed with $i$ has been given by Xiao in \cite{Xi2}. Here the computational algebra system Magma is used to compute equations of plane models of double planes with $p_g=q=1$ and $K^2=2,\ldots,8.$ For ${\rm Kod}(S/i)\geq 0$ and $\phi_2$ composed with $i,$ we show that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is constructed. This last case was a possibility mistakenly excluded in \cite{Xi2}. For the case $\phi_2$ not composed with $i,$ a list of possibilities is given and several new examples are obtained, mostly as bidouble covers of surfaces. In particular minimal surfaces of general type with $p_g=q=1,$ $K^2=6,7$ and birational bicanonical map are constructed. The case $p_g=1,$ $q=0$ and $S/i$ birational to a $K3$ surface is also considered. It is shown that the smooth minimal model $W$ of $S/i$ is a double plane, with a plane model ramified over two cubics.