  For a finite abelian group $A$, we determine the Balmer spectrum of
$\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This
generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders
\cite{Balmer-Sanders}, by establishing (a corrected version of) their
log$_p$-conjecture for abelian groups. We also work out the consequences for
the chromatic type of fixed-points and establish a generalization of Kuhn's
blue-shift theorem for Tate-constructions \cite{kuhn}.
