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documentclass{book}



usepackage[silent,nohug,heads=LaTeX,midshaft]{diagrams}



newarrow{Dashto}{}{dash}{}{dash}>

newcommand{undera}{underline{a}}

newcommand{Spec}{mathop{mathrm{Spec}}}



begin{document}



By the Chain Lemma (2),

the function field of $S(root{ell}of gamma)$ splits $undera$.

If $X$ is a norm variety for $undera$,

there is a finite field extension

$F'$ of $k(S(root{ell}of gamma))$ of degree prime to $ell$

and an $F'$-point $Spec(F')to X$. Forming $tilde{S}$ as in

Lemma~3, this $F'$-point extends to a rational map

$phi:tilde{S}(root{ell}of gamma)rDashto^{}X$.



Recall that the cyclic group $C_ell=langlesigmarangle$ acts on

$X^ell$ by $sigma(x_1,...,x_ell)=(x_2,...,x_ell,break x_1)$, and that

$C^ell X$ denotes the geometric quotient variety $X^ell/C_ell$.

Let $sigma$ be a generator of $C_ell$, 

and let $phi:tilde{S}(root{ell}of gamma)rDashto^{}X$

be the rational map mentioned above. 

Choosing an isomorphism $C_ellcongmu_ell$, the rational maps

$phisigma^i$ assemble to form a $C_ell$-equivariant rational map

$g=(phi,phisigma,...,phisigma^{ell-1})$

from $tilde{S}(root{ell}of gamma)$ to $X^ell$.



end{document}

documentclass{book}



usepackage[silent,nohug,heads=LaTeX,midshaft]{diagrams}



newarrow{Dashto}{}{dash}{}{dash}>

newcommand{undera}{underline{a}}

newcommand{Spec}{mathop{mathrm{Spec}}}



begin{document}



By the Chain Lemma (2),

the function field of $S(root{ell}of gamma)$ splits $undera$.

If $X$ is a norm variety for $undera$,

there is a finite field extension

$F'$ of $k(S(root{ell}of gamma))$ of degree prime to $ell$

and an $F'$-point $Spec(F')to X$. Forming $tilde{S}$ as in

Lemma~3, this $F'$-point extends to a rational map

$phi:tilde{S}(root{ell}of gamma)rDashto^{}X$.



Recall that the cyclic group $C_ell=langlesigmarangle$ acts on

$X^ell$ by $sigma(x_1,...,x_ell)=(x_2,...,x_ell,break x_1)$, and that

$C^ell X$ denotes the geometric quotient variety $X^ell/C_ell$.

Let $sigma$ be a generator of $C_ell$, 

and let $phi:tilde{S}(root{ell}of gamma)rDashto^{}X$

be the rational map mentioned above. 

Choosing an isomorphism $C_ellcongmu_ell$, the rational maps

$phisigma^i$ assemble to form a $C_ell$-equivariant rational map

$g=(phi,phisigma,...,phisigma^{ell-1})$

from $tilde{S}(root{ell}of gamma)$ to $X^ell$.



end{document}

documentclass{book}



usepackage[silent,nohug,heads=LaTeX,midshaft]{diagrams}



newarrow{Dashto}{}{dash}{}{dash}>

newcommand{undera}{underline{a}}

newcommand{Spec}{mathop{mathrm{Spec}}}



begin{document}



By the Chain Lemma (2),

the function field of $S(root{ell}of gamma)$ splits $undera$.

If $X$ is a norm variety for $undera$,

there is a finite field extension

$F'$ of $k(S(root{ell}of gamma))$ of degree prime to $ell$

and an $F'$-point $Spec(F')to X$. Forming $tilde{S}$ as in

Lemma~3, this $F'$-point extends to a rational map

$phi:tilde{S}(root{ell}of gamma)rDashto^{}X$.



Recall that the cyclic group $C_ell=langlesigmarangle$ acts on

$X^ell$ by $sigma(x_1,...,x_ell)=(x_2,...,x_ell,break x_1)$, and that

$C^ell X$ denotes the geometric quotient variety $X^ell/C_ell$.

Let $sigma$ be a generator of $C_ell$, 

and let $phi:tilde{S}(root{ell}of gamma)rDashto^{}X$

be the rational map mentioned above. 

Choosing an isomorphism $C_ellcongmu_ell$, the rational maps

$phisigma^i$ assemble to form a $C_ell$-equivariant rational map

$g=(phi,phisigma,...,phisigma^{ell-1})$

from $tilde{S}(root{ell}of gamma)$ to $X^ell$.



end{document}

documentclass{book}



usepackage[silent,nohug,heads=LaTeX,midshaft]{diagrams}



newarrow{Dashto}{}{dash}{}{dash}>

newcommand{undera}{underline{a}}

newcommand{Spec}{mathop{mathrm{Spec}}}



begin{document}



By the Chain Lemma (2),

the function field of $S(root{ell}of gamma)$ splits $undera$.

If $X$ is a norm variety for $undera$,

there is a finite field extension

$F'$ of $k(S(root{ell}of gamma))$ of degree prime to $ell$

and an $F'$-point $Spec(F')to X$. Forming $tilde{S}$ as in

Lemma~3, this $F'$-point extends to a rational map

$phi:tilde{S}(root{ell}of gamma)rDashto^{}X$.



Recall that the cyclic group $C_ell=langlesigmarangle$ acts on

$X^ell$ by $sigma(x_1,...,x_ell)=(x_2,...,x_ell,break x_1)$, and that

$C^ell X$ denotes the geometric quotient variety $X^ell/C_ell$.

Let $sigma$ be a generator of $C_ell$, 

and let $phi:tilde{S}(root{ell}of gamma)rDashto^{}X$

be the rational map mentioned above. 

Choosing an isomorphism $C_ellcongmu_ell$, the rational maps

$phisigma^i$ assemble to form a $C_ell$-equivariant rational map

$g=(phi,phisigma,...,phisigma^{ell-1})$

from $tilde{S}(root{ell}of gamma)$ to $X^ell$.



end{document}

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