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$\mathit{n}\mathrm{\ge }\mathrm{0}\mathrm{,}$ $\mathit{S}{\mathit{P}}^{\mathit{n}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}\mathit{n}$-simplex, $\mathrm{I}{\mathrm{I}}_{\mathit{n}\mathrm{\ge }\mathrm{0}}\mathit{S}{\mathit{P}}^{\mathit{n}}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}\mathrm{,}$ $\mathit{k}$-varieties $\mathrm{(}{\mathit{x}}_{\mathrm{0}}$, . . . , ${\mathit{x}}_{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{)}{\mathrm{\Sigma }}_{\mathit{i}\mathrm{\in }\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{n}{\mathrm{\}}}^{\mathit{X}}\mathit{i}}\mathrm{=}\mathrm{1}$

${\mathit{H}}^{\mathrm{*}}\mathrm{\left(}\mathit{G}\mathrm{\right(}{\mathit{k}}_{\mathit{s}}\mathrm{\left|}\mathit{k}\mathrm{\right)}\mathrm{;}\mathrm{Z}\mathrm{/}\mathrm{2}\mathrm{)}{\mathit{d}}^{\mathit{i}}$ : ${\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\to }{\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{n}}{\mathit{s}}^{\mathit{i}}$ : ${\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{n}}\mathrm{\to }{\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{n}\mathrm{-}\mathrm{1}}$

$\mathrm{k}\mathrm{\ell }\mathit{n}\mathit{n}\mathit{Y}\mathit{Y}\mathrm{Z}\mathrm{i}$. $\mathit{X}$, Z. $\mathrm{=}\mathrm{0}$. $\mathit{P}\mathrm{=}\mathrm{2}\mathit{n}\mathrm{\ge }\mathrm{0}\mathrm{\ell }\mathrm{=}\mathrm{2}$. $\mathit{m}\mathrm{=}\mathit{n}\mathrm{,}$ ${\mathrm{R}}^{\mathit{n}\mathrm{+}\mathrm{1}}{\mathit{x}}_{\mathit{i}}\mathrm{\ge }\mathrm{0}{\mathit{L}}_{\mathit{n}}^{\mathit{t}\mathit{o}\mathit{p}}{\mathit{L}}_{\mathit{n}}^{\mathit{t}\mathit{o}\mathit{p}}{\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}\mathrm{.}$.

${\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{n}}\mathrm{,}$ $\mathit{n}\mathrm{\ge }\mathrm{0}\mathrm{,}$ ${\mathit{S}}^{\mathit{n}}{\mathit{L}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left[}\mathrm{Y}\mathrm{\right]}$ ${\mathit{K}}_{\mathrm{2}}^{\mathit{M}}\mathrm{\left(}\mathit{k}\mathrm{\right)}$ /2 ${\mathit{L}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left[}{\mathit{S}}^{\mathit{n}}\mathrm{\right]}$ ${\mathit{S}}^{\mathrm{1}}$ A . . A ${\mathit{S}}^{\mathrm{1}}\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{n}}\mathrm{)}$ $\mathrm{\ell }\mathrm{\ne }$ char(k)

$\mathrm{\left(}\mathrm{\right(}{\mathit{x}}_{\mathrm{0}}$, . . . , ${\mathit{x}}_{\mathit{i}}\mathrm{+}{\mathit{x}}_{\mathit{i}\mathrm{+}\mathrm{1}}$, . . . , ${\mathit{x}}_{\mathit{n}}\mathrm{)}\mathrm{,}$ $\mathrm{(}{\mathit{x}}_{\mathrm{0}}\mathrm{,}\mathrm{}\mathrm{.}\mathrm{}\mathrm{.}\mathrm{}\mathrm{.}\mathrm{}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{i}\mathrm{-}\mathrm{1}}\mathrm{,}\mathrm{0}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{i}}\mathrm{,}\mathrm{}\mathrm{.}\mathrm{}\mathrm{.}\mathrm{}\mathrm{.}\mathrm{}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{)}$

$\mathrm{2}\mathrm{\mapsto }\mathit{H}\mathit{o}{\mathit{m}}_{\mathit{t}\mathit{o}\mathit{p}}\mathrm{(}\mathit{X}\mathrm{\times }{\mathrm{\Delta }}_{\mathit{t}\mathit{o}\mathit{p}}^{\mathit{i}}\mathrm{,}\mathrm{}{\mathit{L}}_{\mathit{n}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{)}\mathrm{\left(}\mathit{H}\mathit{o}{\mathit{m}}_{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\right(}\mathrm{\text{-}}\mathrm{,}$ $\mathrm{-}\mathrm{)}$

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