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

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 Quillens plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.