We obtain geometric models for the infinite loop spaces of the motivic
spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are
motivically equivalent to $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where
$\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp.
$\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$,
$\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci
points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$,
and $+$ is Quillen's plus construction. Moreover, we show that the plus
construction is redundant in positive characteristic.
Comment: 13 pages. v5: published version; v4: final version, to appear in
\'Epijournal G\'eom. Alg\'ebrique; v3: minor corrections; v2: added details
in the moving lemma over finite fields