Title P[3]内の曲線の分類と, 関連した話題 Issue Date URL http ......Furthermore, the...

Title P[3]内の曲線の分類と, 関連した話題

Author(s) ロビン, ハーツホーン; 石田, 正典; 佐藤, 栄一

Citation 数理解析レクチャー・ノート (1977), 2: 1-74

Issue Date 1977-03

URL http://hdl.handle.net/2433/216005

Right

Type Book

Textversion publisher

Kyoto University

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Classification of curves in ~3 and related topics

Introduction

The purpose of this seminar was to study the classification of curves

in w3, and look for applications to a) subvarieties of codimension two in

Pn, and b) vector bundles of rank two on Pn for n ~ 3. In particular,

we hoped to read the long papers of Max Noether LlO] and G. Halphen [4]

from the year 1882, to see what their results were, and find modern proofs

for them. We succeeded only partially.

Of course, curves are classified first by the numerical invariants

d=degree, and g=genus. After that, for fixed d, g, one knows from the

theory of the Chow variety or the Hilbert scheme, that the curves with

given d, g forma finite number of irreducible algebraic families.

Ideally, one would like to know which values of d and g occur, and

then the structure of the families of those curves.

Already a non-trivial problem is to determine for which ordered pairs

<d, g> there exists an irreducible non-singular curve X of degree d and

genus g • jp3 in • We spent a fair arnount of time on this question.

can summarize the results as follows.

a) We always have d > 0, g ~O.

b) If X is a plane curve 1 g ~ z-<d-1) (d-2).

c) If X does not lie in a plane, then

g ~

4<l2 - d + 1

4

1 2 4<d -1)- d + 1

- 1 -

if d even

if d odd

We

Furthermore, the maximum is attained for each d, and in that case X

necessarily lies on a quadric surface.

d) If X lies on a quadric surface, then for some 0 < a < d, we

have 2

g = ad - a - d + 1, and for each such a, there is an irreducible

non-singular X.

In addition to these results, we have some information about small

values of d and g, and about curves lying on a non-singular cubic surface.

This does not yet completely answer the question of which <d, g> occur.

There are two further statements in Halphen's book, which taken with the

above would provide a complete answer, but we were not able to understand

Halphen's proof nor find a new proof of our own. They are

e) If X does not lie on a quadric surface,

then

g ~ ~(d-l)(d-2) + ~

f) If <d, g> satisfies the inequality of e), with d ~ 1 and

g ~ 0, then there exists an irreducible non-singular curve X of degree

d and genus g on a cubic surface.

The matters, along with some preliminaries and related results, occupy

the first 4 sections of the notes.

The fifth section is independent. It gives some examples of curves

which have no smooth deformation in various senses (examples due to Mumford,

Peskine and Szpiro, and Hartshorne). It still seems to be unknown whether

every reduced irreducible curve in P 3 has a smooth deformation in P3•

Sections 6 and 7 deal with the notion of liaison studied by Peskine

and Szpiro [11]. In particular, we discuss the equivalence relation on

- 2 -

the· set of curves in IP 3 generated by liaison, and the connection with

3 vector bundles of rank 2 on .IP •

It is my pleasure to thank all the participants of the seminar for

their interest and support. And especially I would like to thank

M. 1'1aruyama for organizing the seminar, and 1'1. Ishida and E. Sato for

writing such clear notes from my disorganized lectures.

i o If~ 6 {)M.cl 7 If

â:t. \7683)

- 3 -

Robin Hartshorne Kyoto March 17, 1976

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'" j) t 1j l,t =t 0 IF '& b o

S--;:-ww-,& 0. 2 IP2 J )< CWV"l (_ 'fl.tPt·A0-.~

-( I;;.< l. t j::' l 1 ?51" ~6.e)d C~fo-n.b>-_;:(, ~ M- ,r: 1j. (}) ()~~X

-= o( 'li 1,i 'l X 0 r'.itJ 'f< [x~, x,, x,] ::> lx IJ.

16

1-& ;t ~ cb.y.-u d..o; i~~~ Â~ + i' Ix =-(f)

t /j it o P"" c)( J -a_ -t~ (X) t ) , °' w l - />t o d (j) .vµ.cf

..âe-~(.Q_

0 -7 ~X --:> ôl-.JPi ~ Ô-x --7 0

1 ~ à} l 1 -z r1 x "' & iP 2 (_- et) -i" t ~ /J' s IL (0-x) ~ X( &w·)

-x(<JJP 2foO) X (f9_w1) = (, t. 0 (rJpi(-cl.)) = ù,.

-A-'(rJrr1 t-J_J) =o 1 li1(éJ-ffe·éd.J) =-/i,0(gœi{::-3-rd.V =(':rJ.t') = (ol; ') J- 1 z ?C((Jy.) = 1-Cj_;')

11,_ (X)= (ol..; 1) = i(oé-l)(cA.-2)

17

2- ô.l § 0) §~li. f 3 (}) cP CD CWw.e. ô) '7J Î~. ti 1 :.~ ~

cc = {(ri)~> 3 X Y\.S. ~ c li?' l rk~ X= ol JMU4 X=â f

i?~og ~ r 1::<5~ o L6' l ::.6Ji-G~'1t-il~~t1f:-t h-:.. L.

1-J !il 1@i JJ ~ 0

~ ô) t,g ~ ~FoY\· ~ l. t 1J t . >k <f) :::.. 'l ? 1) 6) ~~(*À-_s 'l'Lh 0

1 ) lf ~ ~(?~ J:" 3 7i ~~6 \î i) ;t & < clJ ~> 1.: J.:t l . ,X 15~·~.B:i h t li) li:G~

12 'l Al·f 1) 1::. ~ t J7- 1-t <cl, ij-> f_ é( ~ cl2. i o < ~ ~ i_(cL-l)(ct.-2) t 1J: lcl o X c ~ ro3

cl.<G o/. Jl .:: 1i. fi X -7 7C.(X) 15~· 1>1.,d.1tvwl i ~ ~ ~ n.

t fJ ~ JYLojeJ<.cn--. lL ~t-:\\. /·f Ti.f...X) ,, JP1

~ 7C(X} = cl i' ~ X = (jC/rWA Tc.(X) s_ ~([1.J:f))

= ;fCol-r)(cA.-2.)" ::i.) (.: => u -i :J == i_(c/-1>(ol-2) f& > 1-J."

xr~ïftif:t-t~. P~ 1P~(N=(Ji1)) 1~ cl.:,& [? 2

-t À~ "l. lP"107 ~ 1J ~;wi-;Jtutt H ~tL-~ t t<. tf, ~ T~ r::. cl:" ') H fJJP~ X prv /J JPL (i) ~'fWl Q <J) n~ Cw'w-e

18

t fJ !-;; ' &a-~ 0. 2 { 2 '& 1) ~ ::: /l(X) = i {cl-!fll-2).

J' ? -Z.. X~ JP'-c:::__.,;, lP3 ?)~·, ~ ô1 l;t{9'1 'L 1j ~ ,

~ ( :::_ "=' l) l, H~ 1-J . ; k 6) J: -) 1 ::_ JJ;'\' L l ) h 0

o) Xc_,,JP1 ='7 ~= ~Cd..-1)(cl-2)

b) X ~ \P2 =?> < f ~l - d. + 1 cl:-W-eN1

â - l Ql'-- 1 - ol + 1 cl: ad.of /.f

c) :tL 3X ,-0;t j~Cd..-1J(rÂ.-2)+~ r[-:0(3)

~ > ( tcJ-1J(J-2> h 1,2 (3)

~ X1JiQ ~ ~c_e_{~~'à'-~ho ~~C:: ~::: 1- cl-t Ào{_ -;\2.. 3_A ~ Z.,

cL) < â < [ f(ol-1)(d-2) + ~ ol~o (?) 0

- - ( ~ (cl-1)(rl-2) cle.1,2(3)

~ ~(_ ~llCR {~~ ~;j'i_,l,;. rh_J c{ ~ ~ 0) fi~ CWvv-e. ;;j\" 4.f::ri ~ 0

~ O'J 3 -5 c) , ri) 0 t~ IJ :t- 8A i' t/? ~ , ~s L c) ôJ X .?5"'

~ /,t rkc ~œ c:: it. ;1'1..lf'l rf 3 = /- d -r). ol- )..1

t l (-j- () ~ '[ 1t ~8~--Z"-'3 h 0

è( -&~!\· l,}\Kq 12. 1,r;7z 0foq1i~1<- r ? , Proble-m l . 1. V~ '> o 12 t:t L -i ~ ~ OJ

atdw.J: uvw-e fi 4.f:t i ;; '5' ,

CD !Pi!~ i â: h !;, J.e~ ol tfJ non-~M cww-e /~

19

~= ~(o{:-1)(ol-2) -z"

#i ;a ?Î' _; . â 14 /Éi ~ 0)

J:- 7 7J îiii Îi r ~ .

cll / 2 3 Lf- 5 6 â 0 0 1 3 6 10

Q) v rJ 1 v e t= IV 1: f.1 l ~ Tlvrvi. ~ 'J

~tw- cl (f} n&n-4-in~L ~« b''itf:ri 'ho ~ <J)

- ? 11 Fel ~ ~ fJ Z:: ~. -i S 1:: ~ 7~L J:' ') rMF e CîJ ÂM/1.,.-fa.CQ. Fe. i' Fd.. !1 Fe ::=-X lf~· n/JYl:Â/À~l.

CWv0-l l'. '1 ~ -t ôJ~'",;ft;é: i h " r}_g_~ X :::: ole WJP3 = (J {::-Lf-) wFv<. ;; fY-r~ ( J-t.;-) wx = Gx (J.. f e -If) t 'J 2~-2 = ole-ff wx -

cl e C ci+ e - !.f-) &- ) z. ~ = -dz Je C ol f e -4) + 1

l' 1 ~ 0 JP\D tf (J.) Fe<. ~ 1 1 3 If

Fe (J) c~ ~d'on I 0 0 I 3 (]) J-f--11.-M (f)1ft_ IJ/o â) 2 0 1 if ~ 4- .::i r:: n 1;; 0 3 I 4- Io I i

If- 3 q I q 33 @ '(lf!VI.-~ ~(_ ~<.R. Q c lP; -g

~1'-ho G ~ r'xr' i'ifJ 1J' fPLc.[p 1x.ll?1 = ;Itf!g c;·-jo~ 0 ~&r D 6 (!/\-. b) (.:CftL. D: 4 ~;&-

~ o.,,b >o ~, 0 \~ T/vr.1. ~ ~ o._, b '1 o

1J ~ If (9- , b) 1 .è j,-1 h Yi1JY1..-~w.. CWv'V'-e c p'x 1\? 1

20

75'"iftA 1 ~ o x_(f) ~ 1~ o_J:, -o..-b + 1 t 7J ~ o

r;v-~ ~o..ce 1: '.) t• <- ttf(~i'10_ < ~ L < ~~ ~ t (: i "1 • .1 J,- b = J <5J l 1 ~ = a._ - I -L'

X c:. ; J!?'x1P1 ;-,, 1P1

~~ ~\ t fJ 'J x (~ itJr dt-i1-c r IJ -90 ~:fL (~ f ~ If~ > o r.:: JA L . ~ ~ <D u.~ -1~/ii tri h 2- l ?),,,

%-:J-6' Îi. ~I:: ~) / IJ, ~ ~jW"~'- 12 t~~o <'.Thm 1 _ 2> ( HJrfl{);vl_) X~ Jb'vWl ~ ([) ewvV'l

t 1 '!;; t~ ;~rar~(c@. \) "ri (}11_-Aftd IJ ciR.rw- cl 0) ~ C<IVv<pft

~rrV' ;6~' i?tli:i 1 h, iv ol '2: ~ t 3

R~ ) D ~ 'f 6l ?1 r;;v /Jfddc,) 7J ~1»2- d 0

~()'(' l i h l =1 ' ThM 0' 2 J:" IJ D: 'W/"(J {lfti-! ~ V P,G ~ X -/L 0(D- P) ==- ~C(D)-- / { 0(D-f-O) = -h 0(J)) - 2. . c ~ 61.5"' R-R ()) 1î:t~ J= 1) ~0(D) ::::_ ol-~t/ -A0(D-P) ==- ol.-1 - ~ + 1 + i 1(D-P) { 0 (o -P-Q) = d._-2-~ +I +{.'(D-P-0) d:, 2 { 0 (D-P):=

fi, 0(p) -1 i'i-,fa(2/j:. 0-() /1o-vt,-4f-lc,;_J; -Â0{[J-f-Q)

= {0(D) - .2 i'-kih 121-J D-P-Q non-~ùol 7."-bh

2c;)''' .À,l·f/f IJ-::>+1Ji"~/;;o D-P-Q nrm-~Ct.J.._

-i"<V>lh. If D-P nm-4;-edJ -t-;f; iJ75' :S D- P-Q /5,"11

* À~ 1-J«t· ll 0

D-P-Q A-p.ei<_JJ.~ { 1(D-P-Q) > o

~ { 0(k.- DtP+Q) )O

8 3 R1, --·/Rt E X i:= 2â-ri

--<IJ. Rif -- ·+ R t r\__. k -D + P+Q

3tXITT "JIÀC~ ~ , Cf t >n~~ cr: xt+ 2 JrL

~ w (P,Q)R1)---JR-t) 1 > k'+P+O-R1--"-R-1:

'l i ~ 0 t l --t-t-2 < 9r 1& 5 /cl:" k. JJ.. = ~ fS'lj- "5 3 D D 'Î IWI <f ~ 0) D l"d- V p > Q D-P-Q Î1rJY'<4perN,l

-t+2 L~ 6)l=J1-;J. <f l'J ~'.e~ è'._fj, -Z.. v't>

o&.d D = d. /j: 3 P, Q D- P-Q A,;ie~

Q.E.D. 1<..rwtwJ 1 . 3 . X 1i lPi (:; 1 ~ ;fi, 1J l l JP3 6) ?ta>t-

Ahc;J'i'/L e-wvv-e -{ D::: /-/ fJ X ~~ t 1 ~ 'l ~ T~ t 1

) 3 < o&m 1 D \ < ~ ~~ D d="; L

dv > 6 L;J' t ol= 6 ~ D =- k or X : /v;?M­dlft;c

21

No 1 2 3 L/-- 5 6 '7 g q

I 2

3 Lj-

~

b

·o 0 1

1 ----

• 0 • • 0 • • • 0

·--

• • • • D

o~~ 9 Th1n 1. 2 t:: J:' 0 H n X ri61'l-4p!ÂJ

7'J Cvl/\Ne

0 F:i · f3

(0

01

ci =- 6 OJ t :S . ~ .:h ~t-9~ 6l ~ i'f ;J:iftki: L ~ l l ~ t IJ . ;71_ l::

If t 0

îhnt l, 4-. 'f -a ~ <J- ô) eww-e. D 1: X .t <fJ ~ ol ôJ WfaiJ Offe-p_ ~cr i' D 1~ .4ftd..ti/ 20 /~ nrm-A~Gwi i' ~'/Dl L 3 <:'. i ~ t. d 'L. 6 -z" ~(cl -t-2) s_ ~ < 2d.. - o( t 7J h o

~wt ) J 1 l~ -2 fJ > li' { 1(0) -::.-fi_0{f!_-D)

== 0 t 1J 1J D n1m-4p.er;,;.J. à:" , 'Z. 1~ IJi: l:: 1) d <

21j._-2 Lttr"?-Z, ~(c(f2)sS\. ~/Dl>J

i'~/;)6' ~ Uiffo~d Tlvm d= 1) 3 < ~/Dl ~±cl

~., z cl> 6 1l~ l;f <l s2d-g i'-ki~6~', X t JP3 11! (::: Bx(U :: <Jx{JJ) 'l Tj ~ . y_''Jc 11 ;éwn-e. (:_ -ti 1' /f1.

7J ( ) J: -) 1::; -J i:tJ ~ d-1- . ;fa_ OJ ~ut ÂlF Cf. ~ '5 ;Z_ 1.i o

o -7 dx(2) -::> &r3(2) -7 &x (2) --?o

2.::11 r IJ ~J- 4.L~œ

o--7 H 0~x()_))--? H0( 01P,Q-)) 7 H0

((Yx(l))

.?5~'1~ Sk? 7.F H0 ( GJP3Q.)) ôJ)}z_Jtlt /0 i'{IJ) JÎ1o --z."

fJ t )1[ rj:. J]?'()) r;v-wh.J., ~o.ce. (~j_~ L. >l ;;\'t.7)"'

l-f 0 (dxC2)) <JJ 1l ·-c-Jy h ~ l'. fJttx i ~ ~ hVA-(a<-e.

i'.S''X ~i;l ~ 'l t f5l~i'-hh" ~ ~ -i' X CG1 XCG2

01 /J1 ~ F~ Â.Wl.f(!( i' QI =F (),_ ci~ 'l, X/-J ~C//M. 1.:- 1 i ;Ft. lj t J 75' s . Q 1 ~ G .i. 11 Wi..ru:LJ.l 'l i'J. \J x cQ1 rO:i. t ~ 3 ;5"'01no2 li rkr-u- Lf- 0 ~ fi'fJ' 5 ~ te fi 1i 9 4~7J u o ~) L- H0(dx (;!)) 0) >Pit li. ~ 1 I , ;\:- , 1- ..J0.t> J; /Je..~Q. t 1

) Îi, o { fJ.x~)) =

-/:..0(0x(20)) ~ 9 2D /~ n15Y'..-Af€Vl.'ol t'IJ''5 !<-!< t ~ 9 < { 0(rJ1J}.D)) :: 2d., - CfJ + 1 J- ? l. ~ < 2oi.- B

iif't4~ > h h 0

Q.E,D.

::. 6) Thmt r:: J-, l ) L. rl = b t lJ. 1êi (J) 1i ~ T/vn;t

d:" 1) rf:J D"' o ) b) D = K ) c..) x: ~< ?} ? D

= r E 1 s. r i ~-1 i'l) ·f ~J:l' -Z:'~ J.i 75~', o) ri ci t. c, J: ~

i? ~ {~ /j l \ b) \)) j~{; 6 :::: ~ k = 2 ~ - 2 :r \) ~ = 4- --i"

~ h rf', X ~ !P3 rJ ôJ h. t F3 â) UJw-µ ~d'Oh l'. 1 ht o&.~x-::::2,3==-6 â=l·2,3(2+3-4)+1 =:'f

Wx = 6Lx (1) 'l fj ') 6fx(1) = Gx (D) 'li ;;\l 11'. -fl'<P> =­

{ 0(&x) = / rv DAteuJ i'/..19-D)=/v0(:-D)=o

t ') 2 IJ nh(-4f€w.f. :r , l. cl=- 6 ~ = 'f â) t\t RMt, ~ ()) J: J t~ ~OV' ~ :/:1 ? CWl/v'e 11 it+~ i ~ o C) l'J-

(~-1) E ==- k d= ') ~ E ?1éÎ, ~ !HM~ . E ri baM.. jJ~ ~ ···{1;7;6'5 Yt, \r:ol, .. >~-/ -t llH.N(

~rÂ- -c-J; 'J ~~ i1 u, i t d = 2cl- g tfJ 0 ô!l"J:,

li. 0((}.y_@)) = 9 'l /J 0 'l -:1 i". ::. ÔJ 'l :S .!)<: 1 -li,0 (<1-x<?-J)

:= 1 i'11"11-5 . Xtd: 45 G ~ .MA-{a.~ r~ i î ;tt.

ho Tfvm \. ~, X t ~t ~(Q f:c4 ~{'<..

IJ.t> ~ ~ 6J CWvv-l- 'l L. DO>~ d. 6J ~ AA1ph. ~cr -c 2. D li A-pe.C:.J 3 D r# ri!Jl'l-

~u~ ?J =t <f) l".. i f.i 'C ol '2 ~ i' 0 l= 9 ~=-/0 ~1il-J. 2) ri-Z.10 i' ol+t<~ ~ 3J.-/q Li1

25

P~ ) Thrn 1. 't O)~ifef\ r IJ AiJ'·· ~(_, M<ltjw.e ..

( ~ 3 "=J ,fz_lJ ( t ;fl 1 ;f -h 0ldx (2 D)) L. I o 7-'. -;V; ') . 2 D Apeu(;J!.

J: IJ Jim. \2 t::;I ~ ~. 2..J ::::: ot. r- J -z.. cl > q cl:=. 1 'l 1 h â,J 11- 2D :=. /( W t '! -i' ! <6 ==- 2~ - :;_ j:" 1) j :=. 10

:::. :t'Lld- JP3(f)1f 0) F3 t f, (j) <..&r<µ_ ~~ 'l L

·-L.rk7f~: et+ ,a j 1d. ~ 1.. cl 2 10 07 'l ;j . X 12 ff1 (]) r.P f~

O"U) = OxlP) 'l-i;.f;J r 31~ > 't.! 2-J+. ~J 4l~ 0 -7 dx. l3) -7 &_pl3) ----?' r9-;<. (3) -7 0

0 -:> H0Cdx (3J)----'> H0(c9w3(3;) -7 H0 (Üx (3))

-~A~ fci - X c f3 , f-3 1 F> _, F/ ~è A.Wi..fli\_~ F, 4- F;'

l i h 'L . X/;~' F~ .4WL f.C/2 {::; ~ 1. ;fi_ IJ (} 2. L l)' ; .

F; l F/ IJ JÀM_~ 'l t ~ X c F;, () Fj I h fi F, I

oltruJL cr ô) ~ '(_Id. ~ <k:; o è. r o :r 9 . ~ :tt 1i

-;Y} ~«~ 1; u 0 ~) ê -h0(dxGJ) ~ 1 /i0 (&r' (3)) = 20

-i'-J! f;J ;:5' j Ji,D(<'Jx.(3)):::: -f,,D(f3x(3D)) ~ /q R.-P.. ):: ') /9 < { 0

( &x(3D)) == 3 D - ~ + 1 (3 D : nmt-4-µu.J) ~.,

L. ~ < 3 J_- / g ~ t:2_ 2. D AjJUA:J.. i- 'J 2ol < 2~-1 :r., 7- cl+/<';}

G. E. D.

26

3 D ~J, 'li 1J 'L ~ Thm. d=" 1) d;;m /3DI ~ -:} 3c,{_ -i" lJ /;} (/ Tiwt I , $ (f) ~ite~ ()) tf î" i L f. J::- ) {::,

X ;J;;' p-o..k~ ,Uv~ft(_Q / ::.1. î :fL 7'J lî :h t;t." ~ 0(3 D) > 19

t J (_ j g ~ i d i '1 ;f; s cl 2- 1 2 '* iS,q'~· i' f1 h ~

) l:: rd.M:. ,i.wtf~u . .e 1::: i î -ft. 6 Ylon-~;Jw< ûu'vre

{j ol = u + b } ::: Ct. b - i - b î I CL, b t: ,fV o; fi~{.: l lî h ï5~· l s cl~ t l o ~ ;i 5 to (]) ;~@-fl:J: TlvM 1. 2,,

1. L/-, J . !:; ôlf~IID 1::- ;\ S 1:i- U 0) Id: J,::: ~ J::: ~ 01 8î t-''11

-( ~ 6 . ~ i <," 0) ;f!;; "'- Ïj_ :tz I~ L L d-f 1,) 'l

• Thm 1. 2

11 Thiat 1 . Lf-0 ~è@l

~ Thm \. 5 o~C, \j]

~7l-f o! 1 2 3 11- sr:,' 7!g q l10/11[12i13l1~l1s \ 0 1 i 1 '

, ........ . Q ~ Ui.!i/V"€_ _ 1 0 e e " • " e

11•••111111@• 0 -12:'1 ....

2'7

~ 0il *'~oj~ -0 li . 1..;;\'t. i' .;tt dJ ~fr t ~ fti f: i ~ 7J".

t.> h t i if7- 1-i. 1 ~ 1; lJ !i Il{ (~" 1J 5 7J ( ) 2 t 1f 7ï' l --z: () Q ?)\"

tr~: 1:;.4 ,(j -t fJl)' '(; 15' t~h ;5· 5 7J ( ). i 2 -z;· /?

&~;b l:'. l -z. lÎiw. 1 . 4 6) fc 1IB. i rJ ti ~ D µv:J.

2 D Y1011-Aµuo.1 'l 1 h ~ a..ir& dh--tAor /) 1i f~

'/ C-Wvv.e OJiî+li L~~~~(;lill~'> &'! ~==5 t:;. ?Cie

'lf~,f+ft 1 9 ~' '() ,,, gJ~/\" 7 If J:' ? .

@) rti'lij <1)~1 E.~ ~

X f? ~ 5 <J) ewvv-e. D ~ rle.;ru '7 ôJ olv-t/Jôr

t i li R-1< J:' 1) le 0( D) = 7 -t- / - s --f liYD) -c ;fi & ~\" -li,1(D)

= -t. 0(](-D) IJ, ch_~(k.-D)::: 1 1§..11' T~1 1. 2;} :5. D

NeNj WltApk i"-#i Îl 1-f D ! t /JfUc<.J --c cb ~ ïJ 5 /i (D) = / 'l 7J 1

) -li, 0(D) = 4 'l 7J ~ o

[) 4 CWvipi.: ~ V p, Q E.X ~o(.P-P--O)~-fto(0)-2

B ~tb) :-fi,'(D-P) =-/t,1(D-P-G)

"'P,QEX

~ 3 R E-X R=-\<'.--D i"VP,OEX {

0(R)= -{C>(P-tR) =~ 0(P-tQtR)=I

'l '2 3 i' j<f: X ~ fr-- L ~ 'l ~ {

0 (R) = 1 ~ X: 'Î~o-viJ 7-"/J () Yp & )Z ft,0Cl<iP) =- 1 ~ X : lvt;wJt.fti-1., --z,"lJ l J

2B

i.0(R-tP+Q)=I ~ X ~ ~~~,,1 <-''nt J

ewvw. X t$'" -tv:i!ThJ ~ 3 ~or D ~d/)=3 4!DI>\

t , -z X 15~' ~ 5" t5) ewvv--e (j) t ~ X iJJ..3 ~ <."

b;:!z (~· 11j,o; ~or /) dej D = 7 /J ~ ~ -c 7J < 'X ?S'" 1N:.?ô7''J i' (J 1t :11. tf /<€: X ~ -) 1- <-fi 1. -i'

D ~ i< - !< 'l cJ;, 11 li' D f* dt~ 7 -i" ~ aw.;h 'l

lJ h 0 i ~ (,''f J1~1 t ~ t; (j) "bi;,3 f/'flJ, -t 1 ( ) c.wvv-e ;S\",{}Jl i h ~' t") ;)' ('._ \G'J1@-Z:'~ h.

g"P i, 6. ~ ~ OJ -/;,,i,JrmJ i'~( J ut1wt

ef~'t+:f.:t 1 ho Jt'~ X t [llf 11'1 -z:· X= f, · f;. · F; cow-µ ~d..ôn ('.'. i;h,lfJ:"() 0

f2i_.,t ) Wp = (j~ S-)

W F, =- (J {_-3)

WF:i fl = 0 (--1)

W X = (J (\)

'L fJ ') 2 ~ - 2 = J . 2 . 2. =- 'i? ;i:- 1J ~ = s X 7.5' ,,

NaoriJ -z"i>~rf 3 P,0; i< AJ ~/P+G-/-R\ Li R-R r'J ~0 (PfQ-/-Q)=3-!i-r-1-+{0(1<-P-G-!<)

> 2 r., 7- ~0 (~-P-Q-R) > 3 i lJ1? '5 [P'f ())

~ ~~ t '1r;ku- H,; Hl, Hj -z''#

t::. P J G 1 R -ai 1' t 6) l)''',.f+ ifi i ~ o iP = H 1 n j-1, n \-h si

P ~ o / p r 1; 1P1 c r, -F. · F, == x ::: if<. (J ~ 1@ . r , z. 'f- IJ 1/i_;_~ô""-J -( 7J l 1 o

Q. E. D.

30

~ 2 l l ~ -:::> lp ôJ .MA-f~œ (j) _t OJ GWVv-e4

1. 1'1.UM..-4WcrulM. rk0 ~(_Q_ _t ô) uuwe.

Q ~ lP3 15) rirJïv,,;...:;...rf{l/I, ~~ AwL-fu-e l. L Q

J:. 0) i !\" -i ô) n lfh.-AhgJM. ~~ c.ww-e 1/;il~1

f.J ~ 'L 1'5 ;L .?. Q ~ U?'x IP' -c· 1> ~ J5' .5 p,\. Q = z <jl J'.'. ~ L. (1, o)

/j: lP/ = JP17C]x'r ô) dv.M (0, 1) l'i if~::: iAxJP1 (J) ~

'li h 0 (X li IP1 6) !i4;t 6),•?.,)

1) -i!{fe~ ciwvitn D <>~· (o., b) ~ R'c Q (2-

/i i h 'l i .Q 'l ' R ::: D . JP1 > ô b == D . il?,' > 0

2) Dt:: (û, b) lf\" ÛAV'/Â- B D, > o O· E > o

V E :f 0 4~ ~iY/. 'l /J 1J ?j\'' DJ_ =- ). o..b } E =-

(Cl) b) o..'> o, b1,? o t i 1J t D·E =- a.b 1 +be/

J-) 7- D: OM<th ~ O., b >o 3) -.\\\~ t::: 1), ~ a....r6- Dl w Â-a_u_ rtiJ -i J;

1t ( f Di -+ Il. 1-J ~ !Mtt;d l 7j 6 ?j" (1 ) !) <t ~ aayl.( -(,' (1, 0) ( 0, 1) { J -n ü k f,,;.;t ·z." S iJ ;f _;

(}. J b > 0 <J) t. :3 ( ()_ / b) ( 1 /iN!/v'j M>-;~ (,,, ~ h 0

Ll J: -& l Y.. '!!; 6 'L

(y..,, b) .a,,, -1{e~ ~o'L 11{ 7 ~ o..,b >o (Cl.? b) ;?j\'' CIAM-ph ~ (0, b) 1-i''' ~ U#t~ ~ o., b > 0

31

(0, b) b > 0 6J ..vW-~ ~ü"°L IJ i /\"l D =­

~ 1t:tjxJJ?1 O)f~alêUh('.5'; Vw.~ trJ-hOltd.. t."'=.t

b ==- r oJ 'L =t t.·11 -t ~ "!;;., J.t l i' (o., b) E:- 1~ c Q 05'"

n0>1.-4-Wtrk'- /Jvi.ed.,.cJJ-l Gwl!V'e "t 'â <!..>' GJ {j © C<. )) ) o

Q) Cl.=-o, b::.I GD â.= 1, b=o ô)lif';ft.;?,-z.''"t~,

Ca.-) b) (~'th lt!J;t-~ uvvv-e Xli HE(i,1)

t''75' 5 dr;,8 X = a_ -r b <-" t 6 o - J) ~ 11- ~ t~

.v/,/-e~ ~&'L {J(:cjtL 2fl.(D)-2:::-[)·{JJ-tl<)

--c iJJ ') k ~ - 2 H é:(:2,-2) 0"7>' 3

l~ -2 -= GL(_D-2) + b(ll-2)

.:: 2a..b - 2CL -2b

J:- 7 'L ~ = CL b - û. - b t I -(jy IJ o

2 ~ -z," J -a_ F L ~ f}if 6) d- d7 ~;( 1@ t. 5f<. k'J l df ';.

/,, Qtb=ol. à:'J b=cl-a

~ ::: a..(rl- a.) - a_ - (cl -Cl) + 1 = cul. - 0.1 - J_ t /

f'tJ':S cl:~ô)i~,g a.=<1- -(jlt-1~ ~2

-ol+/ r;L: ode{ w ~6 ll = dî_t -z;":fd1@ d~-r- -al t I 'l'. 4

h,

2. rr(._k( {,(J/li!. J: n c.ww-e..

X ~ .\P3 {fJ Fdu:-c C&ne ~ l, ~ ÔJ ~ 0

~ ~ r ::_ t z_ b4w- ""fJ L- t:: t 0; ~ F 'é'. 1 9 o

32

,) c, F I~ 11/J}\-~il/\.

AM-{u.u t (j ~ • t 1,,

kéow u...f t-:2 J: 'J L~.ti .:\'t.

1.1 /I; /

ewwe. ~ Co t <li« o

X~() -aitlJ.tJ

0 ~/M-~ -c· cwi:

· t 1.J 'l , c OJ'lA.è, l'.: 7J ? ;;f-:' I X ~ 0 ?.5' > H { 2

;uc;"<ch'w. t ~ 'l, . ~~ tJ H l OJ ~ 'l 1i ~.

Î3f~I~ F '5' ~ c()'Yl-Ù -" d) /~ m(}..r -a ;,.,..k i ;. L if t ~ :fi. ra. U?~~.t& 0>~111i =t? • NH4, = &Ho) <O""'Uc rJ =- ~!< d:b~! t·~ 6 lf' s

[:" - Co = X - {o'( ~ \/{__(){).))

J r i '

loYV<-G

t, -z.. "F .'.:::' \P ~© &~J) "IL r !P1 i Col.=- -2 -c'

1;~ 0

lP' <l}f:i! 6J,~, 0) fibre i 1- l. i ~ 'l -f z.=O _, Co· f "° 1

P,'c, F = ~Ef) ïZ "i' Q. o) (i Co 1-...J-f -C---'-o

(O. l) (1- + -c+rt :1 *- ('. u !,} t L L t ----JP'

33

lr4:,~. 0) -ejf-vlr-e_ ~M. D a-~ F (~ :ZtL 2fa_(DJ

- 2 = D ·(D+I<} i'-f; 6 2 t °& 1î1u1- k ~ f J<) J:-7 o

k = a.Co t b f t J;· lt If [) =- f Oî t '!

-2 = .Y·(ltCot(_b+l)f) ::: Cl a= -2

D==-f 'l1hc -2= C.·(-C 0 tbf)= 2tb b=-'t

,i:";. '- k= -2Cc-L/-f

~µt~ H-= a_Co + h f /-1 · Co = O t ') H·f == l J:- '.>

J::'.) t.. H =- Co -f2f

b -.2CL =O

Cl -= 1

!) D ::: QCc + b f 'l- th l ~ D Vw_~ -t{~ ~ 5 li D=- f Zlt D· f >o ~ CL)o_,

D = Co ~ 1.2 t;J: D · Co > o ~ -2 Q -1- b ?': o .1 J:. ) L

/) ::: f ) c 0 ~ t:. {l (;(_ > 0 .) b ~ 2 Cl

if D = QCo + hf t i ~ l ~ 0 ~,& 7J- ; 1t'

D·f )O J: IJ a;>o D Co >o J:- 1) b-2et.>o

3) b ::: CL Co -t b -f <$'", U ? ~ r:;vw..,,& ~

/j: ~ 7)' ~"\12 ().

C1Mm. Co + 3 f 1-J: ~ (ÂM>'~

f'1<>-r7 + ) c 0 + 3 f :0 H T f --[_ H t + t

ba.<U p~ ~ -t'-;h ?6' 5 Co+ 3f =t PC14.12

p~ ~ --(-f; h 0 fi/W.. ô)::: ,!t Q_ Mftvtc.:;&, 1 h .2. '(_

1-;l:, - lJ (1) 1~,J\~· F- Lo (2 °ii;h If H c," Mf~ i'1 ltfl

/) 'l t Co O),~, ·("th ;fi_ fi f -L' MfW'~ i' '5. 'h " 11A_.15) i'5,âJ ~ IW-w'cf~ 1i ~ 1~2~/-j_ F-Co aJf.t1t H lèt ~

.4.e..-f vAcVle --T ;\'t -z -Ji· 'J } Co

6) t :~, p lt p i ~ > 7J. ()

3-f 'l- Co ffJ d/v.MrYL Do

'l Co t 1. :\'? _; TJ ( J H t P ~ JÉl G t ff'J oiV.--l4M [>,

C .!:" , l- 412-f evw.h "1 ;\L ~ o

) ). J: 7-" U.M>rc (rj: ~il a~ 1 t_

/2 0

Do

p Co -*1----~l-l--

f

H

1-/ = Co + l { , -Ç (~ bOM. !~ f-w- CD + 3 f /-j: ~ a-w.~ -ci> la -'f' .s a.> o _, b > 2C<. (}) t '3_

°'-Co --f b -f (1 -wNj û-Mp--& --t'1J ~" O/,'A~ ( ::_

id: h ~14' 'L~ -îJ -it (J 'L

e D = Ct. Co -+ h f 1 è ;Z.t L

D : (µMp-& ~ {) : ~ evvv-th- t;;;=; (). > o ) b >2l<--

35

X~ JP3 (i) p-~ c~ 'l L 0 ~ 't 0) ~ t

i ~ 0

C t X (:: i i1t h non-MM.~ ~ t... i J.i t ~

64..e. 1. 0 ~ C m t s. C(;J: F 6) Gw\/V'l 'l ~ ),_

s:tz. c E(o.,b) r t:ttrr c· c~ ==-2a.-t-b= o .'. 2 a.::: b 'C 7J 1) C n... a H -i .. C !;î: cow<µ_ ~&v. X·Fc.. ÛUe- 2. 0 E C (f) L ~ ( .;_ (Ct. _, b) 'l l ~ t '1

C nO'h.-~~ -c,"/hh?.f'.5 C·Co = -2o_+b :=:- 1

.'. b"" 2Q+/ C =. lLCo + Q.o..+1) f C 6) ~

{j d:=:- ( · /--/;::. 2C<_ 1- I ~ (/- {j:

2~-2 ==- C·((.+l<.) ~ (Cl.Co+ 0-o. + l)f) ·(C0-.-2)Co -+QQ-3) f) = ~2 0..(0..-2) t CL(2a-3) + (2a-rl)(o.-2) = -2a.1+ q..ez.. -t 2a.""- Ja...+ 2a..i-+C<.-Lf-a..-2

= 2.0...,l.-2Q-2

.1.- 'J @- = 0-:i. - CL

i<Mrltuvf<. G {L(JYI.-~~ rp-o.ct.M_ ~CQ_ c JP~ -{ ( E (()..1 b) 0) t "3 cJ... == et+ b ~ .::: ()., b -

(). - b t / -( 1> , 0. /)"' b == û + I (J) t ':! ol = 20.. + 1

~ "" a_ (li.+ 1) - Cl. - (o._ -t !) + I == Û. i._ C\. J:'., 7..,

36

cgv-0-Ad..'c li91U. 1::.i '};li_ h YL&it-~~ ~

CD ~ ~ "6'" <o(; ~ > /J. _; non -AWir. pa.du.c. 4wi..fa.c.e (:::. 1 ~ *-,? ?t.&n-4#...rfM. ~ -z_"

~ } ~ l/~' <.d) 'j. > Îj h 'ê (J) -1'4ft1 ~ :! t 1~-.

)J()'ho

3. }lffYL-~F wk .4M,ft..~ J: Ô) ~ JP2

;), _; 6 1'~, R/ · - -1

p, -& t'0 3 / t_- ®#î_t 1_:fi

< I b / 7)1" - 7 0) (~ .1: {:. lj. ( ) J:' ) f .è t h ' >l L L

~M. ~ { C C : cJ-ù.. ewwe. 7 Pi/ ··· 1 p,}

1: à=- '} ./)_~J lttti-.f

JPJ. ~ lf3

~ ft; 1} /Àna.y 0) ~ t F t_ t ~ (f F lt 11.m.­

Ah-~ ~ ~ce. i'1)h.

[

fit:: ~ ~ d~coJ4 ci(j'UJ 'fdJ l L t J 'L ';\' ' i /{ (., 0) lW7!,-~M_ AIA/i.. f<.u. c JP .3 (;t .

2 ôJ J-) 1: Lê {(,lî .SAL ~

~if BR lt N~ If) ~ F()1,M,(J / 9 17 'I-f 111"'- f/2G

F 1 ntJ>t-Affi;uJ.M ~ ~\.e t t 'h t 1

F ft__ IP:l. d) 6 I~~' R / - - 0

/ R ~ bk- "'-f L t_ f_ OJ t t ~ ; ;n..; ï5' .s R:c. F ô:- 7L7 i' ..1. e1 - - · eb ~

/ / /

37

pa,4ÙJ l ~ t 1L .-5 o 1 E L Jl 1 * 1[f <D P / - --; P 6 tZ: i i-~ (1 k e, t* P.: <n ~~ t1i5 o 1~ 1,

e~:::-1} ,le,'=o, e1.e;' =-o <1 ,'ti), t?.t_, LU ~o

6 H = ~fM-~ M-~ = 3}_ - t; el

k-:::: (J-(<i-'-1-)-=:: &(-1) {è ~t.t-t;ft li" {,

k = cevn&-rvt,cJ ~IYL "" -3}_ + :Z e.:. ,:::.1

D = o.,,f - 2 Dt' e~ r_ th t ~ o&d- D = D· H ==- 30- - L. b\ .

2~ -2 = D -(p -~) = a. (a-J) - L. hdbl'-1) ;r V ~ D = ~ = ~ (a_-1)(a.-2) - Z 1.b~ (b,'-t)

ü D ~ _vff~ lit,; 'l:,!zO>=---; OJ

±~-5 i)"'~ fJ c

<D D = e~ !Y [ D-e,, 20

D· 1- >o 2) D ~pie "i' lti;fL rf

D·J.. = ()_ >O D -e~ = b,· > o

b2 ::: 0.,1

- L. be > 0

38

~~ ~~ ~ (~JI) ' \:L O.è 1" \ i < ).L Q~ 1 \L. bt\

l =- \) ... , 6 ) CLè = I I:'. ~- (1 (f

\2- b~ \2 < 6 \:Lb!\ a1 > b( z· ~;li_ ti L. b,· < Vl ()... 'l 7J h.

3) H = 3 .Q - .L. e~ 1~ '"V'V"'J ~ ... /;,l _i_ - e,

21- - e 1 - e, - e) - e If 3 .X - 2 e1 - el. - e.3 - e '+ - e,- - e6

1-J Î'll} ~ ,~ -{j; h 0

Applicd:O'V\ cl ~ ~ L~ l'i ôi F 12i

j ;ft)~ 'flfJYl-A#i~ Wu;_~ Cww-e (j) 'J/1~d,

jhwA ('J.

~ =- f Col-1)(cl-2) + ~ = f&!--1)(cl-2)

;); rl =. 0 (Wtod. ~) i,(. ol = 1. 2 ( mod3)

t ?J 1) ' ~ Ill :fR. t: {~ ~ ~ ~ (:: t. h C-ww-e. Jf";J'f ifi i h 0

6

P-i~ ) D = al - fi b2 e2 l L t: 'l "1 d,

= 30. - Ibl·, ~:::: i(Q,-~(o..-2.) - zd b"(b"-1) ·'( i , f;. '5~· 'f4 ~ ~ f2 of... (2-J1 L "2 o..._, b'J · ..

1 D6 e

~~ 1R tt<i ·fV;?5' l t::.810) ~ OJl'x.*-11@ !-st~ -z. ;11- J:-). J. ·f ().., ! -{-vf- Lis. L.. ~ :Lb~ = 30...- d... J:- IJ

39

~= dC0--1)(a.-2) + l(3ct-cl) -_i :L b~1

2-:l'Lv'"IJ::.L~tsh0tà b,=-··--=-b6 ô)l1o ;r, 2-iût::

7-~t 6 ~j: {@ li

~;:::. ~(0.-1)~-2) + ~ 0o..-cl) -1i (.}ec-ci).,_

= J.. 0.1 _ i. o... + 1 t 3- a... _ _g_ _ ;?__a....,_ + 0 .. .1.J.. _ oC .2. 2 2- 2 'f :;2. /2

=- --' ri' -+ o...ol - ol'- - oL + I 4 2 12 2

2- ~ 'i' C<. 1? 7}J ~~ :S t:J.;jt If Ut= d_ IJ) t ~ ~J;:_ --z·, :Îc ti> ~

:il j: {@ (;). a - - _1 oL i. + c1._2 - d} - ol + I <l - 4- l 12. 2

d.,'--Jri+2 2 - 6 -l-3 - (r}_ -1 )(r;l-2) 2 - 6 -t- 3

t, c. ri=o (MoJ3) <Dt'1

â ~ ~ (oi.-1)(c:,{_-2) -t ~ dv = 1, 2 (14tocl 3) 6) '(_~

~ { f (ol-i)(cl-2)

t lâ 1,J 0 2- ()Jj!._A_ ~ ~ 1:Î ~ 1:: t ~ ; t (~

CD d._ = o (J'rt-oJ. 3) 6J 'é ~

J._, = 3 M (_ m > o ~) 'l~ 117.i ;;y· 11t.H =

3 ml- - vt-t z:_e~ 1t ~ Mvtr&- i' ~ tJ:.

3() ~t) - 6 W.. = 3 M =cl (, , ~ (J.

L/-O

~ = i (3Yrt-1)(3 rn-1) - 6 · i_ ltt(/'l'l-1)

=: fGm-1)~rn.-2) - ~fC3»t-1)(3»t-2.)-2}

= : (of.-t)(ol-2) -t ~ @ d., = 1 (»t-ool 3) 6) t_ 3

d = 3 m. -t 1 t:!1t h ?J·" 31 - 2. e-: ~ ~11-21- - e1 - e:i.. - e.3 - e..'1- no~ 1~ J: 1

)

(rit-1) (3t- 2 Q~) + (21- e1 -e1 -e3 -e4-) + (21. - e, -e2-e5"-e6)

= (} Vil+ 1) J'_ - (wt + 1)(21 -t é:'2) - jll,l(e~ -tf'.4- té'. s + f s)

(j ~ aw.pÀ- i' cÛpz (j:

3(3M.+1) - 2(\t\'\.ti) - 4-Wl = 3WL+ l

~t

d- == ±. 3W\.(31'Vl-1> - 2 - i(~+'> m - +· i 1M-cm-1> = i_ · 3 Wt(3Wl-I) - \IV\. (3WL- I)

= ~·3\'V\.(3Wl-1)

= -t (cl-1)(c{_-2)

® cl= 2 ÜvtoJ 3) (J) t =li

cl= 3 »L -t 1 t. l 1-t ,61)~'

l')t (3t-z:e~) + {Jl - e1 - e:i. - e3 - e<t)

=(3m+2).f -(m.+1)(e1 -r ei i e3 + e1.1-) - wi.(es- -reb)

lt ~ (JVW.p-f.e ?,' ~ ( ;j:

L/-1

3(3vrt+2) - 4-('Wl-+1) -:im. = 3 m + 2 = c(

tt:_ ~ ~ ~ (3m+ 1)(3 M) - 4-· j (Wt+I) Yv\ - 2, d rn(m-1)

= ~ (3 /'Yl +1) (3 m.)

= ..L (çl-l)(cl-2) 6

Q,E.D.

:i,t rl= 1,2,'I- (2/(JZ.,fd-~(§i~!(df-J-.

L/-2

X t JP3 <5) ~ -u.~ GWl/V'{, ~ JP1 -z·· de~ cl '(_ -f la l. :i ' ;}2. û) J'Lo µd.M ~ !t 1- ,t.; 0

Ro~ (1) !P3 (J) ~ H -z; X(\ H ;f ?to 3 ,~

~Cl/L 1 ol am OJ l. l. 7J h t ô) lf~·if.}A -t t,; 0

P-w~ C2) f m ~ ~ ~ ,t P, O t;Jz 0) à:) 7J t ôJ 15,.-

4fr i~o P,o ~fID~4M _,t 1;1Xt P,Q -r; mwét;­~~ I i"~ti ~. 1. 01t!"!ôJ1~ -z"iJC-i>; llr;"

~cvilz 3. 1. U) ~ (2)

R~ ) (1) ~ l2) 1~ tiA :ç J5'.

(2) ::9(\)

-Ç ~ X X. X -LX G"tCl<W (1 '3) JJ w

P; o "(P,0 êi~L_t) L'. iif ~t ~ 'l Ili\ -f =:;'if)_ 12x-t L X * JP' ~·75, _; 1 1 ::i 'l

.îz_ /,; )( IJ) ,~J~~ r!J(~@o ~ ? L f 1()) (J ô jJZft_, I / 1: 1)

JiM lw-1 = l r c Xx '>(X I(M f -t .,,.dùmtl IYlap X >< X -7 I~ f 0 rw-tÂ- t i 1ci t -1 .

r ~ v (_--J-,,.1,_,,.tJ 1J. Â_,? ::1.,,1,_ 'Cf;;, Z\1~.

(2) à:: '} ' 117>? 1 (:jt l (_ r _:L) ~ f Ô) {1bre li ~tt â ~ ki ê 2 -;_; t; /;; ZJ' !, f ([) ~ (J 2 -[' 1.i h

1./-3

2-l7l'"tt?l5'~. T = { .1. E.q(l,3)1 #l,tnX\ )3 (

t<t,<t dkm-1 <\ J:,, z.{Hp/am.e,3-1_ H:>_l_} 6))1_7L rt 2 J.'J. f. L t_-b"·, l Y JL~ T H :3? 1 iJ â

µ~ (-/ 1f·'if-ft_ th . H //' )1°µÂ0 (/) 6)4f1t12 )fti t 1 .:. t 1 t 8 A :s -'?'.

Q. l::: .D. ~~ (3) .~. A~ IP'-c X 6)f±)[ 6) nun.+

1~ P (j) fa,.-..;eJ ~ -tp t/ A 12 it i- > 7J: t O! ~'"lÎf ~ -f 1;; 0

~ 3. 2. (__J~) G)75'f>Î1T-t1J c ~ Xr;J:~y. --z:·;t6tc17o

;l/dz, 3' 5. "./ (2-) ='? (3)

11-z ~ ) {)) -Z:' 7J c ) t -th '(_ . ;;VJ ~ r- > 2 7), .. ~ , 1

li y_ ;... ·c-t -"'' -z_ <JJ 4-U.cw:i 11 r 1&-id. Â.P.-ccuwt t ;J 1ri •

0

Lf-lf

J', z, xxx ffi ~

;J f~ (f,Q) t:).q l

p, Q ~ ~ t kuz. _R. fi .t 4 'j.. {f) ;(f: <!) '"' 0 ~ ,,,_ j;; 0

~~1: 0 'l tp~{i;:!f

Œi H ô%1.~'l. tL~ lf'" H 1 ~ 1 ~ ~ 7j 11 :tz Id'-",

~ fJ p1 E:'X !~)1l ê 0 'L p' Œ iffih@ht.L- XôJ

~f.J 1J rt r- 1 r;._ -r::. Y.-/i, !J iJ' s ~ rJ" H 1:: î 1: tfi. .t? tf Ji s

111 " :i:- , c_ tp t ia f;J: -tl ~ i'1t A -t~ :!? f.>. 1t: If)

~ c.,f 1J X OJ ,~, R 6J 1;~Wl +R t I~ r I!f ~ -,:· tp g ·V-"

i-0 t i:b ~ ?S~'. A ; t 1f01 f., ·t ié :fJ J,i t 1 1il t r<- r if- H ( :;_ { 1- ;te 6 :_ r r.: lj '6 • X ~ [P1 c" Yi 1;; li' ; . ~ OJ 'd: ) 7J R fî#i~~{@ L 15' ~ 1) 1~ Ji r l. t-, z. ~J. 7j R "t' -+p. 1~ A -a ûil h. :__;1>c t 1J mlfe4,!l:J5' > x 0 i /\;' 1- {)

1111N~ ,~·dJ ~~ ~ t-J A ê ~-Jv· ~ t ( ~ ?j_ h • l ;( l '"i' (}) .f" ~~ ~ -:1 :.n. ~ :-_ '(_ 7}" ;IJ 6' 1$ 0

fÀ op 3. 'f. cl 'fl == 0 (]) l'..- '1 (3 l (J- J\ 1/ ~~ /j_ ( l o

~) A 1J' ; 0 rojecA..'~ ~ ~ ::l ~ r X 0 1 /\'--CôJ1~,i' -'1.4J L.!J!d. c1 f<=-o 0ii{"J.. t1

? -t .A.e.~ 1nec.1, 13.'1;, s ::. :h 1~ i 1) f~ !J t 1.

Q, E .D.

[ ~ / /8qo J X! d~A c~"µ ?t{1Yl-MVL~M ~ 'l L ,.J2 t ..z. ([) J: (J} ~()./\. ~ r=~;J2 i1fd. lt6)'l1 c),~~o ,15 l'd-";f< d) P/[oM è ~[4~ -f o

p,_oM Clf-) ~1t CD ~Vif- P) ~ ~J-1

@ ;Jl- P 1~ b~ ld !'f4f:-fj( l. ~ ~~/211'~, P G X .ï5'"/l}it.1 ~ t -1 , / < 'v'A < 'f- 1 /2 }:-! L l.. p \ J - .. 1 PA E X i'

1) ~(J--P,-·---P,o) =~tY-A ;_) J - p, - · - - B tJ: lMe ;~ t 14 t ï1 () Q

'l ~~ f:_ 1 -:fç,â) 7.if4i1i 1 "1 0

~~ cl fi_ = p > 0 6) i ~ (lf-l .:1''"~ :à_ 1 h .?f"( -) l.5' ;!? 15' 5 7.i li '

© (if) =='?> (2) J.)~-u ~;?. t~i- X y \P3

U)'l~ J-== /Hl t.'"h-f11i"(4) O)fl1t~1itL A=- 2

0) ~~ ~®[-fi t:h 11" 3 p ) Q , J- - p - Q : ~ !~ f-.u_ t,1ho ::.~rt P,Q 1@h ~ t X 6)~tJ 1J rt P) Q i'ôJ 1_1t 'ti ~jt_·c.:4- I f:," 11-Z''"i? ~ _: '(. ~fi; l l. l 1 /5 • t ) l (2.) 75"' (1 J._ h 0

Th~ 3. S, CÂ~) X ti 1P 11 1~1i 1 i~ ?11J>l-~~ ~ 'l-f,? l-=! {3) l5'"J.t 'l JL l.ti" )<. ('J.:,}1_

6) l 1 -r;h 7f' -z--~ ~ 0 [13]

i) X=-IP 1

Il) cL { = 2

X = C~c.. c ll?2. 'ef.=X2

d-1 =- 0

{

X= t f c Ji\3 '<} == -t /j\_

~ = --e-r 0.:f~ti~i (j: ::z rtib ~ "f ,f(-z"'-;Y; J,) 7)\". .: ô) C<-v'VV'Jl (;}: JP3 -z· !"J

~~ ~ i(f;~o ~0-tY+ ) cJ., i :=. 0 7J ~ if i) ;)•"() ;\ ~ ïf !:,

d { ::: p > 0 t_ i ,Q 0 Èif ~fi (j M\f Q) fi ~ X ~ .lP' UJ~ 1& t" 1t/ff 7J ? , JP2

::::> Al.2 6. ~1z OJ J: 7 (: t J,; o

1) A IJ- X-t?/;Ù-0 6) p~ d ~ 2J t. L A ~ x t. s 1·r tA :t 1 w / tA + x-~ 3) O ~X

~

A

@ RtcJi H~ X ) y "&' YLOft--~L VWvv'-e. --t ~ X -7 'Y

] k (X)/ k &') lf~· MfAJ,e_ t 7J 6 J: 7 7J htn;~ 'l j "1 o X ::;i P ! ; 11 L 7-- -f{P) ôJ ~Vv;f.@

If 7

,~ ()_ 15~· p OJ ww/~d /Mcw~ -t (~J.1L -f*U = a__tJ- -+ l,~ ~ , cc:~ c :i!-f ~ t :t , t > I -( ii ~ /J" { f'J P , .. 4J

/> / 'j ûJ t ':f ~ /l.4J '{_ L 1 ? 0 1. dl t.

~if~t'

~~x -2 =- (d.tg-F) (2~y -2)-+ :L (er -1) P6X

( ,(.@ l ep = rvr ( ~11()-t- / )

?)~·~ IJ il. J o

Tlim (i)~Jti~1~t '(~ 0

,\t A k 5 ;i-~ 1\ <D

ro~4 ·~,f 1'.JJ t X ~ '4-C14M # (3> J" ~ ~ 1cvi Ji,. '-.. -Ji .? , t là .t s f< Cxi ::::> '{{(y) - -5 o ?J, s 1.<!X? 1\ (j) r o ~ ~ ~ :Z. ~ r x ----7 100

(J X-Cùû4 O),\t,.!J-?f (-;\: ~~J_ ·( H~ <Di\~ J: 1)

2~-2 :::::L(ep-1) - 2cl fEX(l.z-~ Pt-A

P~ X n :k~ P=fA C.f<il 7- 2p~;li~ J.- Jo ) -:::. (Cr -1! -t- Cr+1-t"+

1+ -· - ) p Cr t o

:X ==- a.+ -t

2~-2 == z:rn - 2o1...

1) ol = 2.. P r~ Vir A t X 2) d-= Z P rt -t 1 if A ~ X

1) (Î) 'L ~ 2 ~ - 2 = 2 d- = 0 ,> cl= 2 X: ~ 2) ôJ 'l ~ 2 ~ - 2 = o{- 1 - 2 o(_ t 'J 2 ~ -t ol -:=o !

~ = 0 ) J_:::. / X = iPI Q. E.0.

~o._ 3. 6. JP3 ? X -8_ ~ ~ rh;rw-cL ô) non~ uvvw. 'l L X {-). {.) lJ 7J h ~r-F r::. ti" 'if*-4 ut: t /.J 0 ~ 0) t 1 rît > * {0.,-3) iJ) If fJ-1:-M) li rtŒYL/3-jU-wj i'~ /;;. P~) FI'/\= \j.nm.!-1/ 'l~,~ Fw

/Jp-€--w.1 t 1ii~ ~ 0 7 ~ ~ fi°(k-fM) > 0

J" , L. X ôJ +~ ~m. A -i' Fit\ -+ A "-" k ~ ~ t O"'J 11'"# ft-t 0 • ~ OJ J.- 7 7J A ~ - ? \Il Jt 1 Jj •

-1 -z. X ~ .lP2 -(j, ~ii' .5 \Hl ~ P 1" D (]) '{(])3,t

t- ®#,~J: 1:.;J <) J- ) ?J tef)lJ~·4-f:t -t i5. ex= ~ l kl

~lk-D) 'l"*< 'l

cfMr.. 1 k. - D l = -{ 0 (K - D) - l

= { 0(D) - cl-t â- - l -1

?_ 2-cl+â- (-fi_0 (D)'t._4-)

J, ~v:' o{V,,, !KI = d'. - \ j:- 'J

o( ~(i-1) -(2-d.-r'i) = cl-3, -- -· I ~t_ f> =- ~IF~I- ~IF111-D\ t-~<. 'l~~1:!0 ~('._/~~I~-t;hli {j~D( ~.f~ho

© 1=" ~~ ,X7Pt_M;.ffA t-lï3'l 0 ~ -ft,0(E)-{0(f-P) < { 0(f+A)-{0(Ç+A-P) ~ 1

:_\ L D =:. p, + --. -t Po<_ t i l:J 'L 1 WI. L ~~ -3)

i- 1) 2fh L. cl-J .:_ ~t.."~ t Os 1 ~ c;(-3 (2.

~-:\L L E~ é FIM --{. E~ :7 P,-'---/ Pc· / Et_~ P.:+1 1 ·-·, Pot 't. ts j t G.l ~~;tt.A t ~. ~~ t=-2)., if ~ t-i' 1- < \'V\ <'..' i; ~ '4-r-1kv L ',; -- -..1 LIM ~ L1 il D ==- iP. 1 Rj, Lin D =-~fi ... P'f-{, - - - - / L.R. fl D = {P.ü-1) P12} ) L,e.+1 /) J) :=. - - .. ::::- L/-}\ Il D = ri> 'l

IJ ~ ~ J (_' 'l ) z E~ =- l1 u ---V Lw. 'l -;):,' (T If J: l 1 L

1. -=- 2.1. + ' li > n" L ~ lt\-1 -t ~ ~ ~f1Plz--,IC0!.o4 L !! · - -> Lw- ~ L, n D =- ~p,, g_ ( , Li fl D =- [ f;/ 171- i - - - -) L_,e_ (]/) =- ~&-,, Pi.e r ) L_,e_+, () t) = !Pz..e-r,r

1

.to

L.t+i (l D:::: ---- = l.,,. 11]) = ~ t.fJ; .i:- 7 (.:: "t? -z: Ec' = L , u · · · u Lw. 'l 't· r t If ~ ( 1 ~ ~ :l'i l ~

k- IF~l - ~IF!!\ -81 == 1

~ \F~-l~l - ~/Fm - fi-Pi/ = 1

~JF10-P,-·-·-Pot__,l-uL/~-ri-----Pc(.)I = 1

~ ïÎ L -z. cl)' 11 $ = ol,,.,.. /Fw.1-~ IFw.-DI > ol- 2 - ··JI ~ ~ c< "i' i, ~ -".)' s T ) "] ( ;t jf@j D t ) "2. c9-x:(VVI)

/~ ')J{Tf!. .Aj~

Q.E.D. ~ ~ô)~a_3.6. lf"~

!~ t fi '51f'I X = Fi · Fe t. t;,, 'é ri=2e r.dx = <Jx &-Lf- -r 2 --r e) = c9-x ~ -2) z:· ()Jx ft Affwd i' ~ 'J IM = !i_ (of.-tf-) -z," lY-x(M) -4jU-c;;.J c4~o .

Thm 3. ri. (C~) Xti P Œl de~ ol ~ ~ ff) '11cm-4#irk CWvW. -z:·. ( )

;f 1 6 ~r ,KM'-€ r..: t -1 i n 1 ( 1 'l i ~ r: g S: ~i - oL + 1 ~ ol ~ o ( Wl od. 2 )

· i 5 d~I -oé t I :{ d_:::. / (moo<. .2)

tl

Re4) F111 = lx n Vt'\ 1-11 r. "à)·< , {tE L 1-1 1t

p3 6) 4-r--/M'l-€. ~ 0 è'. ::I f; ;;> D,, D = p, + · -·i Poi.

-i' R_,·-·,Pol O)l'."073::>{~-W~lli.::;JutOJ(?'l~o

s t11 = 4 F~ - ~ (Pw. - )) ) t ~· < • x. ([) <:: ~

s"" 2. 2wi + 1

1: -e·;J _; r== o; .. "J::il"\ l2J.tl :::i El" E Ff\\ ,.!_t,

Ev- 7 p,_,--·_, Pv- > Cr~ Pr+1, .. ·.) P~ 'llJ ~ =ê: ôPV~ ~.

~.::!\. (i 1r ( r+1

~ ( f w. - L. P~·) - k hl'\ - L P~) ~ \ i -=..t ~=- \

for- Y' ::: o · -· ) w-, /

~ t 0 7\: t ~ tJJ z" div" \-W1 - L (F~- D) > .:2.WL.+ 1

Fm -D = F~-1 12 ~it t~ l

~ Ff'-1 = ~ F1 t 2-.Gol"F{- ~ ~-1) il\ ~~l l'-'-

> 3 +~2{<.tl .

= \IV\ (f'l-1. + 1)

Gae. f. © cl=- O (!'1.oJ.2) IJ)'( ~ W..=iécl-J.) r t ~ '( ~o... 3. b t \) Üx('M) !t ')?!frt.-t1-(J€d

J:", z R-R .r 1J -/_ 0((JxCW>!) =- 111ct - d- -t ! --.. I -?j --t_ 0 (62;x(W1))-/ = ~~ 2. f\'\Qvtt-\) - --- ··]

I, JI d:" 1J Mol-~+ 1 7 Wl(Wl.+ 2) + 1 Wll"

1 <1--2) ~1-fÀ j;;uf

~ rlCrJ-2)- ~+ \ L_ ~(cl-1J(olt2) t-1 ~.t.. ~ 1)

~ < ~(o('--4-ct-t4-) = tl.-ol._+ 1

Gue. 2. @ cf._= ( &cod.2) IJ) t ~ V\1= }irJ.-1) 'lt Îcl 'C ~'°" ?. 6 J:" ~ 0<C\MJ (t n(j)t-A~CÂ.J t ) 7.. i(- R J.- ~ /,/ ((Jy({:A)) = wo<. - ~ t 1 - - · T

- 75 -/,_, 0(CJy_l.!{_t\) )- 1 =- ~ FI" / 01. ( 141 + l) - - . TI.

1, 1[ J:°9 VV\o\_-~t 1 ) \11(11\+2) f- / Ml~

i ( ol- 'J ~ {f'A -t ;11_ ri' ~ of_(~ - 1) - ~ f 1 > l (_cl-1) (rA-3)-+ 1

.l.

d- < J (o{l.-q.o{_ +3) - o<_ ;r -of-+ I Q.E .D.

~~ CM-€_ 1 , {a,4..L 2 l :t'$~ ô)hY 1>if--::;ôJft- kFh\ == Vl\(Wl.-+ 2 ) O)t~-z" d > !>

(t'-1)fJ.ll 0

D -? J__x.C2) -3> e1P3(2) -7 8x(_2) -::i 0

(2 :Zi L .»x:acl '9 1 \

o -7 H0(tdx9l) -7 H0(<Yp<?l) -7> H0(!JxC2l) ~~~~'l

~ H0 (fJJP3Q.))-=- /0 1

ol&w, /-( 0 (61-x &->) = ~ t 1J H0(dxl2J) "::\= o :;...;h.rt X ?f,,·ii J jv-~ ~f.c.e 1.:-i° ~ :\\-h ~ l. ~ ~ ~71 1 h

0

Hd~ r-;t :;z 0) cr) t 2-ll,"':· -z u 15 0

1 ~ S -f (pl-1 >(d-2) fJ > tf cit? c{ ~ J­O) 1ZOh.-~~M ~ X c. P1 -i' n-=>

1 6J ev.k ~Le 1::_~~~1;; =ê <f) lf'''/(J. fi 1 ~' __J

:::. O)ip ~O) ~;\~ IJ 4~/~-Z:' ~ ~ .:5'" >7z ô) ~wvk /!' ; .Œ l

ut l -c =t ':i(J) ~ A-wt.P,~ r;t 1/Jjn-~F 'l rt 'l :\l'L?J- t i 11'/ JJ'" ~ ~ ~ r Jf" t1 ?5' h.

~cvvk X t ~ â- ~J. <D

11..(frl.-~~a//._ ~ c. .JP3 -(,' 11&>!-~i:vt eul-t,c,

.MM,{ccu_ 1 :~ ~ ~ lJ L i h /:'.: â > ~ (ol - 7)

I HES *JG 112~ 11

IJ) fl~/rt'Vh 11

~~ w-14 ~y~ ~dcM 1:::.rht..

X ~ ~ ~ UJ ?1o-Yt-~~wt ~ 1.: 11rm-~rîwt.. A.MAjcu .. .e F f.>i

l ::l1t, (_ t ) ~ '(.. 1 h. l , /}?_ O} t) , 2.) Œ> 'l "t, ) 75' -( ~ ~ 0

1) F li X x ./P1 /~ b,'r~~ 2) X2 < 4- °J +) -{ t{ .,q 11. X ;&' JP'- ô)

lL ~ ~ -z:_· â--= 1 Ô) '( =3: _JJ

. { 2 -z; X ?J, .. ~ ~ i n/7>1.-~7.fwi. ~è. ~

F'C pi (::'.~ ~ *- ê l) iS. '(_ -t"' l ~ AWL--fo,c.e (-J

~hi.J 1,' -fi IJ.f 5 Ct) f* ~ =-o J;<9~ Ji· 2 11 f~tJ- t).

o/;_J-x =cl r 1 J.) ('.'._ W-x = X·(X-1- W-F) CUF:::.éfi=tO t:"7>~ _s 2~ -2 =- x1

- o1._, x2 = ol+2j.-2 j t o 7J; r1 X2 < 4-~ -t !:,- -5'?) ;

o{,f2~-2 < 4-~+-> ol-7 < 2~

~-, 2. i 2 1 (cl -1)

~ Lf- A/m,~ MoVMdt'erc ~ Î~ fs iJ_ ( ) CW1;1fe .

r t I\'' 7.. ü) /,JMc~ ewvv-e (J 'Ylon -4htF ewvv-e ôJ 4-;uud c CtA.R. -t"b '8 1' ..J 'L 1 ) ) fii1 IU! '8 ~ "!- fJ o

: 6'J Ftl!rn l ~ / c l -z. IP3 0 Wt.P.dv..~ ~ced ôWv1Ae

-i 6) & if'I r1 ~ ~, ~ '/ JJ' ., 'Z'. u 1J. 1 J if~" 1 =à- ·J" ~ O)'ft~ -z.' · O> §.'19y

~ ~ Â ~ ~ 'l ,i)''"z" -'.f ~o

~ 4 . / . JP3 Œ) rr()rt llJlclcecl CWvV-e. -C

Mn~ fd- +db ~:f#~ 7j_ t 1 t o;x~·{};ri i lii.

~r .,,e 2 o ! ~ c · s::: i [!:, '4, 'i, w-]

0 J.~o~~ t41J I ~ I = < :X: / x ~ / lj.,_, z.z..,,e - ia wL >

('.'._ -t .-5 ~ .Jl ()) 'l .::! y= ~Oj s / I c:: p3 L t ~ l ..-Ù<-ff y = i :X.=-~ = 0 } = JP 1 c lP3 -z,". i tê y lt. -VlM-fudo{ C~~ ~::(~, l. 11 lJ ( J. rk~ j = 2

7.-" 1i h ~~' Po,_ 1f :#. ~ t -) 0 y (f) f/,'/ ~ p/p l)?IU,J, 1 ;J.

-,f ~i'.9yl'AJ) -== c{ rt + 1 - Pa. i"t ~ J5'' ol == 2 ·t n lf" i 7i 7t 6) '(_ '1

o -? k ~) 'f<[x, ~J z, w-J)\ -7 H 0WyfllV-:> o i 4 ~ o li_ ~ -z· (Sir\, ~ l :Z. ~ t ~-"'t'.cf &ï- .Aptl,~ t l -z._ (J) b~ Œ

:X. 1 .~11-1 X~ w-i\-l.. x.~,W-11-3 __ . X 2:.1.-1 w-Yl-...l IN.1 1 .J )

1:1- ~ 11-1) i z:.V\-2w I ~ 23 h-J w-~ - ' ' I 1;J w-yt-/

Z:11. ~i'\-1W- 2'.:h-~w-'- --- w" ) .J J )

L 'l~ ~ J5' .5 (S/:r ))'\ (J) ~/<. ft.JJ- .)_ -+ YI. i nt 1 = 2 Y1 i 1 +1

~., c_ P0- = - )_ -;t.vt rk{()'VM1.~~ rcJ: J,, > Po_

~1ff.? ;>),·- of.efW- 2 15) non-~~M Cw'-vW. /1 iP 2 Ô)

CCtn<.(;. ~'X rj ?t&rt-~r-ltvl.. o-v-a~'c ~au.. .l: if)~';!? 5 1 ( ) .=: ~fi~~ L ?J' ,(+ fi L 7J. iJ

0

o c OYkc 0 t ~ . R = o -t SxM ... ~ âJ -1 = o 6) :t~ ë1 " . /,._ - .-," "\' 1_ 0 - ..4.;t-eùa/'-Ar,J"- t,.. - '<Jo

0 = @~If< OJ L ".\ fi ::: -1 -( j_ = / (}) #}, § 1 2 fip-l -

u4 i ~~o

.1 > 2 (J) 'l 1 ~if) ~ph. (J M11ov-t/c fld ~,,;â_IYM 'f2:fij f; lJ: ( J ~ '{ ?)~· ~ 7f' h o

~ 4. 2 . ( S;;wi-15' ) f 3 Ô) M-riA.uz ol

11iYf ~ ~ -( .MrlMY-i/. flcVt 4t/VV(lD: -u~ ~ t. f: !;; ( ) =t 0) y 't lP' c JP3 ô) ~rw- 4- r5)

'flfirl.-.AM..ptvt-~ L. L 1._ ~ IP 2 t Y (J) - ,~, -c~;p-?

ku ri~. YuJ..~*1.~'l ol=s -ci'>?lV !{ ô) 7s (;} -~~ (è y,) Y; i ~ 'l l 1'. y_-::j_

il (Y, u Yi)= Po-(Y,)+ P0c(Y,)+t!(y,nYi.)-1

Po. =- Po. (Y) -+ Po..{),) -+ 1 - I =3-tO+/-)

:: 3

l ~ ~ 1)" § 1 0)0 ~~:t<_ ]- ~.:ft If:\? -1' 1j ~ 7 1: ~ 4-

<f) ?1tm-~~ Curvv'-e. 0) ~ (d: 0, 1 ) 2 ) 6

6) t ) ,.-~ 7.5' i' i ~ _,, _; y u )__ /~ .Mn~ ftcJt ~~ ~#t.1 () 2- r.>J~'h·f 'ho

~pie_ 11-. 3. c H(Mflforol) ~d-"vvurk..~ ~~ i' /.Vl1Urtrif{ µ ~ a tt ~ l) t. q;i

Il . 1. Il X -t;. ' ,,., tf ;vn-&!Wit'â- VW\JV'.e. e. 'l/1711-.41,,.._~/Vi. ~

'l L P ~ 10)-? 6),ff, t f 18 t ~ p OJ --focd ~~ C9p 14t.O) ~J R 2 f< -(-i? ~ YL 1~J.1L R ::> Wlp

clJ h ~ 61 t~;Z::t L J.LVt t L 7.. Id. )( 'l{S/ L' Culw.e X /

i X ~ X1 ;y· p >;<.4~ -z" i-oir>"M~ p C).ife1

6 ,f, 1 p' 'l t.. t:. t 1 Opr == R t fJ l;i t. 0) ;y·if} ii t 1o o

_:_. 6) l =i

o --7 ()x 1 -7 -f-i1 tJ>, -;;, +* 6x / rJ-x' -7 o Sl

é9p/R

"!i;if ~ (f~ 6' ~ ~ ) (~

P0- (X) = P<>- éf, l-+ _j_(Gp/p.) t.;$ > ('. lJ b 0

X ~ oÂ.dJw.d ncm-,M-,..~<M.. ~ ~ ~L2 P ~ X <f) 1,f, 'Yl >o 1 0-vt.?'1- --t ~ é'.9-p ~

wrn.f ~f! /CVL~ '[_ J ~ o )i (f) ~ ~ ~. tY\ -+ i<--ttl -t ... -+ f< -t:Ylt 1

(j) ~ -y..e_A ,4-ftld \T ( ~ g_q L R =- ~ -t V+ !M.pVl t

1-i < r. R 1-J: ~d i 1 1J . >'z. it 1.: r ~ ;Wnch;.-.o ewwe.

?i: Xv t t <. ~ è: c-t Go ~ 2 -t; t ~ ~ L -Z

o4w- v ~ 1- ôJ 1'~ t.s' rt ~Ji. ~ 'l. Po.. ( Xv ) = ~ -t i Yl -1 - i

t-.1-?o V~V1

6)c~ X--7Xv;X.-7Xv' (.:' i}tl L XïT '{_ X171 lt ~1R 1 ~ -1 11 ~~ ïJ (). ~ 1.: ~c)t

?J.Fl~ Xv- -7 X"' ?5' .• ii?;fi ri· X OJ ~~ q--

15~· ti ') X --7 Xv ô J, 0'- i. X -7 XIT'

l-lJ ~ /)"' ~ ~ 2 "f-' 15' S X OJ a..de-m01--~ f;J:-1) P~(@o

+, "2-11{07 vr.:J-:1 c~11~1@()) ~. ~)IÂ~ V\,··: Vr l5'""(0, z \114 Ve (c=- 1,--·,Y) -('i;:/1_1f XIJ '<:'.

Xq' (;!. ~d ~ 'l L t ~~ -;,· iJ < 7J 6 o T = lr/l.tl.44 (n / t) 'l "Ji \ t If dwn T = t (Yl- t) -z;· -Y,~ ,

X :::. 1." î1 ~ 1~!;;. } = ~ 't. i ;fi ti 4 T = !}j +" , l.. 11 ~ +-1i -k ~ <., t;ll li

~ T > 3 P°'- - 3 = 3 ('J + 2vt - 1 - 6-) - 3

::_ ô) c ~ °&i;; V 1::)-:t L Xq (;l: ~ fi µf_ ~~ ~=tf2/j() _: '( lY'~)~ *o

<1? c, < rt ~ol PcJV~~ W ~Jt~9 !Jk6-* 'l L 71t;?<_ô)~6jf75~-~ & o

Tbn Lf. 4-. ( /'vl. Ç~ ) X ~ A3 @ ~d. 7J: ~ '(_ 11cJ t ~ y. /'J: ~ j!J q~~ ~ =t?,

66

~ t p1 .1:. :) U.uweA f1) .e.:~ 1::. .., " ï

~ Y ~ 1;! p3 1: ~ 'I - ·J: X., <!) J.,ou,J.. .u.As ~ l"'

Wedd.e.J. """i'~ k t~ K. 1j " t 4 l • 4 y, 'I' ~

;t.~ 5. 1.

t. 1 ;I , }: •. 0 2 .., fJ) -!J_,(!f e :J!i r ~ 1;: t j •

y t Y ' 1-J. *:~ f1) i.f:/1~ JJ;(,-fi é ~ t: 4 " • ( L~ e J.W.. ·c y., y'J

::; 0 )

2. 2 .,, "' cJ.<A11..to.U!.-O ( ~ J J.e J. c,m,..; onJJÂ t: :f'!t ç_ /j ")

H1 , l·h fjr Ji 1) Y U Y 1 ~ H 1 /\ H ~ ( S ~ 'C L 1: J •

~~ S. 2. Y~ y' t. rJ UA/lAl'tA Y. ( = y) y 2 .... ; '

;: "' ~ ,. ~ 1 1t IOJ i'1 ~J.J {~ r- ;li ? •

H 'I . y f iiwui..._,;.B..11. 11.e..l.u.c..e. J ~ 'L. 1' ~ o Y "I

~.. rviWl t. I[ t r ~ 'C, e~.-< ~ t9q_ ~ ·L &J c- &;. à

il' .; f. 'il 1~ tt tr.- l -C. Ps 0> 2 ? " M !!Il® ·H., H,, é. t 3

H H 'f. "- H ""-'"''• !'..., ,() ~=YVY' t.;; "0 1>"-1: I I

191 Y = P' t. f 3 • Q \: t 0 2 :l i.!IJ JÏll t L . Y ·f.. ( o, 1) ~-.

61

( ç; k ( Xo, X,, X2, x.,JJ , 0 0) ~ 1· 0> i:t:_ ~ 1 'T" T IL- LQ ~

( lC a )(. - ')(, )( 3) t 1 ~ 0 :'. O> ~ '2J ~: $Jt e 1 : ;If L , ;,t !;( e ~

(âl ~.!'. ~ ~ ;\; f ( <\ I Q ~,, G ly ) /J''" #.fi t î . Q fl He

:: YvZ ( He•~ f.ô-··~c!1J ~ e ;;Zd!Jtlu) t. t;i 'J, Z r;J (e,e-1)

iJl0 Q.0 ~ 7: $,~ 0 1!1~(1:-ft.0 Q.OJ (e,e-•J f9,/q

~ z 1:. tf ·l ( e- 1 :z: oJ r3 tf) e. :,Â_ dh.W He q·.(.j..(t

l , 0 0 He= Z V Y E ~if, il • ( ;: ~ 11 H0(P~ tlp1(eJ) ~

H0 ( Q, <941 (eJ) __, o J: 'J 9JI ;, tJ\ ) t < 1-;: e- 1 .> o {ï b,

t 'J z J;t '

4-" ~ !i.J '/ r ;J:J~ rJ.. 11) i.MaJ.M.cAJ,.R.u. hAll"- SIM.~

cwwe t. ;J: 3 • 't ., ~ ~.t "> +.ï 1( l :~ f::- ~ e ;J.. lll1 ffll H,, th

6\ Ji 3 • ( rJ.:f-1 :; e. ) :

H1 <)Ha~ Yvy' î.. Y' /1. ifwi_.Wv:t.f.o. >Wh-si'M~

~ \- If' ..,. Y \::. y' l"J. ~-wtl.S .J/. î- h 3 ,

~it llfl) ..9 = { H : ~ e Il) tl!J tln j H 2 Y j t f 3 •

x=iJt:L Y~~t.l"'î .Mln.r°1'1~.'(ok.E ......... ""- * 1 1f : X ~ )< t. L J> = 1T J) @ (9 X' (- lî- (''()) t if 3 t

cto...:.... : .J 1.t ~cJ. ~ .,; : x '--? r"' t:

· lit.~ 3 c 5~ ~ r~~)

62

i<x) ,;i rN0 ""'"-so.«~ ,o.,J,..v~. ~ d) ~:vr.t

'I p"' "> ~ .lf IÎ!l H, , H~ t/•" ;g. ,(j_ L 1 >< 1 H 1 · ) H, ~ ,,,.,,_

s~ Üv.n_J.,.,..~ ~ j! t, !.lb) ~ 0

- /.Î "ln rr-'(YJ

1 ;I ..il... Sn oc.. ( 1t J 1 ~ jâ] ~ (" , Ti 0 (;] [· fi l>.A. I~ 2 .~. Ji à :: t -r .;...(,t;, .-....... -

li 'Î " o H.; ~ JT( X AH.:)

J:: OJ 1-li '.{__ e z dt 1 /J' .. tr .. t ~.K °' J: :> 'J. & 4#1 ti·· h ;} o

19• I • 4'A70 - s°WI s-J.M- inM..lu.dk'.. ~ Y t _iE !l lx 0.,, (). l

l:J"" /À'. E ~ 1 J: j r: 13-îi. Î à 0

ll(f $ ll!J i~J"H.; . ( ~· ~ 1, z) <!.l

)Â. t;i. 1t 11..: î·· H' (\ 1-h. = y V y I Î"' 1, ;} 0 't. 0) ~ -:. ') J;. ) {1

-& l.!f i! ;f.:i 1 H,, 1-f,_ 1: t:'"' J: :> 1~ t .,, î. t y'11 iJ *~ î- ili,

t-.. t i. rf." y 1: rJ../lAL c..v.J,.,:.c. u..vv>-e ~ 1 ~ • o.,= 2

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References

1. Bertini, Interna ad alcuni problemi •.. Atti Ace. Sei. Torino 1§. (1890)

2. Castelnuovo, G., Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica. Rend. Circ. Mat. Palerme l (1893)

3. Enriques, F. and Chisini, O., Courbes et fonctions algébriques d'une variable. Paris, Gauthier-Villars (1926)

4. Halphen, G., Mémoire sur la classification des courbes gauches algébriques. J. éc. polyt. ~ (1882) 1-200

5. Hartshorne, R., Residues and Duality. Springer Lecture Notes 20 (1966)

6. Hartshorne, R., Curves with high self-intersection on algebraic surfaces. Publ. Math. IRES 36 (1969) 111-125

7. Kleiman, S. 1., Geometry on Grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math. IRES].§_ (1969) 281-298

8. Manin, Yu. I., Cubic ferros. Algebra, Geometry Arithmetic. North-Rolland, Amsterdam (1974)

9. Mumford, D., Pathologies IV. Amer. J. Math. 97 (1975) 847-849

10. NRether, M., Zur Grundlegung der Theorie der algebraischen Raumcurven. Kon. Ak. Wiss. Berlin (1883)

11. Peskine, C. and Szpiro, L., Liaison des variétés algébriques, I. Invent. Math. 1§. (1974) 271-302

12. Saint-Donat, B., On Petri's analysis of the linear system of quadrics through a canonical curve. Math. Ann. 206 (1973) 157-175

13. Samuel, P., Old and new results on algebraic curves, Tata Inst. Lecture Notes

14. Schaps, M., Non-singular deforrnations of space curves, using determinantal schernes, Thesis, Harvard Univ. (1972)

15. Serre, J. -P., Groupes algébriques et corps de classes. Hermann, Paris (1959)

16. Severi, F., Vorlesungen liber algebraische Geometrie (tr. E. LHffler) Leipzig (1921)

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