  Let $M$ denote the space of complete conics. We compute the cone of effective
and numerically effective $k$-cycles of $M$, $\mathrm{Eff}_k(M)$ and
$\mathrm{Nef}_k(M)$, respectively. In addition, we compute the
Bia\l{}ynicki-Birula cell-decomposition of $M$ with respect to a
$\mathbb{C}^*$-action and compare the cone generated by the closure of these
cells to the cones $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$.

Algebraic Geometry