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We obtain geometric models for the infinite loop spaces of the motivicspectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They aremotivically 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 lcipoints (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$,and $+$ is Quillen's plus construction. Moreover, we show that the plusconstruction is redundant in positive characteristic.

On the infinite loop spaces of algebraic cobordism and the motivic sphere