  Tensor triangular geometry as introduced by Balmer is a powerful idea which
can be used to extract the ambient geometry from a given tensor triangulated
category. In this paper we provide a general setting for a compactly generated
tensor triangulated category which enables one to classify thick tensor ideals
and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak
g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a
Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate
that this topological space governs the tensor triangular geometry for the
category of finite dimensional ${\mathfrak g}$-modules which are semisimple
over ${\mathfrak g}_{\bar{0}}$.
