数理解析研究所講究録924 巻 1995 年 188-197

188

Generic quotient varieties$\mathrm{A}1\backslash ^{r}\mathrm{I}\iota 1\mathrm{I}\mathrm{K}()$

Introduction. The

$1$

)

)

Theorem 0.1.
gebraic vari $ety$. overopen dense$X_{0}\subset.\iota^{i}$

(1)

(‘2)

(3)

$l\cdot$,(.

$C^{\mathrm{v}}f$$(\mathit{0}.\mathit{4})$$.\mathrm{q}\rho,\}.9\mathrm{c}$$a$a va riety,$\phi$$1^{\neg}\mathrm{a}17\partial lg(^{\neg l)\mathfrak{l}\partial}i\mathrm{C}\mathrm{a}ll1-\backslash$$|/1_{\mathrm{t}}’$$]_{\mathit{8}^{el)}\mathit{9}l)}r\partial \mathrm{i}\mathrm{t}\neg 1O1l$

and

),

$a/\tau 7\mathrm{t}$)].

closed fiield, .X an irred

acting on X.
)$\prime l1$]$l\rho’ 1tl_{1}cr\mathrm{r}$$:.\iota_{\mathfrak{c})}^{r}arrow 1^{J}l1|^{(i\mathrm{t}\mathit{1}\ulcorner}l1tl1$$rl^{)}l1i‘\epsilon_{\mathrm{i}11}?\subset^{},$$\mathrm{b}^{\neg}c^{\tau}l_{1^{\gamma}}$$\mathrm{S}\mathrm{l}\mathrm{Y}\uparrow Ootll$$\mathrm{s}i_{\mathrm{J}1\mathit{9}}l_{C}$a$\mathrm{I}|(^{\neg}j()l‘’\partial \mathit{1}-$oxists an$t$) $it$ ,non-sing 1’ la ,$r$

$k[\mathrm{M}^{J}’]1|arrow \mathrm{A}\cdot[.\iota_{0}^{-}\simeq]^{(:}$

)

$l$

.

$\iota\cdot(\iota\uparrow’(1)arrow \mathrm{A}\cdot(.\iota_{0}^{j})\simeq(,’(=l\cdot(.\backslash ^{\gamma})^{cj})$$not,r’1t\mathrm{i}()\mathrm{J}?.$

G-or

is preci, . a ,is ,$\mathrm{i}^{\mathit{1}}l_{0}\mathrm{r}?l’ C$

of $IJ.Mn$ mford$\ulcorner 1^{\urcorner}1_{1}\mathrm{i}\backslash \mathrm{s}$)$\mathrm{T}^{\gamma}\mathrm{t}_{\mathrm{l}\mathrm{e}\mathrm{o}}\mathrm{r}\mathrm{C}111\underline{)}$,$\phi:x_{0}arrow \mathrm{I}^{\mathit{1}}|_{\mathrm{t}1}’$fori.s to review [Rl.

$\mathrm{t}_{}11\dot{|}.\backslash ^{\urcorner}[]\mathrm{o}\mathrm{t}_{}\mathrm{e}$llcl (; an a

. fibre of$\phi^{*}:$

of$l_{z}\mathrm{r}t\mathrm{A}\cdot/$$e\mathrm{t}^{7}\mathrm{e}\mathit{1}y$$(\prime 5)\phi^{*}:$(6),$GAYo=-\mathrm{x}’\mathrm{r}|$(4) $X0$ ancl

$0.\backslash ’ \mathrm{C}$$\iota 1\Gamma 1$

GYOIA) $.l$

theorem can be$d$.

$l_{l?_{l}})\mathrm{a}rfj_{(^{\neg}\mathrm{I}ll}j1\Gamma,$$\mathit{1}^{\mathrm{M}\mathrm{u},\mathit{1}}l$, an

is a geometric$c,):.\backslash _{0}’arrow W_{0}$$c_{l^{11()}}\mathrm{f}ieI7f$in$tl_{l}e$.

$\iota 1^{\mathrm{C}}.$}$\mathrm{e}\subset 1$

in the$\mathrm{t},]_{1\mathrm{e}\mathrm{o}1\}},-$

of )
$1$

$\mathrm{r}\mathrm{e}11\mathrm{O}|\mathrm{n}\mathrm{o}\mathrm{g}e11\mathrm{t}^{\mathrm{Y}}\mathrm{O}\iota 1|9$vector$.\mathrm{q}1)_{\zeta}\urcorner c\mathrm{e}\iota \mathrm{s}$a$‘ \mathrm{q}$follows. Put
77$:=111\mathrm{r}\urcorner\backslash$

Then this lnaxiln\iota lnl is attaincd by.}above$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{n}$, we$\mathrm{g}\langle^{\mathrm{Y}}\mathrm{f}|$.

$.\mathrm{s}\mathrm{I}$$\iota \mathrm{n}G.r|.r\in X$)($\}1_{01}1\mathrm{g}\mathrm{i}_{11}\mathrm{g}$t,o a$l_{}c\dagger$

notation ) as in
$l$

$e$

$(\mathit{0}.|)$.

}.
$\mathfrak{c}1_{\mathrm{C}1}1\mathrm{S}\mathrm{e}$the following results concerning

Corollary 0.2.
$\mathrm{t}\mathrm{r}.\deg_{k}k(x)^{G}$

’

{di

$n7$

subset of X. From t,he.

$\prime J^{1}l_{\mathit{1}\mathrm{e}}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{i}_{1}11.\mathrm{v}-,77=\mathrm{c}\mathrm{l}\mathrm{i}_{\mathrm{I}11}\ovalbox{\tt\small REJECT}\nu_{\mathrm{r})}^{r}=$189

Corollary 0.3. l.et notat ion$e\mathrm{q}_{1\mathit{1}}i1’\partial l\epsilon 1\iota lt$

$l$

as in

) $e$

$(\mathit{0}./)$.

conditions are$r_{I^{1}l\iota e}f\mathrm{t}\supset llo1\mathit{7}il1g$.

an open(1)$Xl_{1}a‘ \mathrm{s}$’(2)$\mathrm{t}\mathrm{r}.\mathrm{d}\mathrm{c}\mathrm{g}_{\mathrm{A}}.\mathrm{x}\cdot(.\iota’)^{(j}=0$(3)$\mathrm{A}:(X)C^{\gamma},=l\cdot$$d\mathrm{r}’\uparrow.\mathrm{q}e(_{-0}^{r},,.l)i\dagger$

.

.

.

0.4. Convention and Notation. We fix an algebraically closed field . and$\mathrm{A}\cdot$

we always assume t,hat aIl algebraic variety is$\backslash \mathrm{t}’\mathrm{e}$byident,ify a(resp.

$l\cdot[X]$if.X isfor all$\mathrm{A}\cdot(.\backslash ’)$$\mathrm{i}\mathrm{r}\downarrow\cdot \mathrm{e}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{C}\mathrm{i}\mathrm{l}$$\gamma\in 1^{\neg}$.

$(l\cdot-)_{\mathrm{V}}C\gamma 1^{\cdot}\mathrm{i}\mathrm{c}\mathrm{t}_{\mathrm{V}.\}\partial}_{\vee}‘ \mathrm{y}.\iota^{j}\mathrm{c}$)$|\mathrm{c}$)$\mathrm{t},1\mathrm{l}\mathrm{e}$ri ng of$\mathrm{t}\backslash ’ \mathrm{i}\uparrow|1\uparrow_{}|1$’$1’ \mathrm{c}\mathrm{g}\mathrm{l}\mathrm{l}\mathrm{l}\partial 1$) on.X. For a

)

(

(

$\mathrm{t}^{-}[_{\mathrm{C}}\mathrm{f}\mathrm{i}\mathrm{I}1\mathrm{C}$set of$\}$over . unless
$l$

$-1$

rat,ional$\mathrm{i}\mathrm{t}^{\iota},.\dagger$$|^{1}$

act,ing on a set,.

$\mathrm{p}_{\mathrm{o}\mathrm{i}_{11}}\mathrm{t}_{}:\mathrm{i}’ X(\mathrm{A}\cdot)$

funct,ions (resp. t,he filed of$\mathrm{g}\mathrm{r}\mathrm{o}\mathfrak{l}1\iota$$o\mathrm{t},11\mathrm{C}1^{\cdot}\mathrm{w}\mathrm{i}_{\mathrm{S}}\mathrm{c}$$A,$stated.

)$\backslash l\backslash \mathrm{c}\overline{\mathrm{t}}1(\}11\mathrm{o}\mathrm{f}_{}\mathrm{e}$ns$\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{l}\mathrm{i}\mathrm{o}11_{C\gamma}|\{_{11\mathrm{I}1\mathrm{c}\mathrm{t}},\mathrm{i}o$$\Lambda^{\Gamma}:=\{\mathrm{r}’\in./1|\hat,‘’=c\ell$

}.

\S 1.
1.1.

Flatness.

$r_{1’ 1_{1(}\mathrm{B}}\mathrm{t}^{\neg}\mathrm{O}\mathrm{I}1$(,$(^{\}}1)\mathrm{f}$of$‘ \mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}_{1}1\mathrm{c}\mathrm{s}.\mathrm{q}1$)$\mathrm{l}\mathrm{a}.\mathrm{v}\mathrm{s}$an$\dot{\ddagger}\mathrm{t}111$)$\mathrm{O}\Gamma \mathrm{t}_{}\partial \mathrm{I}\mathrm{l}\mathrm{f},$$\mathrm{r}o$algebraic geomet,ry [EGA]. A concise account, can ) found in [Mi,le in t,be$|$$\mathrm{c}$recall t,wo$s\mathrm{t}^{\backslash }\Lambda \mathrm{e}\mathrm{m}G(/.

there exists a$/\mathrm{E}\mathrm{G}\mathrm{A}$\mathit{2}.l_{(}.\backslash ^{\supset})$

fini

,

. and

$(l|, (_{)_{-}}^{\tau}..\mathrm{Q}.])]$)$l^{)e’ 1dt}c\cdot,\mathrm{q}C$ $\{’\subset\}fl\iota’ cl_{1}tl7$$\dagger_{\gamma}e_{l}$)$\mathit{1}\mathrm{E}\mathrm{G}\mathrm{A},$},$\Gamma \mathrm{c}$$\mathrm{S}et\mathfrak{j}t_{1}r\gamma$

1.4. Hilberttion$(I|, \mathit{2}.

$(\mathit{1}, ()7.\mathit{2}.\mathit{1})$$\mathrm{S}_{1)G}\mathrm{c}(\mathbb{Z})$(2) a closeda.

be a loca$\}’$$\iota 7?()\mathrm{r}j)l_{1}$a $t?/:$

l.\mathit{6})]$.

$l_{/}et_{}$.

$rj^{1}l_{7e}\downarrow\tau.\dagger$is 1] of finite.

$t_{c}1^{J}l^{)}G$$(l, (\grave{)}..\mathit{3}.\mathit{2})$)$.9\mathrm{C}11\mathrm{C}\iota 111\mathrm{C}$.

\overline{/}.\mathit{1})$,$\prime l^{1}l_{1e}n$

is $Hat$ .
) $e$$ll\mathrm{v}$. It, is kllown

open$(ll’, \mathit{2}.

$\mathrm{f},1_{1\partial \mathrm{t}_{}\mathrm{f}\prime}|1\mathrm{C}\mathrm{r}(\tau$whose connected}$\mathrm{t}l_{\mathbb{Z}}’\subset \mathbb{P}_{\mathbb{Z}}^{r\iota}\cross Hilb_{\mathbb{P}_{\mathrm{Z}}’’}$

flat, overl.\mathit{2})$.

exist,$(’ \mathrm{Q}|111)0|\mathrm{l}\mathrm{e}\mathrm{I}\mathrm{l}\mathrm{t}.\mathrm{s}$. anrl$\mathbb{Z}_{-\mathrm{C}}.\mathrm{i}\mathrm{l}\mathrm{l}1$$(/, \mathit{2}.

: X $arrow\}’rl:$ a flat morphism ofis universa$\mathbb{Z}-.\mathrm{q}\mathrm{t}\neg 11(^{\tau}|11\mathrm{C}H?ll)\mathbb{P}’’\mathrm{Z}$noetlt crian$ll.)^{\text{ノ}}$$\iota/-1(\iota \mathit{1}^{7})arrow I^{f}r$$l_{\lrcorner}et.f$$\mathrm{s}\mathrm{c}1_{1}\mathrm{e}\mathrm{l}\mathrm{n}\mathrm{e}.[\mathrm{M}\mathrm{u}, 1)1^{)}\cdot 2122]$(1) a locally noetherian

project,ive over.

$?/:.\backslash ^{I}arrow\}’$$\mathrm{t}\mathrm{l}‘$Lemnia 1.3.
$l_{oCa}ll_{\}^{\prime \mathit{0}}}.t\cdot$]. Wefrom [EGA].

$1e\mathrm{m}11\mathrm{l}\mathrm{a};\dashv$

Lemma 1.2.

in tegral$\mathrm{t}_{-}^{l}|_{1\mathrm{a}}\mathrm{p}\mathrm{f},\mathrm{c}\mathrm{I}’ 1$$H_{\dot{2}}lb_{\mathrm{P}^{\prime r}\mathrm{z}}$

.

are

190

such that for(3) any locally noet,(t) ally

there is a$|1G$$\mathbb{Z}$-scheme ,and,$\mathrm{q}_{\mathbb{Z}}^{1}$.

$\mathrm{t}|\mathrm{a}\mathrm{t}$$(^{\neg}1\mathrm{o}\mathrm{s}(^{\mathrm{Y}}\mathrm{t}^{-}1\mathbb{Z}-.\mathrm{s}\mathrm{c}11(\backslash \mathrm{t}11\mathrm{C}/_{\mathbb{Z}}r_{J}\subset \mathbb{P}_{\mathbb{Z}}’’\cross,\underline{\iota}_{\mathbb{Z}}^{\mathrm{t}},$:$\mathrm{U}11\mathrm{i}\mathfrak{c}_{1}\mathrm{u}\mathrm{C}\mathfrak{l}11O\mathrm{I}^{\cdot}1^{)[\mathrm{i}^{\mathrm{c}}}1\cdot\{\mathrm{I}\tau 1.f_{\mathbb{Z}}$1.5. If we ), we may)$\mathrm{t}_{1}^{1}\mathbb{Z}arrow H?/\mathit{1},\mathbb{P}|\mathrm{z}1|\backslash ^{\epsilon}$) $\mathbb{P}_{\mathrm{A}}’1:=ff\uparrow/l$)

tlcIl$|\iota \mathrm{t}Hill$$1$$\mathrm{s}_{1}\supset \mathrm{e}\mathrm{c}(\mathrm{A}\cdot)$

rian

wit,h

$\mathrm{r}\mathrm{e}_{1^{)}}1_{\partial C\mathrm{C}}\mathbb{Z}$$\mathrm{A}$over$S_{\mathbb{Z}}’$,$\mathrm{t},1_{1}\mathrm{a}\mathrm{t}/_{\mathbb{Z}}^{r_{\text{ノ}}}=(1_{\mathbb{P}_{\mathrm{z}^{1}}’}\cross.f_{\mathbb{Z}})^{*}(1\cdot 1_{\mathbb{Z}})$$\mathbb{Z})‘\supset_{1^{)}}(^{\mathrm{T}}\mathrm{e}\mathrm{C}(k)$$\mathbb{P}_{\acute{\acute{\mathrm{Z}}}}\cross_{\mathrm{L}}\mathrm{b}_{1)\mathrm{e}}^{\tau}\rangle\langle\langle$

and$|/l_{k}^{r}:=[|_{\mathbb{Z}}’\cross_{\mathrm{h},l1)0^{\mathrm{t}}(\mathrm{t})}\mathbb{Z}$. everywhere in (1.3).

\S 2.
We start,$\mathrm{t}_{\mathrm{t}}1_{1\mathrm{i}}\mathrm{H}$sectiowitl] ,he$\mathrm{t}$$\mathrm{t}\mathrm{l}$Lemma 2.1. $l,eff$ :a col 1$\mathrm{a}\mathit{1}1\subset lZ\subset X$

with$\mathrm{r}e_{l^{J}}|\mathrm{q}6C.t$$\overline{.f(\prime\nearrow/)}=\}^{\prime’}$to}$.\backslash arrow$$l$$s\dagger\iota\cdot \mathrm{I}l(.t\mathrm{i}l)ie,\backslash ^{\urcorner}1’ l)|\mathrm{q}_{(^{\neg}}t(i.c..a$$t_{6}l\mathrm{i}e/_{r\gamma\Gamma}^{r_{J}}i.\mathrm{s}ki\mathrm{f}_{l}ol$) $ol1$ )$g_{\backslash }\iota j.$)lemma.

) $\mathrm{r}C\tau t1or_{l}$ ) $l_{1}isI1ll$ ) $et_{\mathrm{Y}\mathrm{r}}eoI1$

fin$ite\mathrm{r}l$$\dagger f’cfi_{I}r\mathrm{f}$isjoint 1 ion

lter

$lIl$

$ot$

.

alge )$l$loca$a.\mathrm{S}^{\mathrm{C}}‘ \mathrm{i}\mathrm{t}l\iota?1\rho t]_{\mathit{1}a}t\mathrm{J}/_{d}’$$ll_{1,\backslash }$ties,$\Gamma/\prime 1\mathrm{i}(^{\neg}1^{-}i1\Gamma i\mathrm{r}$. closod It$s$is irrcd$\mathrm{I}\mathfrak{l}\mathrm{t}^{arrow}il\gamma$$l,‘ \mathrm{s}’ \mathrm{r}ts$] $e$ and. Then(1)$\}_{0}.\cdot:=\{y\in]’|\overline{.f\cdot-|(?/)\cap/_{J}\prime}f^{-}=.\cdot(\mathrm{l}y)\cap\overline{7_{J}}\}$

cont ains an open
$Co\mathrm{n}t_{:}ai_{\mathit{1}}\mathit{1}\mathrm{e}d$

in

.
]

$1\lrcorner$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}\mathrm{r}\iota \mathrm{l}\mathrm{c}\{,\mathrm{i}\mathrm{b}$irreducible
the

(2)
(.3)

$cI\mathrm{c}^{\tau_{l}}\mathfrak{j}$‘se($\mathrm{t},$$/_{0}’\lrcorner$$\mathrm{e},$$/^{r_{d\int 1}}$is$\mathrm{C}\mathrm{o}\ln_{1^{)\mathrm{O}\mathrm{t}}}1\mathrm{e}11\mathrm{t}^{\zeta}.|$$\mathrm{t}_{1}\mathrm{o}\mathrm{t}‘ \mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t},\mathrm{y}$of $y\in$ }$\mathrm{d}\mathrm{i}\ln$$l\mathrm{s}\mathfrak{j}\mathit{1}l$of}. () $Sot$$I,c\mathrm{t}\}’\#$

denoto$\mathrm{t}l1O$

large.st opon ,$\mathrm{S}1ti$)$,\backslash \neg \mathit{0}\mathrm{f}$

of}’

)

$l_{0}^{\prime’}.$$t^{)}roof$is$\dagger 01|_{(\supset}\mathrm{Y}\backslash ’ \mathrm{i}\mathrm{I}\mathrm{l}\mathrm{g}‘ 9\mathrm{i}_{1}111^{)}1\mathrm{e}$

be
$01$)

of

$\mathrm{s}\iota\iota \mathrm{c}\mathrm{l}\mathrm{l}$$\mathrm{t},1\mathrm{l}\mathrm{c}^{\mathrm{Y}}$largc.st$\mathrm{C}11\mathrm{d}\mathfrak{c}\cdot \mathrm{t}\mathrm{l}\mathrm{S}\mathrm{c}$in$\overline{/^{r_{d}}}\backslash /_{d(1}’\mathrm{s}|1\mathrm{c}1_{1}$$\aleph \mathrm{t}1|$,$\overline{/^{r_{d}}}$).

t,hat,$\mathrm{s}\mathrm{c}\mathrm{t}_{!}$of 7 which is$0_{1^{)(}’}11$$[_{\mathit{4}(^{\mathrm{Y}}}\mathrm{f}_{}\{’\nearrow_{\lrcorner}i\}_{i\in}l(\Gamma \mathrm{e}\mathrm{s}_{1)}.

$\overline{.f(’/_{J}i)}=\}’$(resp.

in$\overline{/^{r_{J}}}$\{^{r}/_{/_{j}}^{1}\}_{j\in}, )$$\overline{f(’/_{\lrcorner i})}^{\subset}\neq 1’$).

t,hat,( $f^{-1}(y)\cap$ ノ’Ji)$f^{-1}(y)\cap/_{j}^{r_{J}l}=\varphi$$=\mathrm{d}\mathrm{i}\mathrm{t}11/_{i}^{r_{d}}-\mathrm{d}\mathrm{i}_{\mathrm{I}1}\urcorner\}’$for all $j\in.J$ .

for all$i\in l\mathrm{U}\{0\}$

, and

.

Since

$J_{J}’$

bc all t,he$[_{i}\mathrm{c}\mathrm{t}\}_{1}’$

be

191

$r_{1^{1}\mathrm{h}e\mathrm{n}}\iota_{1}^{r}$is a const

while (3) gives a.

$(\cdot\downarrow)$dcnse .S}$1^{\cdot}11\mathrm{c}\mathrm{f}_{}\mathrm{i}1_{1}1\mathrm{c}$$\mathrm{c}o|1.\backslash ’ \mathrm{t}\iota\cdot 11\mathrm{C}\mathrm{t}\dot{1}|$)$[_{\mathrm{C}}$$1|_{)\mathrm{q}\mathrm{c}}.\mathrm{t}$oIleof}’. ($\mathrm{i}\mathrm{t}1\circ\circ \mathrm{c}|1\langle^{\tau_{\mathfrak{l}\mathrm{a}1}}.$$1\urcorner|_{1}\mathrm{c}$condition (2)

) For

,

$y\in 1\downarrow$$f^{-1}.(y)\cap/_{d}’-1(\subset.f\cdot y)\mathrm{n}/r\subset d.f^{-\mathrm{I}.\prime}.(|/)\cap\overline{/_{d}}=$$\mathrm{g}\mathrm{i}\mathrm{Y}^{-}G^{\mathrm{C}}.t$an)$\mathrm{q}\mathrm{e}\mathrm{t}$$\mathrm{o}_{1^{)\mathrm{C}\mathfrak{l}1}\cdot}\mathrm{q}\iota
11$.

we havc$(f^{-1}(y)\cap/_{i}’)/\cdot$$\cup$$i\in l\mathrm{u}\{0\}$by (3). Especially.

$\mathrm{d}\mathrm{i}\downarrow 11(.f\cdot-|(y)\mathrm{n}\overline{/\prime})\lrcorner=1)1\mathrm{a}\backslash \mathrm{d}j\in l\mathrm{u}\dot{\mathrm{t}}^{\overline{\mathrm{r}}_{\mathrm{I}}}\mathrm{I}\mathrm{i}\mathrm{I}1\gamma(.f\cdot-|(y)\mathrm{n}/_{d}’)i$$=|117i\in/\mathrm{u}^{c}\dot{\mathrm{t}}^{\backslash }(|\}(\mathrm{d}_{\mathrm{I}|1}.1/r_{di}-\mathrm{d}\mathrm{i}\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{I}’)|).\mathrm{V}(2)$.

$\}’=\mathrm{d}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{l}\overline{/_{\lrcorner}r}-1-1\mathrm{i}_{111}\}’$$=\mathrm{t}\mathrm{l}\mathrm{i}\mathrm{I}11/_{0}^{r_{d}}-\mathrm{d}\mathrm{i}\iota 11$(Indecd.

for all $i\in/.$ ) In othcr words,$\mathfrak{c}1\mathrm{i}11\urcorner/_{\lrcorner i}’<(1\mathrm{i}\mathrm{I}11/_{d(1}’$at,tain t,he lllillitl\urcorner lllll (are of$f^{-\mathrm{I}}(y)\cap\overline{r_{\text{ノ}}/}$$\mathrm{t}11e$

relat,ed gclieralit,}.)$1\mathrm{i}_{\mathrm{I}11}\mathrm{c}\mathrm{t}1.\mathrm{S}\mathrm{i}\mathrm{o}|1$sa meHence allat, $.|/\in\}_{1}’$ .
)$\mathrm{t}^{-}1\mathrm{i}\mathrm{I}11\mathrm{C}\mathfrak{l}1.\mathrm{S}\mathrm{i}\langle-$$\mathfrak{l}1\mathrm{t}^{-}1\mathrm{i}\mathrm{I}11\overline{/_{\lrcorner}r}-\mathrm{d}\mathrm{i}\uparrow 1\gamma$$f^{-}1(y\cap/_{(1}^{r}(\subset$\mathrm{t}_{11_{1\mathrm{a}})}\mathrm{t},$$\mathrm{r}\mathrm{e},\mathrm{s}\mathrm{t}\mathrm{l}[\mathrm{t},$

i.e..

,

)

$\overline{/’}arrow$$\mathrm{O}\mathrm{t}\mathrm{l}\mathrm{C}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{s}$}. (Scc [EGA. (I $l’$ . $13.2)$ ] for}of$\mathrm{t}_{1}11\mathrm{C}$$\}_{1}\subset\}_{()}$.

. a$\dot{\mathfrak{l}}\mathrm{I}1.f^{-1}(y)\cap\overline{\ulcorner/_{\lrcorner}}$

dcnsc itl.ff^{-1}(y)\cap/^{r_{J}})\dot{\iota}.\mathrm{s}$–rlim}’$(-\mathrm{i}_{\mathrm{I}\mathrm{l}\mathrm{c}}\mathrm{t}_{\grave{1}}e\}_{1}\mathrm{i}\backslash ^{1}\mathrm{t}^{\neg}\mathrm{O}\mathrm{t}1|\backslash 1\mathrm{t}\mathrm{t}\cdot 11\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{I})1\mathrm{e}$$-1(y)\cap\overline{r_{J}/}$a nd dcnse$\mathrm{i}11$.

$111^{-}1co|1.\mathrm{s}\mathrm{e}(1^{1}[e|]\uparrow \mathrm{t}|\backslash ^{-}\backslash ([)$$\mathrm{H}_{\mathrm{C}\mathrm{t}\mathrm{l}\mathrm{c}\mathrm{c}}\overline{f^{-}|(l\backslash \cdot/)\cap/_{d}\prime}=$

}, wc

thc

$\mathrm{g}e\mathrm{t}_{\mathrm{t}}$$(1e..\backslash |\mathrm{i}_{1}\cdot \mathrm{c}\mathrm{d}$. I2.2. Let$\mathrm{A}\cdot,$

( .

$.l’$

$;$

bc as in thc introduction. alld

varicty cont,aining X a an opcn dcnsc$.\mathrm{c};\iota\iota|$$‘\backslash ^{}$

G-act,iot] on

be t,hc$1)\Gamma \mathrm{o}\mathrm{j}$:$\iota\grave{1}(\mathrm{i}|1ce$

for $i\in/,$ $f^{-1}(y)\cap/_{\Delta}’i(?\in l)$ are nowhcrc tlensc$f^{-\mathrm{I}}(y)\cap\overline{r_{J}/}$.
$\Gamma e.\backslash ^{\epsilon}\mathrm{t}^{-}\mathrm{J}\mathrm{f}.f$$\mathrm{t},1\mathrm{l}\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{c}\mathfrak{c}11\iota \mathrm{c}\mathrm{i}1_{1}1\mathrm{C}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{l}$$\zeta 1_{\dot{1}\iota 1}1(.f^{-}|(y)\cap/_{i}^{r})=\mathrm{d}\mathrm{i}\mathrm{n}1/_{J}’i-\mathrm{d}\mathrm{i}\mathrm{I}11\iota<\mathrm{d}_{\dot{1}111}\overline{7_{J}}$yields)$\mathrm{t},\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{i}1$$r/,\mathrm{a}\mathrm{r}\mathrm{i}.\mathrm{C}$‘lct,io11.

}$-\backslash ’$ca Il Ilot$\mathrm{k}\mathrm{i}\mathrm{C}\mathrm{l}\mathrm{o}.\mathrm{s}11\mathrm{l}\mathrm{c}$$\dagger$of$\mathrm{I}\mathrm{n}\mathrm{t}_{}11\mathrm{i}\mathrm{t}\mathrm{i}\nwarrow;\mathrm{c}1_{1,\backslash }-$.

)$\mathrm{c}(^{\mathrm{Y}}\backslash \prime \mathrm{f}_{}\mathrm{c}\mathrm{I}\mathrm{l}\mathrm{r}|\mathrm{e}\langle 1\uparrow \mathit{0}\overline{.\backslash \vee}$.

)$‘ \mathrm{s}\mathrm{c}\mathrm{t}$. (($\mathrm{t}^{\backslash }e[\mathrm{S}]$

( $3$

}

:

$\pi^{-1}(.\backslash ^{\prime i}’)arrow.l^{d}’$

is

for

$\dot{|}\mathrm{l}\mathrm{r}\mathrm{c}\mathrm{d}$uciblc)$|_{)11}\mathrm{t}1$

)

, and
$\mathit{0}\dagger$$\pi$$:/_{J}’arrow.1’$

orbit,$1^{)1}’ \mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{f}_{}\mathrm{i}_{1^{-}}\mathrm{e}$$\mathrm{O},\mathrm{S}^{\mathrm{C}}’.$;ibly the$\mathrm{r}\mathrm{t}\mathrm{i}o\mathrm{I}1\aleph.$$\mathrm{e}\mathrm{r}_{1}|1\mathrm{i}\mathrm{Y}- \mathrm{r}\gamma \mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{f}\mathrm{l}\mathrm{t}^{arrow 0\mathfrak{l}}111^{\mathrm{I}}$,hc family$\mathrm{t}$aIlexist,s [N].

$\mathrm{i}_{\mathfrak{l}1\alpha}\overline{\backslash ’}\cross X$$\{((J\cdot\}..\mathrm{t}\cdot)|.\}\in.\backslash .\langle/\mathrm{L}\in(:$$\pi$$|-11\iota \mathrm{c}^{\backslash }\mathrm{I}1\overline{.\backslash \prime}$$\overline{.\mathrm{x}’}$)$|_{f\mathrm{e}}\mathrm{f}/^{r_{d}}$thc.sccond$\mathrm{t}^{\neg}|\mathit{0}.\backslash \tau 111’ e\backslash \backslash ^{1}\overline{(_{l\cdot 1}’-\cdot}$in$\overline{.1’}$

192

by.t $\in.l’’$ .
.
: X
and
$\cdot$$\mathrm{p}_{\partial\Gamma \mathrm{a}111\mathrm{e}\{1}\cdot \mathrm{i}_{7_{J}}\mathrm{e}\mathrm{d}$in)$\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{c}$

$1$

of

such t,hat,$/_{\lrcorner}’$

$\pi$

:

rc

$\mathrm{w}\mathrm{h}e$$A\backslash ^{\mu}l$$arrow\}’$

$.f$

.

is

Let,

is Hat$\pi^{-1}(.l_{11})arrow.\backslash _{\mathrm{r}1}^{r}$as Il 2.

$\langle^{-1\mathrm{c}\mathrm{f}\mathrm{i}\mathrm{I}1\mathrm{C}1^{-}}1$$\mathrm{i}$

$($

bc

$.\iota_{\mathrm{r}1}^{f}$$(\mathrm{t}^{arrow \mathrm{t}}.

$\mathrm{t}_{1}11\{^{\mathrm{B}}$(|.\underline{\cdot)}))$.

.

$\swarrow^{r}/=\pi-1(X_{0})(\subset\overline{.\backslash \cdot}\cross.\iota_{\mathrm{t}1}’\subset \mathbb{P}_{\mathrm{A}}^{11}$ $\cross.1_{(1}’\mathfrak{i}01^{\tau}.\mathrm{C}\}\mathrm{O}\mathrm{I}11e’ 7)$

which makes t,he$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\backslash \mathrm{v}\mathrm{i}$ng diagram ca$1^{\tau}\mathrm{t},\mathrm{c}\mathrm{S}’ \mathrm{i}\mathrm{a}11$$|)\iota\iota.\mathrm{s}\mathrm{i}$largest,ng

a nd$/_{d}’$:$\pi$$\overline{.\backslash }\mathrm{X}.\backslash arrow.\backslash ^{\nu}$(dense)$0_{1^{)\mathrm{C}\iota 1}}$of$|\dashv\iota\iota 1,\aleph \mathrm{r}’ \mathrm{f}$$r_{1’ 11e\mathrm{I}1}\mathrm{a}_{1^{)}1^{)}\mathrm{y}\mathrm{g}}|\mathrm{i}\mathrm{n}(|..\ulcorner))\mathrm{t}_{}(^{-,\mathrm{h}^{1}})=.\mathrm{X}_{(1}’$. we get, a$\iota\tau\tau \mathrm{o}\mathrm{r}\mathrm{l}$

)

.

:

$1\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{I}\mathrm{l}\eta.f$$.\iota^{L}$

,’

and

$.l_{0}arrow H?/\mathit{1}$ )$\mathbb{P}_{\mathrm{A}}’’$.

$r_{\iota}^{r}-\mathrm{I}(.\iota_{\mathrm{r})}.)arrow$$[\iota’$$\downarrow$$\downarrow$$rightarrow j\cdot H?lb_{\mathbb{P}_{k}’’}$$.\backslash _{0}’$be the$\mathrm{I},\mathrm{e}\mathrm{t}1_{1}’$

(1.2)

$)$

$\partial 11\mathrm{d}$)$1\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{c}^{\mathrm{Y}}\mathrm{s}\mathrm{t}l\mathrm{o}_{1^{)\mathrm{c}\mathrm{I}1}}\mathrm{s}\mathrm{t}\mathrm{l}\mathrm{l}$surjectivc.

(.3) $f$ :$\hslash l$)$\Gamma \mathit{0}_{n}\mathrm{q}$t-tl$.\backslash _{1}’arrow\}_{1}$Proof. (1) isof$\overline{\backslash f\cdot(.\iota_{\mathrm{r})}’)}\mathrm{t}.\{\mathrm{t}\mathrm{l}\mathrm{c}1_{1}\mathrm{t},1_{1}\mathrm{a}\mathrm{t}.f.

$1^{)}1\iota \mathrm{f}_{}.\backslash _{1}^{f}=.f^{-1}()_{1}’)$

Lemma 2.3. (/)(2) The$.\mathrm{q}\mathrm{e}\uparrow$

$f$

is

$\iota|\dot{|}1(\neg\Gamma,\mathrm{b}‘ r1\prime ll_{1,\backslash }-$$01_{)\mathrm{V}}io|1\mathrm{S}$$41_{1}’|)\mathrm{c}\mathrm{t}\mathrm{h}_{\mathrm{C}c}\mathrm{o}\Gamma \mathrm{r}\mathrm{e}\mathrm{S}1^{)}0|1\mathrm{d}\mathrm{i}$a$Al_{1}’arrow\}_{1}$the$rc_{l)}\mathrm{r}ecj‘ \mathrm{q}(^{)l1}\backslash$$1$)

is flat (cf.

.

$l^{)\Gamma t}\mathit{0}_{t}\mathrm{q}(\neg\cdot\gamma\cdot \mathrm{r}d/).1^{-}(-;$

in$.\iota_{1}’$

.

open.

$o\mathrm{i}11\uparrow$.

$r_{1^{\mathrm{t}}}$hcn we get,$arrow\gamma^{-\mathrm{I}}(\downarrow.\backslash _{1}’)arrow$$\overline{(_{-\mathrm{J}}^{\ell}.r\downarrow}$is$G-()\Gamma bj- t$. (3) follows from (1.3). For.

ng gcomctricf^{-1}(1_{1})arrow\}_{1}’$.

$r_{l^{1}l1r\cdot(l^{)cn}}d\Gamma’ ns\circ,\backslash ‘|ll-).\mathrm{S}e\mathrm{t}.l_{1}l\subset.1’)$::$\mathrm{l}\mathrm{c}\mathrm{t}|$$l\cdot\in.\backslash ^{f}\rfloor,$:?

$\mathrm{t}\mathrm{t}_{)}\gamma 1^{)}\mathrm{C}(^{\neg}(l\cdot)arrow$$\mathrm{c}\mathrm{a}\mathrm{l}\cdot \mathrm{f}|(\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{q}\dagger 1c\urcorner \mathrm{t}\mathrm{c}\mathrm{s}$$11^{\text{ノ^{}r}}\downarrow$.
$\int_{rightarrow}$$\prime \mathrm{t}_{\supset_{1)\mathrm{c}\mathrm{c}()}}^{\tau}\iota\cdotarrow i_{1}$$.l_{1}’$Hence for.

$r,$$x’\in.\iota_{\mathrm{t}}^{r}$$Hilb_{\mathbb{P}_{k}}?1$.

$c_{\tau=}.(,|.\mathrm{t}’.\Leftrightarrow\overline{C^{\mathrm{v}_{\Gamma}},.}=\overline{(^{\mathrm{t};},.r}\Leftrightarrow.f\cdot 0\uparrow.\cdot.\cdot|=.f\cdot \mathrm{o}i_{r}$($\mathrm{f}\mathrm{o}$see$\overline{C’,.\iota\cdot}\cap X$$\mathrm{t}_{\text{・}}11e$

first

. The

. note$\mathrm{f}_{\mathrm{t}}|1\mathrm{a}\mathrm{f}$$e\mathrm{q}|\iota \mathrm{i}\mathrm{Y}^{\cdot}t\urcorner|\mathrm{C}|1\langle^{\neg}e$$\mathrm{s}c\mathrm{c}\mathrm{o}\mathrm{t}1\langle^{-}1$

equivalence follows, . is t,hc$(’$$-\cdot\iota$$,$$\Leftrightarrow f(_{7}.\cdot)=f(_{l)}.\cdot’$$11\mathrm{I}\mathrm{l}\mathrm{i}\mathrm{r}_{1}\dagger\iota eG- \mathrm{O}\mathrm{t}\cdot|$),$\mathrm{i}\mathrm{t}$,$\mathrm{t}_{\Gamma \mathrm{O}}11\urcorner \mathrm{t}_{}[]\mathrm{C}1\iota 11\mathrm{i}\mathfrak{c}1^{\iota}1\mathrm{C}\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{s}_{1)\mathrm{a}}\Gamma \mathrm{t}$.

which is$01$)

in $(|.[).)$ I$\mathrm{c}\mathrm{n}\mathrm{i}$

Il

193

\S 3.
somc

(

$- 1$

$\backslash ’\backslash ^{\gamma}\prime \mathrm{c}|1r$$1$Lemma 3.1.
$,\mathrm{q}\mathrm{I}\mathfrak{l}(^{\neg}l1tl1$

at

ancl

$\mathrm{S}11c1_{1\mathrm{f},1_{1\mathrm{a}}}\mathrm{t}$among such

algebraic.

, t,hc,cn$t\downarrow \mathrm{b}_{c}\urcorner \mathrm{f},$$\mathrm{t},\mathrm{r}.\mathrm{t}^{- 1e}\mathrm{g}_{l\iota’}N$$\mathrm{e}$$.\backslash ^{}$

ions.

tra

a

$\mathrm{R}e_{1^{)}}1\mathrm{a}$$1^{)111}\mathrm{c}\mathrm{l}\mathrm{y}\uparrow_{}\mathrm{l}^{\backslash }\mathrm{a}$$\mathrm{i}\mathrm{o}\mathrm{l}1$. i.e..

,ranscendental.

$[_{\lrcorner}\eta,/_{\mathit{4}\backslash ^{\gamma}}’:.\mathrm{V}]\leq[l_{d} :of a$1^{)\mathrm{t}11’\epsilon}\backslash |\mathrm{y}\mathrm{t}_{\Gamma}\mathrm{a}\mathrm{t}$)of$\mathrm{I}\mathrm{l}\mathrm{S}\{^{\neg}e\mathrm{I}1(1\mathrm{c}[]\uparrow_{}\mathrm{a}|\mathrm{c}\mathrm{x}\mathrm{t}_{}\mathrm{C}11\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{I}1$

ci

wit

$l_{1}’’$$\mathrm{n}\mathrm{g}$$|1l_{1’}$

.

nsccn

$‘-|\mathrm{c}\mathrm{I}1\{_{}\mathrm{a}|$. cont,aincd in$/\mathfrak{i}’$$\backslash \backslash ^{\mathrm{v}}\prime G$

may a

me hat .1$\mathrm{t}_{\text{・}}$$\mathrm{S}\mathrm{S}[]$ext,ension of [ , cont,a$\mathrm{t}’$$[/_{J}$:$’\backslash ^{7}]<+\infty$$\mathrm{O}11$”’$‘\dagger l$,

is a 11$l//\iota’$$\mathit{4}b/yg‘$

ental$r1^{1}11e\mathrm{I}1/_{d}/\wedge^{\mathrm{r}}$the othcr$11\mathrm{a}\mathfrak{l}1C\iota,$$\mathrm{i}_{\mathrm{I}1}\mathrm{c}\mathrm{d}\mathrm{i}$is a Il$\mathrm{d}\mathrm{i}\iota \mathrm{s}\mathrm{j}\mathrm{o}\dot{\iota}11\mathrm{f}_{}$.

$\mathrm{a}\mathrm{t}\mathrm{g}\mathrm{c}$is$^{\dagger}\tau//\mathrm{t}^{r}$$r_{1^{1}1_{1}e}$

Il /,,})$\mathrm{r}\gamma \mathrm{t}11^{-}1$$\Gamma C\gamma \mathrm{i}(’$$\tau\gamma[\mathrm{g}\mathrm{C}$}reforc.

such$\mathrm{t}\mathrm{i}11\mathrm{i}\mathrm{f}(^{\tau}[\vee\backslash \cdot$$)1_{C}’\gamma \mathrm{i}(^{arrow}$.

$[.\dagger l :$.\backslash ^{l}/l\backslash ^{r}$/_{\mathrm{t}}’]=$. I$\prime\prime.\backslash - \mathrm{r}_{l^{r/’\tau}}$$\mathrm{s}\mathrm{c}(^{\mathrm{Y}}11(- 1$.

thcy are lincary$’\urcorner|1(^{-}1[_{1\mathrm{C}11}\mathrm{c}\mathrm{t}\neg$\backslash ^{r}]<+\infty$.

3.2. Separably generated extension.
is called

a ticld$l$.

$11\Gamma \mathrm{C}[\mathrm{y}\mathrm{f}$$\mathrm{t}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{l}$$fi$itoly gen era $ted$ .

$\backslash ^{1}11\mathrm{C}1_{1\mathrm{c}}.\backslash \mathrm{t}\mathrm{C}11\backslash ^{1}\mathrm{i}\mathrm{o}|1.\backslash ’${$1$$\mathit{1}//1’i,\backslash ^{\mathrm{t}}al,\backslash ()h\mathrm{f}1$$|\iota\downarrow(^{\tau}1\backslash \backslash \cdot$

tecl old extension. and 41$jf_{C}l\tau’\cdot \mathrm{g}\backslash \cdot \mathrm{c}\tau nc\Gamma i\mathrm{i}$is maximal a mong.

$gen\mathrm{c}r\mathrm{a}t,e\mathrm{d}\mathrm{e}\mathrm{X}\mathrm{t},\mathrm{c}\mathrm{n}^{\iota}.$is )) $c\mathrm{r}\gamma 1$ )$\mathrm{t}_{1}11\mathrm{C}o\mathrm{r}\dagger^{r}.\cdot$

aximal$\mathrm{e}\mathrm{x}\mathrm{t}_{1}G\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}|1$)

is a finfield$\mathrm{t}_{1}1_{1\mathrm{C}}\mathrm{Y}$$\uparrow_{\mathrm{I}c\gamma \mathrm{I}1},.\backslash 1\mathrm{c}e[]\mathrm{t}1e|1\mathrm{t},\mathrm{a}[(-\mathrm{l}\mathrm{c}\mathrm{g}\mathrm{l}\mathrm{c}e\uparrow_{\mathrm{I}}\cdot.C|\prime \mathrm{g}’_{\iota}\cdot/\mathrm{t}^{r}’(<\uparrow 1’.\mathrm{d}e\mathrm{g}_{l\mathfrak{i}}/_{d}<+\infty)\mathrm{i}\mathrm{S}111$$\mathrm{e}\mathrm{x}\mathrm{t}$Let $N|-$minary from. th cn.l$\mathrm{I}{}_{I}\mathrm{C}\mathrm{t}l_{1}’’1$

Oof$$\mathrm{r}\mathrm{c}\mathrm{l}\mathrm{i}$$lt\cdot/_{J}/l\backslash$$/\mathrm{t}’\subset.1\mathit{1}\subset/_{I}$$/^{\lrcorner}?’)}}$e\backslash ^{\vee}\mathrm{t}e$$\mathrm{t}$ratrd r.rtr }’$.\backslash -io$}’$([\mathrm{W}, 1).\rceil l])$if it is a

A fi nit,[ cx-$\mathrm{e}\mathrm{I}\backslash ^{\tau}\mathrm{g}\backslash e|1\mathrm{c}11’\iota C\mathrm{t}_{}e1^{-}$)$\mathrm{s}^{\tau_{1\mathrm{y}\mathrm{g}.\prime}}‘$$\mathrm{a}1’\partial 1\supset|\mathrm{r}\gamma|\mathrm{c}|_{)1’ \mathrm{a}}\mathrm{i}_{1’}\mathrm{c}\backslash \uparrow_{}e|1.\mathrm{q}\mathrm{i}_{01}1$

nsion.

. we Ilcct-l thc$(_{-\mathrm{o}\mathrm{n}}^{\mathrm{t}}\mathrm{c}\mathrm{e}\Gamma 11\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{I}1|.\backslash \mathrm{c}\mathrm{o}\mathrm{t}1(’ \mathrm{c}_{1^{)}}\mathrm{t}$$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}_{1}$easier half of$[\mathrm{W},$$({}^{\mathrm{t}}[]\mathrm{a}_{1^{)}}.|$,

Prop.l 9].
Leninia 3.3.
fixecl a $lgel$)is a$se_{l}$ )$l\prime etl_{J}//\backslash ^{r}l$) $S$ (

$r_{r?\mathrm{i}\Gamma}All_{1}.\cdot c]_{(}$$\urcorner dt\mathrm{i}$$ar\partial l$) $]_{1\prime}.- ge\prime 7(\neg r’\gamma tcd$.

$/^{)}1’ \mathit{0}\mathit{0}f$old.

cxtension on$\iota,$$\cdots.r_{\gamma\iota}$?

$(^{\mathrm{B}}’$$lt\cdot/_{\mathrm{t}^{r}}\mathit{1}$tin$it\mathrm{r}l_{1^{\tau}I ,\backslash }.07(\neg\downarrow \mathrm{r}\Leftrightarrow(^{\neg}\tau tecl$) $-1i117\subset//_{d}$are$\mathrm{i}_{11(}1$$\gamma$

}.

ficld

$cx\mathrm{t}c\iota 1‘\backslash ‘ i_{\mathrm{t})\downarrow}1(0arrow l$

arc littcarl. disjoint$1^{J}$$\dot{1}(J\mathrm{I}7$$(1^{-}c’\cdot l_{1}$.

faincd in a$t/7cl|/_{d}//\dot{\backslash }^{r}$.

$(_{1}\iota_{1}1\mathrm{c}\mathrm{x}\mathrm{t}_{\mathrm{l}\mathrm{a}\mathrm{C}}\mathrm{t}\mathrm{t}\mathrm{r}\mathrm{o}|11 [\mathrm{W}].)$$\mathrm{t}1_{1}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{a}|_{)}\mathrm{y}\mathrm{i}_{11}\mathrm{c}|_{11\mathrm{C}}\mathrm{t}$where.r)

Lct,$|,\mathrm{t}_{1}/_{\text{ノ}}=f1’(C/1, \cdots , \mathrm{r}l_{1l})$. and lct lls prove$/:=\{.f\cdot\in l_{1}’[.r_{1}, \cdots, .r_{r},]|.f\cdot(r/_{1}.

etcrminatcs. If $/=0,$$\mathrm{t}_{}1_{1\mathrm{e}\mathrm{t}}1$

our conclu.q\cdots.\mathrm{r}/,, )=0\}$$\mathrm{i}_{\mathrm{o}\mathrm{I}}1|_{1\mathrm{O}}|_{\mathrm{t}}1,\mathrm{s}$

. If

$|$

)

$\mathrm{o}\mathrm{t}_{\ell}$

,

, lct

194

a polynomial of minimal degr .

$P(.\cdot l_{1}..\cdots , x_{n})\in/\backslash \{0\}\mathfrak{l},\mathrm{c}$If $P_{1}=\cdots=J$ ),$’=0$ . t,hcn(($Q\in\{g\in I\mathrm{t}^{P^{-1}}’[x_{1}, \cdots.

[$\mathfrak{i}\prime\prime=l\mathrm{t}^{I’}’-1)$.\}.,l]|/$, we can see

Since( $1e\mathrm{g}P_{n}<(^{-}|\mathrm{c}\mathrm{g}/),$$I_{J}’:=J\mathfrak{i}’(a_{1}, \cdots , \mathrm{r}/_{t-1},)$

generated cxtellsion

generat,eed over$I\mathrm{i}’$a$l_{n}^{J}((/_{!}$

thc$\dagger$)$\mathrm{Y}$

the

and

U. This Ill$/_{J^{/}}$)

t,hat,$e\mathrm{a}|1_{\square }‘\{$$‘ \mathit{1}_{7l}$arc linearly clisjoint,.

$\mathrm{o}\mathrm{t}\backslash 11\mathrm{c}\mathrm{S}\mathrm{i}\mathrm{s}$$\mathrm{i}_{11(^{-}}|_{11(^{\neg}}\uparrow\mathrm{i}\mathrm{o}|11_{1}\mathrm{y}1$

Hence

.

$dr?SS\mathrm{I}|$$l_{J}\mathrm{r}^{i}f/_{\text{ノ}}/l\iota^{r}$mea$tl?,\mathrm{t}tJ_{\wedge}$and 1$1”/l\iota’l$)( held
$l1d/\mathrm{Y}’$. ancl ,$\urcorner,\backslash ^{\urcorner},\mathrm{q}\mathrm{l}ll$})$\mathrm{r}\mathrm{x}tel\uparrow Sj_{(}$$\iota 1S$

are $lin$ ea rl.v disjoint over’$\uparrow Cj\mathrm{f}(\mathrm{r}\prime_{\mathrm{I}}.\cdots , \mathrm{r}/_{n})=0$$\mathrm{I}_{I}\mathrm{c}\mathrm{t},$$\{\wedge\cdot,\},$

$|)$

(

a 1

$\mathrm{t}^{r}-\mathrm{I}\mathrm{i}\mathrm{t}1\mathrm{c}t\gamma 1$’

basis of$\mathit{1}\backslash ’’$.

.

$(\underline{9})\kappa_{\iota}\in l\mathfrak{i}’’\mathrm{a}\mathrm{l}\cdot \mathrm{c}\mathrm{l}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{C}\mathrm{a}\mathrm{l}’|\mathrm{y}\mathrm{i}|1(\mathrm{I}\mathrm{c}\mathrm{l})\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{d}\mathrm{e}\mathfrak{l}1\{1$$g_{l}$(1)

it follows$T_{J}$(a 1,$\cdots.\mathrm{r}/_{n}$and$\mathrm{t}1_{1\mathrm{a}\mathrm{t}g}$,)$l\backslash ’’\mathrm{a}\mathrm{t}\cdot \mathrm{c}$$r_{I^{1}hen}tl_{1}cr()$$\in/_{d\eta}$lin$.\{\mathrm{c}_{1})_{t}\gamma 1^{\cdot}\partial\dagger$is$/^{l},//\mathrm{t}^{r}$isin a tixed$Ji^{r}$. l.et$[]$)$|_{(}\mathrm{Y}$

.
.

over
’

$\mathrm{s}\mathrm{c}_{1)\partial}\mathfrak{l}\cdot \mathrm{a}|11;$$.\aleph e_{1^{)\mathrm{a}\mathrm{I}\mathrm{a}}}\cdot|_{)}|\mathrm{y}$$\partial l_{\delta}\circ_{(}\tau l,r\partial i_{\mathrm{t}^{\neg}a}ll\mathrm{t}’\vee$$c/_{1}.\cdots.\mathrm{r}/_{l},\in l_{J}$ancl $g(.r_{1}. \cdots..r_{1},)=$$e\mathrm{a}$

over

can )
$1$

$(^{1}1\iota|1\mathrm{i}_{1^{1}}(1e|\}’(^{\iota}\mathrm{X}1)1\mathrm{e}.\mathrm{s}\mathrm{S}\mathrm{C}^{\mathrm{Y}}C|$. Since,$l\backslash ^{r}$,rly$\mathrm{t}^{-}1\mathrm{i}^{\iota}.\dagger \mathrm{j}\mathrm{o}\mathrm{i}\mathrm{t}\mathrm{l}\mathrm{f}_{}$

over$l_{1}’$.

awcon irred$d(l1\mathit{1}jlanfm()rl^{)}ll\mathrm{i}" \mathrm{i}1?\uparrow l)ct$$\mathrm{q}\rho l^{r}\subset.\backslash ’s\mathrm{l}|ch\mathrm{t}hat.f\cdot|l_{-}^{r}$it$\cdot$(resp. a. gonora tecl$\partial I\mathrm{a}l)l_{1^{-}}$$\mathrm{r}$cxists an open den,$(\mathrm{J}\backslash ^{r}, \S/_{\overline{i}J}]$

)

(wit,

and

$l_{r}e\mathrm{t}f:.\backslash ’arrow\}’l$)$S\mathit{1}l1Oot\downarrow l1)\mathrm{f}^{\mathrm{E}\mathrm{G}}\mathrm{A},$$‘ \mathrm{s}e_{l}$$o\mathrm{w}$$(\mathrm{r}l\mathrm{l}.\cdots , rl_{\mathrm{J}},)=0.1$

Lemnia 3.4.

variet.ies.

$’[|_{1\mathrm{C}\mathrm{t}1}$$\prime l^{1}hentl1e\Gamma^{(^{\neg}}cxj.\backslash ’ t,\backslash -_{P}h\cdot i\in l\backslash ’’$$r_{1^{\urcorner}[]\mathrm{C}11g}$$0=g(’ \ell_{1}.\cdots.’\prime_{\gamma})|=\sum\prime h,\cdot g\ell(\mathrm{r}l1, \cdots, \mathrm{r}/_{n})$

(3)

is

.

.

$g= \sum_{l}\kappa_{\iota}g_{\iota}(\mathrm{f}\mathrm{i}\uparrow 1\mathrm{i}\mathrm{t}_{}\mathrm{C}.\mathrm{q}11|11)\backslash \mathrm{v}\mathrm{i}\mathrm{t},11(j_{\ell}\in \mathit{1}\backslash [’.r_{\mathrm{I}}, \cdots, .r_{\}\downarrow}.]$(1).

$\mathrm{t}_{\mathrm{t}}11\mathrm{a}\mathrm{f}l^{l},’\neq 0$$/_{\text{ノ}}=/_{d}’(\mathrm{r}/_{r\prime})$$g_{i}\in l\mathrm{i}^{r}[.r_{\mathrm{l}}, \cdots, .r_{l},](1\leq i\leq‘\backslash ’)_{S\mathrm{I}1\Gamma hh}\mathrm{r}t7t\mathrm{J}‘ i(\prime 1\cdot\cdots.\mathrm{r};_{tl})=0$

Proof.

$l^{j}/\partial.1^{\cdot}j$$\{\sum_{i}\lambda_{i\mathit{9}i}|\lambda_{i}\in l_{1^{\int)}}’-|.J‘ i\in/\}$

$l$

)

bcl

$\mathrm{s}\iota 1\mathrm{I}_{)}|\mathrm{C}\mathrm{t}11111C\urcorner$

clcg ). Thcreforc, we may assumc$-\mathrm{t}$$l \mathrm{t}’\int$$|1\mathrm{d},$right, hand side is$\cdot\cdot.r/_{rl})\neq$4, a$Q\in l_{\dot{\mathrm{Y}}’}\mathit{1}’-\mathrm{l}[.l\cdot 1\cdot\cdots..7_{l},]$).

$\sum_{i}\kappa_{i}gi(_{T|}., \cdots, .l_{\gamma 1}.)$as$=Q^{\mathit{1}’}\dagger \mathrm{t}^{\vee}\mathrm{i}\mathrm{f}_{}11$$(\prime \mathrm{I}\cdot\cdots.\mathit{0}_{r1})=0\}$. Since$ndg\in \mathit{1}\backslash ’’[x_{1}, \cdots, .l_{1\}}]$

and

.

some

)

( $1_{\mathrm{C}}\mathrm{g}Q<$$\mathrm{I}_{1}\mathrm{y}$Sublemma.
closed fiield, an$/$$\mathrm{t}_{r}\mathfrak{i}_{1\mathrm{a}}\mathrm{t}\mathrm{f}_{}11\mathrm{c}$This is impossible since$1^{)}\mathrm{t}\mathrm{t}\uparrow l_{i}^{J}:=i$$e\mathrm{c}$$a\downarrow \mathrm{i}\Gamma/()nl1^{\text{ノ^{}\vee}}i\{\backslash \cdot \mathrm{x}\cdot(.\iota^{r})/\mathrm{A}\cdot(\}’)$$\mathrm{r}^{\mathrm{Y}}xt_{C)}\prime 7s$

ion).

is a sepa ra )$l$$l\mathrm{t}^{\text{ノ^{}\vee}}$is) $lc$$\mathrm{I}lCj/$.

$\acute{c}t\partial l\mathrm{r}\cdot(\iota\cdot c,\backslash _{j)}^{\neg}$

algehraic extens ion

195

is 11 ecessa$\backslash \prime \mathrm{V}\mathrm{b}\mathrm{a}\mathrm{f}_{}$

easy exerci se. For[$\uparrow_{0}11\mathrm{c}^{4}\mathrm{o}|1$

and

the

$li’=\Gamma_{l}^{Cj}$, are linearly disjoint,

ovcr$/\backslash ^{r}$.

$(_{-}^{\tau},]\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}$be the

Hold

so

‘

so dct$\sigma_{r}\mathrm{f}\mathrm{i}\backslash c$

exist,. $3$ ] and [SGA.

$‘ \mathrm{g}()t$

.

atltI

.

fiieldall eletlell[R2].)$\mathrm{E}_{1^{\mathrm{v}}}(3.3)$$h\cdot i^{\mathrm{S}}\mathrm{a}1e0$} $>|$.

. it$\mathrm{s}\iota\iota \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{e}.\mathrm{S}$I I]. 1’$e\mathrm{x}_{1^{)\mathrm{O}\mathrm{b}\acute{e}}}.$.

$[. \uparrow(.81)^{\mathrm{F}_{)}}.\cdot$

hxod$Ll_{\mathrm{t}^{\neg}}t\dot{t}$t,o show$\downarrow(.9]$.
,$l_{\overline{J}_{\backslash }1J^{\vee}}c.a\mathrm{c}l_{l}$are lin. we have$\mathrm{c}\mathrm{a}1[)\mathrm{i}t,\mathrm{r}\mathrm{a}r_{\mathrm{Y}}.\cdot$

forso that$\sigma_{\Gamma}\in G$.

.

$\sigma_{1}$

.

ancl

$\mathrm{w}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}\cdot \mathrm{C}$$e\mathrm{a}\mathrm{t}\cdot \mathrm{I}\mathrm{y}\mathrm{t}1- \mathrm{e}1^{)}\mathrm{c}\mathrm{I}1\mathrm{d}\mathrm{c}11t$),$\sum_{i=}^{tl}1\bigwedge_{i}\sigma_{j}(\lambda_{i}^{\mathit{1}}’)=0$$=()$ .

$\mathrm{d}e\mathrm{t}(\sigma_{j}(\lambda_{i}))_{\mathrm{I}\leq i.j\leq},,$call assrnle$\cdots,$$\mathrm{t},1_{1}\mathrm{a}\mathrm{t}\tau_{1’}\iota’-\rceil$$\sum_{i=1i}^{\gamma}\prime h\cdot i\lambda^{\mathit{1}}’=(.)$$\mathrm{t}_{}|1\mathrm{c}|1\lambda_{1}.\cdots.\lambda_{n}$and hcncc

$()$

$=$

”

.

$\cdots.\sigma_{rl}\in G$$l\mathrm{f}\cdot\sigma_{1},$$\mathrm{c}\mathrm{I}\mathrm{l}\mathrm{O}\mathrm{o}.\backslash ^{1}\mathrm{e}\sigma|,$$l^{J}1\cdot\ldots.//_{r}\in/_{J}.\mathrm{q}^{1}\mathrm{l}\mathrm{t}\mathrm{t}^{\neg}1_{1}${ is aIl$a\mathrm{I}l\mathrm{r}()’()’ l^{)}l1\mathrm{i},\backslash \urcorner n1,\backslash \urcorner()t\cdot/_{J}$

of

$fS$

$1^{\backslash }o\mathrm{o}$. generated.

(1. and 1et $\sigma_{\gamma+1}\in G\}$ )

(1)

$ot$

)

) $\partial\Gamma’?l$) $l_{1’}$$\mathrm{t},11\mathrm{a}\mathrm{f}_{1}(\sigma_{j}(_{/}\backslash _{i}))_{\mathrm{l}\leq}i,j\leq n$$\lambda_{i}\backslash ’$

grou upconsistin$(\sigma_{j}(\lambda_{i}^{\mathit{1}’}))_{1\leq}i.j\leq$rank$\mathrm{s}^{}\mathrm{c}1$, i.e.. that, if wc have a relation$\mathrm{t}11\mathrm{a}\backslash \backslash -\mathrm{t}‘ \mathrm{a}1^{r}\backslash \mathrm{c}\gamma$

Reorder$\cdots,$$\sigma_{1},$$\mathrm{t}\mathrm{l}\tau \mathrm{e}\mathrm{r}^{\mathrm{Y}}$we$111\mathrm{a}\mathrm{x}\mathrm{i}_{1\rceil}1\mathrm{a}1$$1\leq?’<7l$ .

$\backslash ^{-}\mathrm{c}\mathrm{r}/_{1}’$is t,hc ‘if part,’ who.

$(^{\mathrm{t}}1_{1}\mathrm{a}_{1\dot{s}}).]$$(_{-}/c7$

and where not all$(j=1, \cdots, n)$ ,$r$)$\subset$[Mi.

$\mathrm{S}\mathrm{t}\neg \mathrm{c}$$r_{l^{1}l_{l\mathit{0}}11}/_{\text{ノ}}//\iota’\mathrm{i}.\mathrm{s}.\backslash \urcorner el$$\mathrm{f}_{\Gamma \mathrm{O}11}1$$\mathrm{t}\downarrow 1^{\cdot}1^{)\mathrm{o}\mathrm{S}\mathrm{C}}$a tield,

) $c$

$I$

$\lambda_{i}\in T_{\lrcorner}$)$1^{\cdot}\mathrm{e}\mathrm{S}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{t}1$[cmma of M.Rosenlicht, [R2, 1).

$o11_{\mathrm{o}\mathrm{W}\mathrm{i}}\mathrm{n}\mathrm{g}$. (An extract.

$1)\mathrm{a}\iota\cdot\uparrow,$$l_{r}c\mathrm{f}/_{l}l$$l^{\supset}\}’ oof$$\kappa_{i}\in \mathit{1}\backslash ^{r}$if$91’ l$ ) $hc/d\mathrm{o}\mathrm{f}/_{I}$of G.

$\mathrm{a}\iota\prime t\mathrm{o}\mathrm{f}?l()|’ l)l1\dot{|}.9\mathrm{n}\mathit{1}$$1$$\mathrm{y}$We also 11ccd thc {Lemlna 3.5.

for our )$\iota\cdot \mathrm{y}$$\cdots.\sigma_{n}\in(’,$Let; t,hcn$\mathfrak{c}1\mathrm{e}\mathrm{f},(\sigma_{j}(\text{ノ}\backslash _{i}))_{1\leq\gamma}i.j\leq\neq 0$$r_{\mathrm{J}^{\urcorner}]_{\mathrm{l}1}\backslash 1}‘ \mathrm{d}\mathrm{e}\mathrm{t},(\sigma_{j}(/\backslash i))1\leq\dot{\iota}.r\leq‘+1=0$

.
,

t,hat

$\sigma_{\dot{J}}(,\backslash ’+\downarrow)=\sum_{i=1}^{r}\mathit{1}li\sigma_{j(}/\backslash _{i})$for all $j=1,$

determined
choice of$\cdots.$”$+\mathrm{I}$$|\leq j\leq?$$0\uparrow 11\mathrm{y}\mathrm{I}_{J}\mathrm{y}(|)_{\dot{J}},$$\sigma_{r+1}$

isHere (1).

, and } ence
$1$$\mathrm{v}\backslash ^{\tau}\mathrm{e}$

have

.

$1^{\tau}\mathrm{C}\iota 1|111\mathrm{d}\partial 11\mathrm{t},$$r_{111\mathrm{c}\mathrm{r}}$

eforc

,

and $l’1,$$\mathrm{t}_{}11\mathrm{C}\mathrm{S}e\int’i^{\mathrm{S}}$$\sigma(\lambda_{r+\rfloor})=\sum_{i=1}^{r}/li\sigma(\lambda_{i})$$\cdots\cdot l’$arc, are uniquely)$\mathrm{c}\mathrm{t}1(-1\mathrm{c}[]\mathrm{t}$$\mathrm{i}_{\mathrm{I}1}\mathrm{r}1_{\mathrm{C}}1$

for any$\sigma\in(;.$of t,he$\mathrm{I}\dagger\tau\in(’,$,we have$\sigma(\lambda_{r+1})=\tau(\overline{f}-1\lambda_{r+1}\sigma())=\overline{f}(_{i=1}\sum^{r}l^{r}i$By$\mathrm{t}_{}11\epsilon^{\mathrm{Y}}\iota \mathrm{l}\mathrm{I}\mathrm{l}\mathrm{i}\mathrm{q}\iota 1(\tau \mathrm{n}\mathrm{e}\mathrm{S}\mathrm{S}$of $l’|,$

any onc of (1) yiclds
linearly$\cdots\cdot\int’\gamma\cdot$. we have$\lambda_{r+1}=\sum_{i=\mathrm{I}^{\prime/}}^{l}\prime_{ii}\backslash$$\mathrm{d}\mathrm{e}_{1^{)\mathrm{e}1}}1\zeta 1e\tau_{\mathrm{l}\mathrm{l}\mathrm{f}_{1}}()\mathrm{V}\mathrm{c}1’/\mathrm{t}^{r}$. I$\overline,$.

$(/\iota_{i})=/^{\iota}i$

wit,h

.

$\overline{l}^{-}\sigma(\lambda_{i}1))=\sum_{\rceil r=}^{r}\mathcal{T}(\int\ell i)\cdot\sigma(\lambda i)$, so each$//\iota\cdot\cdots,$ $l’r\in J\backslash ^{r}.$$l^{l}i\in/\mathrm{t}^{r}=/_{d}^{C^{\tau}}’.$

Hellcc$\lambda_{1},$Hcllcc$\cdots.\lambda_{\mathit{7}1}$

are

196

3.6.
$l\dot{\backslash }^{r}=L^{(f}$$J^{)}t’ \mathit{0}of$. By 3.

. ($arrow \mathrm{T}l’\cdot\underline{\supset}$$\tau_{l}l’.$

:

.

}

$\backslash C$. is ti(,$01$ )

(2)

$)$

on

is

$1\dagger_{1}$.

But each$c_{-\mathrm{o}\mathrm{r}}|_{)}\mathrm{i}\mathrm{t}$$\mathrm{f}\mathrm{i}1$) $re$

of}$r||1$)$\uparrow_{r}\mathrm{r}\mathrm{i}\backslash -\mathrm{i}\mathrm{a}1.$en$||_{-}’,$

is an$\mathrm{t}_{}|1e\mathrm{r}e$domi)$1.,\mathrm{e}\mathrm{t}.l_{\underline{J}}^{-}1$$\subset \mathrm{I}1_{\mathrm{I}}$ng (cf. (1.3)).

: I$l’\underline{)}arrow\}_{1}.$’la)$[_{\backslash }\prime \mathrm{e}\mathrm{o}1$

(!-cr

-l

)

$f=\iota\cdot \mathrm{o}\mathrm{o}\mathrm{i}:,1$$\mathrm{I}^{\cdot}$($\mathrm{f},1_{1e\mathrm{a}}.9.\mathrm{q}$$\mathrm{w}\mathrm{i}\mathrm{t}_{\mathrm{t}}[1\acute{\varphi}:.\iota_{3}’arrow|r,’.?$$|_{\lrcorner}\mathrm{c}\mathrm{t},$.

($.\backslash \mathrm{I}\mathrm{o}1^{\cdot}\mathrm{C}\mathfrak{c}$)$1^{\cdot}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{I}1:\mathrm{i}$)$\backslash ^{\tau}\mathrm{c}1\backslash t’$.

$G- 0\iota\cdot|$)$\mathrm{e}\mathrm{a}\mathrm{c}1_{1}\mathrm{f}\mathrm{i}\mathrm{l}$$\mathrm{i}\uparrow$)$\mathrm{r}\mathrm{e}$of$\mathrm{t}_{\mathrm{r}}\mathrm{o}\mathrm{p}_{\mathrm{I}\mathrm{O}1}.- \mathrm{c}(0.1.$$k[.1’,]]^{c;}\backslash \phi^{*}\mathrm{A}\cdot[W_{4}]$$1/\beta$is.

$r[|]\mathrm{e}\mathfrak{l}]$$1^{\cdot}\mathrm{c}\mathrm{g}_{11}1_{\partial}\mathrm{r}$.

)$|_{)\mathrm{O}}\mathrm{t},\iota 1\phi^{*}’\beta$

and

(6) .

of (2.3) rcmain valid$:l\cdot.$$\mathrm{f}\mathrm{i}1$)$|[11\mathfrak{l}O[]$of$\mathrm{I}^{\cdot}\mathrm{C}$

$)$

that,

$o$

of

$(,- \mathrm{Q}\Gamma 1)\mathrm{i}\mathrm{t}\mathrm{s}$is also$1$).

$1’ \mathrm{C}\mathrm{C}\mathrm{i}.9\mathrm{C}\mathrm{l}\mathrm{v}$. is smooth. and$‘ \mathrm{s}^{1}\mathrm{a}\uparrow \mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{c}(^{-}|$)$\mathrm{A}\cdot(\mathrm{I}l_{\}}^{\nu}.)=l\cdot(.\backslash _{\gamma}’,)(\int$$|1(|1)11\mathrm{f}_{}.\mathrm{x}_{4}^{j}:=o^{-1}(|\{_{1}’)$a( $J^{*}i$

$=$

of a

$|/\mathit{0}^{*}.\prime i$

$1$

)

wit

is a

$111^{\cdot}\mathrm{c}$$|1$$1|’ \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{t}1$SOIII$\mathfrak{l}10\iota\cdot|11\mathrm{a}$thc cont,rary.

$\mathrm{i}.\mathrm{c}..\mathrm{t}1_{1}\mathrm{a}\mathrm{t}_{\mathrm{t}}\mathrm{f}_{}1\rceil \mathrm{c}\mathrm{I}^{\cdot}\mathrm{C}$$i\in \mathrm{A}\cdot(\iota\uparrow\prime 4)\backslash l\cdot[1\prime 1_{4}’]$$C\gamma 11\langle^{-1}$

$e$

${ }$

gct ({$.|.|$$r_{111\mathrm{C}11}$all.

$\mathrm{I}_{I}\mathrm{c}\uparrow_{r}\mathrm{I}\uparrow^{r}|/\}$

.

)

cxist
$\mathrm{c}$the$\iota_{\{}\mathrm{n}\in$$1o\mathrm{C}\mathrm{t}\mathrm{l}.\mathrm{q}$[ variety, $\nearrow^{r}\lrcorner:=\{\iota\iota’\in\iota\iota_{4’}^{r}|i(\}(’)-1=0\}$ iscodi IllcIlt ioll on .

are reglllrl’l’ on

.

.

.

.

$\mathrm{c}b^{-1}(\mathrm{t}1_{\mathfrak{i}}^{r}/,)$$r_{1^{7}\mathrm{a}\mathrm{k}\mathrm{e}.r_{0}}\in\lrcorner l_{4}^{-}$

. and
.

we get, a colltt’f1\Gamma ($(^{\backslash }\mathrm{e}\mathrm{d}$. and$1=(\mathit{0}^{*},ii)(.r_{\mathrm{t}1})\cdot(_{\mathit{0}}*\prime i)(_{l}.\cdot(\{)^{-}|=(\varphi^{*}’ \mathrm{t}^{\int}\mathrm{j})(.\Gamma 0)\cross 0$$\mathit{7}\mathrm{I}^{1}\mathrm{h}\mathrm{t}\mathrm{l}\mathrm{S}$) .

$\mathrm{w}11G\mathrm{t}1f:.\backslash _{1}’arrow\}|\mathrm{i}‘\dashv\ulcorner \mathrm{c}_{1^{1\mathrm{a}}}$} $arrow 1l_{J}’$$1e\mathrm{t}_{1}\iota\iota.\mathrm{s}\mathrm{a}\mathrm{s}.\backslash ’ 11|1\mathrm{l}\mathrm{c}$$)$$|\iota_{1\dot{|}11}^{1}\mathrm{c}\mathrm{e}l1_{1}J$$\mathit{7}1^{\cdot}\mathrm{i}\mathrm{c}\mathrm{t}.\backslash \cdot$$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{e}|11\mathrm{P}^{\mathrm{t}_{}11\backslash \mathrm{R}}\mathrm{y}_{\mathrm{S}}1$

Thcn$(3.\cdot\ulcorner)$

.

is a

$\phi$$\{2^{*}\mathrm{A}\cdot[|\iota_{4}^{r}]\subset l\cdot[.1_{1]\subset}^{\cdot}’(j\mathrm{A}\cdot(.\mathrm{X}\prime 1)^{C!}=_{\mathrm{t}^{)}(^{[}\rangle}*\int\cdot|_{4}’$aa Iltl$|)$ncd

Hctl ce we get a lIoIlli nant,. Hcnce eachnon-si ngular locus of 1 \dager ’,1. a$\mathrm{I}^{J}\iota_{4}^{f}\mathrm{I})\mathrm{c}\uparrow 11e$the conditions (0.1, (1) (.5)) arc

where$\mathrm{t}|\mathrm{c}\mathrm{f}\mathrm{i}$.

Hence allIn ordcr

is.

$1^{)}\iota 1\mathrm{t}.\iota’,’:=\mathrm{t}^{-1}’(\nu\iota\cdot.\})$).

$\mathrm{t}_{}$a$1_{\mathrm{o}\mathrm{C}11}\mathrm{s}\mathrm{w}1_{1\mathrm{C}}\iota\cdot \mathrm{c}o$$e|1\mathrm{t}1_{\mathrm{C}\mathrm{I}19}.\mathrm{c}l\mathrm{t}^{r_{1}},\subset[\uparrow^{\gamma}\underline{.\lambda}\partial 11\mathrm{d}1_{1},\subset\}_{1}\iota_{\{\mathit{0}}$$\iota\dot{|}\mathrm{Y}_{C\gamma}r\mathrm{l}\tau \mathrm{i}_{r\urcorner 11}$$\mathrm{Y}1^{\neg}\mathrm{i}\mathrm{s}\mathrm{C}[.\backslash \tau$$\mathrm{i}\mathrm{t}\cdot \mathrm{l}\mathrm{c}\langle 1|11- \mathrm{i}\mathrm{l})\mathrm{l}\mathrm{c}\mathrm{A}\cdot-\backslash \mathrm{r}\gamma 1^{\cdot}\mathrm{i}\mathrm{c}\uparrow_{}1^{\cdot}$

is opcn densc (cf. (3.

$\zeta\gamma 11\mathrm{d}(_{-}/.\backslash ^{j}.-)=.\mathrm{X}’\cdot)\sim$and$1^{)}\iota\iota \mathrm{t}/_{d}=l\cdot(.\backslash _{\mathrm{I}})$$11\partial 11\uparrow 1^{\cdot}r\urcorner\uparrow \mathrm{i}0|1_{C}\gamma 1|11\mathrm{o}_{1}\mathrm{I}^{\cdot}1\mathrm{I}_{\mathrm{l}\mathrm{i}\mathrm{s}11}1$thc$\mathrm{c}$$\mathrm{A}\cdot(\}_{1})\subset l\cdot(.\backslash _{\mathrm{I}}’)^{(}’=l\cdot(l\mathfrak{j}\mathrm{I})=\mathrm{A}\cdot(|\mathrm{t}_{-}’.,).$$arrow 1\mathrm{t}_{3}’.\mathrm{i}.\mathrm{q}$$\lrcorner\backslash .’$ret,urn t,o (2.3).

$(_{-^{l^{-}\mathrm{c}\mathrm{r}}}’1|1\mathrm{i}\backslash ^{-\mathrm{I}}\mathrm{r}\gamma\cdot \mathrm{i}\mathrm{r}\urcorner \mathrm{I}\mathrm{l}\uparrow$$l\mathrm{t}_{3}\prime r.arrow\}_{3}’,\mathrm{i}_{\mathrm{S}\mathrm{S}\mathrm{l}},1\iota\cdot \mathrm{j}\mathrm{c}\mathrm{c}\mathrm{t}_{}\mathrm{i}\lambda\cdot \mathrm{c}1^{\cdot}\mathrm{c}_{\mathrm{o}}\circ\cdot|\iota \mathrm{I}_{\mathrm{r}}\urcorner 1^{\cdot}|11\mathrm{o}\mathrm{I}’ 1^{)}\mathrm{I}1\dot{|}\mathrm{S}\mathrm{t}1$:$\mathrm{t}\mathrm{l}\mathrm{S}$

gcncratcd. Hencewc get a$\partial \mathrm{I}\mathrm{l}\mathrm{t}1$$(^{\backslash }|1111c\gamma 1)1)\mathrm{i}$.

. NNow let,$\uparrow 7\mathit{0}./$$11\mathrm{i}\uparrow \mathrm{e}[\backslash ^{-}\backslash$$\mathrm{l}1$$|\mathrm{n}\mathrm{o}\mathrm{r}_{1^{)}}1_{1}\mathrm{i}_{\mathrm{S}1\urcorner}1\iota,\cdot$$|(\mathrm{i}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\acute{\phi}$a}$o_{l}(.\backslash .-,)=:[|_{\sim^{\lambda}}’.$

is a Il$\mathrm{E}\mathrm{y}(2.3,$

rational$/\mathrm{t}’/l$$\mathrm{I}1_{1}\mathrm{c}$and smooth. Put$\emptyset:.\iota_{-}’.,$$\prime l^{1}/7\mathrm{r}o$$J\backslash ’=l\cdot(1\mathrm{t}|)$$\iota\iota/’1\mathrm{S}’ 1C\mathrm{I}1\mathrm{t},\mathrm{h}\mathrm{a}t,$$\acute{\varphi}:.\backslash _{1}’arrow|/l_{1}^{r}!$

.

$|)$

$($

of

$((\mathrm{i}))$. I

so that$o(.\}_{(1})\in \mathrm{Z}$

.

197

3.7. Reinark. In ordcr t,o si$l\backslash \cdot$

is algebraically$\mathrm{c}\mathrm{l}\mathrm{o}.\mathrm{s}$essent,ial. and indcc‘1(

and$\mathrm{t}1-$not,$\mathrm{a}\mathrm{l}\cdot \mathrm{c}$$|111^{)}1\mathrm{i}\mathrm{f}\backslash \tau \mathrm{t}1_{1}\mathrm{e}e\mathrm{x}\mathrm{l}\vee$is$\mathrm{t}_{}11’\gamma \mathrm{t}I.\backslash$$\mathrm{i}\mathrm{I}^{\cdot}1’ \mathrm{c}\mathrm{t}-|\prime \mathrm{l}c\mathrm{i}|$))$1_{\mathrm{C}}$$\mathrm{O}\iota.\mathrm{i}\mathrm{i}\mathrm{t},\mathrm{i}\mathrm{o}\mathrm{I}\mathrm{l}$.

$\}_{1\mathrm{t}1}\mathrm{f}$, wc$\mathrm{a}\mathrm{s}\mathrm{s}\dagger 1|1\urcorner e11$t,hesc)in (1). )$1$$\mathrm{f}\mathrm{i}\mathrm{o}\mathrm{I}\mathrm{l}\mathrm{s}$$\mathrm{a}.\mathrm{s}.\aleph 1\mathrm{l}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{l}$are$\uparrow,|_{1\mathrm{a}}\iota$$|1o\mathrm{t}$ill [R1].

$\mathrm{a}.9^{\iota}\mathrm{i}11111(^{\}}\mathrm{t}|$References[EGA] A.Crothendieck,
(

[SGA]

-ノ,,

$/|o_{l)}tc\}^{\backslash }.\backslash -‘\sim^{y}$$h^{1},/\mathrm{r}’$$1^{)}1\mathrm{l}\mathrm{I}$’}}77rfi.if a lgibriqnr., (hap$77(\dagger.\overline{\backslash }d\mathrm{f}g\mathrm{r}\mathrm{o}\prime\prime$I H ES)|.

ct al.. ,$\mathit{4}\backslash .\mathrm{t}_{-^{1}\mathrm{r}\circ \mathrm{t}}\mathrm{t}\mathrm{h}_{\mathrm{C}\mathrm{t}}1\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{c}1\backslash$in Mat,h. 224[Mi] J.S.Milne.

$(1.9^{-}||)$$F,’ to/_{\Gamma}$[Mu] D. Mumford$\mathrm{c}\uparrow n$.

$\mathrm{s}_{1)\mathrm{t}}\cdot \mathrm{i}$inairr$d_{\mathrm{r}g\mathrm{r}’}\mathrm{r}’ol7f$$(\mathrm{I}^{\langle}.- \mathrm{I}- 70\mathrm{I},\rangle-)1^{)}1\mathrm{i}_{1}1\mathrm{g}\mathrm{c}\Gamma;\mathrm{t}\dagger$}’ir$o/g\zeta b\text{ノ}$}’$iqu\mathrm{f}\text{ノ},$$6|(:A/..\mathrm{I},(^{\mathrm{Y}}c\mathrm{f},\mathrm{t}11^{\cdot}\mathrm{e}.\backslash \tau o\mathrm{t},e\mathrm{s}$ngcr.

cohom ology. $(|(.)80),$

al..

l,,

1967).

$(|.\iota)6|-$$\mathrm{S}^{1}\mathrm{r}’ 7t\text{ノ}$$t_{\mathrm{f}1}’$$(^{\mathrm{v}},-\mathrm{r}on7\mathrm{r}f$}’.

$1^{)}1^{\cdot}\mathrm{i}_{\mathrm{I}}1\mathrm{C}\mathrm{e}\mathrm{t}_{1o\mathrm{I}1}1^{\gamma}\mathfrak{l}1\mathrm{i}_{1\prime\cdot \mathrm{e}\mathfrak{l}}\cdot \mathrm{s}- \mathrm{i}\mathrm{t}\mathrm{y}^{\mathrm{p}\mathrm{r}}\mathrm{c}.\iota;.\mathrm{q}$ir. inrarirrn t th ro}’ y, ,3rd r.d., (1991). Springer.

[N] M. Nagata, }}7l) $cddi_{1},g$ of an absfiact rarir ty in a co’7l)$/$$/ct\epsilon’\tau io\}^{\tau}ic\dagger y,$.J.

$.\backslash 1\mathrm{a}\mathrm{t},|1.1\backslash \backslash ^{-}0’\sim \mathrm{t}\mathrm{o}$1-I niv. 2-1(1961). I I .
$\mathrm{t}\dot{\mathrm{J}}$[R1] M. Ro:ielllicl]t . So

$n7’‘ l,a.\sigma?c$

${ }$

101

$\sim 1\prime \mathrm{t}3$$\mathrm{H}.\mathfrak{i}^{\urcorner}\supset \mathrm{t}\mathrm{t}\eta \mathrm{i}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{r}\mathrm{o},$A. ($\backslash$$^{l}C^{1}\mathrm{i}1,$$(^{\mathrm{t}}01- 1\mathrm{o}\mathrm{q}$.

}$- 7/\cdot\backslash of$$\mathit{0}\}^{\backslash }l^{/_{71}}’.\backslash$on$0/\mathrm{r}b_{l^{\backslash }}a|cg\}^{\backslash }\mathit{0}?lp.\overline{\backslash },$function$f\prime^{1},q_{\mathfrak{l}/}j_{?)}a’\cdot i\mathrm{r}’\}\}\dagger r\cdot \mathit{0}’\}7l^{)}/rf|\mathit{0}n$//, ibid. 15 (, 1975). 5736$[\mathrm{V}’’]$.

$\prime 77^{\mathrm{q}}$Alllcr. .I. Math. 78 (1956),.

[R2] M. Rosenlicht,. A utom
[S]

th corr$fl_{ol}^{1}\}d\mathrm{r}\prime fi_{\mathit{0}},’.\backslash -$. Publ.

of$(\tilde{\rfloor}_{\mathrm{J}}^{\ulcorner}.$,l. .J.

$fi_{\mathrm{f}/d.-}\backslash ,\mathrm{r}_{1}r$$\mathrm{K}’\mathrm{I}\partial \mathrm{t}_{!}1_{1}$

als
l

A

. Kyoto$l\uparrow_{\rangle}^{(},’ 79$$[\uparrow|]- \mathrm{i}\mathrm{v}$(1955). 1 11.

. 14$(\mathrm{I}9\overline{\prime}|)$. |28;.
)’)$al_{\backslash }\prime\prime\prime l$$\mathrm{Y}\mathit{1}|c_{J}\prime\prime \mathit{0}\mathfrak{l}7$’ $try,‘\sim’$

nd r’d., (1962),$\Lambda_{\mathfrak{l}11}e1$. Mat,h. Soc.