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(1)K?? ?? ?????? dimP(E) = dimE1:(1) ????? ?0?
P(E) =hviv2R2n f0g??hvi=fvj2Rg
P :
En f0g !P(E)
v7! hvi P(E)?
P(v) = P(v)82K:
P(E) = (En f0g)=K
dim(A\B)dimA + dimBdimP(E): P(F dimF + dimGdim(F\G) = dim(F + G)dimE dimA + 1 + dimB + 1dim(A\B)1dimP(E) + 1
A = B?
??????? ???? ???? ??????? ??KP2?? ??????? ?? ?? ?????? ?? ???? ?????A\BA? ??dim(A\B)??????1?? ?????? ?? ????? proj(S) = x6=y2P(E)? ?? ? ?projfxg=fxg?? E???f? ?? ????? ?????? ??? ??????? ???? ?? ????? ??f? ????? ??? ????? ??? ??????? ? ????? ?????? ????? ??? ?????? ??F? ???? ?? ????? ??P(F)?
P(f) :P(E)!P(F)
=hvi 7!f() =hf(v)i
En f0g
P f //Fn f0g P P(E)
P(f)//P(F)
E??????? ?? ??????f()??F?
???P(fg) = P(f)P(g)
Aut(KPn)?
i=0bii? i?? ?????? ??i2K??? ??? ??? ???f(bi) =ibi? ?? ? ???? nX i=0f(bi) =nX i=0 ibi ??????b=Pn ibi) = P nX i=0f(bi) =nX i=0b i
PGL(E)'GL(E)=K
??? ???'(x) =y? x x ????(x0;x1;:::;xn)2Kn+1n f0g?? ???? [x0:x1::xn] = P(x0;x1;:::;xn) =f(x0;x1;:::;xn)j2Kg 2(Kn+1n f0g)=K ????? ?? ?[x0:x1::xn] = [x0:x1::xn]???? ????2K? ?? ??????p=hx0e00+12 p
0;p1;:::;pn+1??? ?????? ??P(E)???? ??? ?
???pi=2projfp0;:::;^pi;:::;^pj;:::;pn+1g???? ????(i;j): ????? ?? ?????? ??? ????(b0;:::;bn)??E????? ??? pi=hbii??0in p n+1=hPn k=0bki ???b0i=bi???? ????i? ???pi6vectfp0;:::;^pi;:::;^pj;:::;pn+1g: ???? ??????i2 f0;1;:::;ng? ????~bi?? ??????? ??? ??? ?? ?? ??????pi? b n+1=nX k=0 k~bk: k6= 0???? ??? ????k? ?????? ?????bi=i~bi? ?? ???????(b0;b1;:::;bn) [x0:x1::xn] =f(x0;x1;:::;xn)j2Kg 2(Kn+1n f0g)=K ?? ?????? ??? ?? ??????[x0:x1::xn]? ???? ???? ????? ???? ??? h(1;0;0)i;h(1;1;0)i;h(1;1;1)i;h(3;2;1)i: (1;0;0);(1;1;0);(1;1;1) ??R3? ?? ???????(x;y;z)??????? ???? ????? ???? (xy)(1;0;0) + (yz)(1;1;0) +z(1;1;1) ?? ????? ??(p0;p1;:::;pn+1)??? ?? ?????? ??P(E)? ????? ???? ????i?? ???pi+1=2projfp0;:::;pig 8i < k b ?? ??? ????(b0;:::;bn)??E?? ??????bn+1=Pn
0bi? ????? ?? ??????? ???
?? ??????'= P(f)?? ? '(pi) = P(f)(hbii) =hf(bi)i=hcii=qi ???? ?? ? ???? ????in? q i='0(pi) =hf0(bi)i=hc0ii q n+1='0(pn+1) =* f 0nX i=0b i+ nX i=0c 0i+ ????k? ?????? ????A???B????? ?????? ????? x 0b 1b nb n+b1 b n+b0hvi
H (xn= 1)
F (xn= 0)
U = P(E)nP(F)
??? :U!H hvi 7! hvi \H b
0;bn+b1;:::;bn+bn1)?????? ?? ???H? ?? ???? ?????? ?????? ?
: [x0:x1::xn]7!x0x n;:::;xn1x n [1x0:1x1::1xn]7!1x0
1xn;:::;1xn1
1xn x0x n;:::;xn1x n ??B? G????? ???? ?????? ?? ?A = P(G)? ??GF? ?????A\U =????? ???
G = vect(B)(A\U) = B
: U = P(E)nP(F)!H P(E)? H`(H) x 0`(x)
1(x)(1(x))
1(x) ?? ?????0 B BB@ 001 C CCA B BB@t 0 M??? t n1 00 11 C CCA 0 B BB@t 0 M??? t n1 00 11 C CCA0 B BB@x 0??? x n1 11 C CCA=0 B BB@M x 0??? x n1 01 C CCA+0 B BB@t 0??? t n1 11 C CCA k i=1H i6=? ??????? ??? ???? ????i? ?? ?proj(p0;:::;^pi;:::;pn+1) = P(E)? v 0=0 @0 1 11 A ; v1=0 @1 0 11 A ; v2=0 @1 1 01 A ; v3=0 @1 0 01 A ??????? ???(hvii)06i63??? ?? ?????? ??RP2?? ?? ?????? ??? ???? ??FqPn? ?? ???????U? (d+1)(d+2)1???? h(x0;x1)i=f(x0;x1)j2Kg 2KP1??? ???? ????[x0:x1]? ?? ???
KP1= (K2n f0g)=K(1)
????? ??[x0:x1] = [y0:y1]????? ?? ?????? ??2K??? ???y0=x0?? y ?????? ????? ????? ???????bK??????? ??K?? ???? ????? ????1?
KP1!bK
[x0:x1]7!( x0x
1??x16= 0
1??x1= 0
x [x0x10: 0x10] = [1 : 0]? ??????? ?????? ????? ?? ??????S1???? ?????? ?????? ??? ???? ????? ? ?? Hx 1 1 x 0x 1x0 a b c d x y =ax+by cx+dy (3) ?? ???? ???x y = [x:y] a b c d x y =x y a b c d 1 0 =1 0 a b c d 1 0 =1 0 ?????? ??2K??? ??? 0 0 1 1 =1 1 ?????PGL(2;K) = GL(2;K)=fIdj2Kg? ?? ??????? ?? ?? ?????? a b c d a b c d ;2K ?? ??????? ??PGL(2;K)??? a b c d x y =ax+by cx+dy ????? ?? ? ?a b c d 1 =db c a
PGL(2;K)?
a b c d z 1 =az+b cz+d =8 az+bcz+d1 ??cz+d6= 0 az+b 0 =1 0 a b c d 1 0 =a c =8 ac 1 ??c6= 0 a 0 =1 0 bK? w0 =1????w2K;a1+bc1+d=ac ???? ????c2K a b c d z=az+bcz+d ??????? ????? ???? ????a??? ??? ?? ????b?d???? ? a1+b01+d=a0 =1 bK??????? ??????? ?? ??????
1 x=1 8x2K
1+x=1 8x2K
bK? bK????? ? ?? x=1? ?????a b 0d ??????x???1???? ????a;d6= 0?? ????b? ?? x6=1? ?????a b ccx ??????x???1???? ????a? ????b6=ax ?? ????c6= 0? 0d 0d 0d c d ???(1;0;1)? ????? ' 1= Id? ???? ???? ????'= Id? ????? ??'??'0 [1 : 0];[0 : 1];[1 : 1] ????2? f: E!F????? ??? ???? ????v2En f0g? ?? ???'(hvi) =hf(v)i? ?? ????'= P(f)?
Aut(D)??PGL(E)?
D?KP1? ?? ????? ?? ?????? ??? ????(b0;b1)??E????? ???p0=hb0i? p bL?? ?? ???? ?????? ?????01??? ??? ???? ?? ?? ?????z!"z+b??b2R??"2 f1g? ?? ?????? ??? ????R(x;y;z) =zxyx
0y0z0y00z00x00
???? ???? ???????(x;y;z)???? ???? ?????? ???(x0;y0;z0)???? ???? ????'????? ???'(x) =x0?'(y) =y0??'(z) =z0? ??? ?????? ?????? ?????? ?? ???? ?? ?? ???? ???? ?? ??????? ????? ?'(z) =z0? ????(4) '(z)x0y
0x0=z0x0y
0x0 ?? ??? ????? ????'(z) =z0? ????b=ax? ??ay+b= 1????b= 1ay? ?? ?? ??????a= (yx)1 ?? ????b=x(yx)1? ??????? ??z??? ???? '(z) =1yxzxyx= R(x;y;z) ???? ??? ???? ?? ?????K? '(x) =1; '(y) = 0??'(z) = 1 ?? ?? ????[x;y;z;w]? 'D!D0??? ???'(x) =x0?'(y) =y0?'(z) =z0??'(w) =w0?? ?? (x) = 0(x0) =1; (y) = (y0) = 0;?? (z) = 0(z0) = 1 ???? 0'= ? ?? ?? ?????? (w) = 0(w0)?? ????[x;y;z;w] = ['(x);'(y);'(z);'(w)] = [x;y;z;w] D (w) = [1;0;1; (w)] = [ (x); (y); (z); (w)] = [x;y;z;w] ='(w) [x;y;z;w] =wywx zyzx ':!yx zyzx [x;y;z;w] = [y;x;z;w]1 = [x;y;w;z]1 = 1[x;z;y;w] [a;b;a;c] = [a;b;c;a]1=11= 0 [a;b;b;c] = [a;b;c;b]1= 01=1 ?????? ?????? ?? ?????p2CP1??? ???'(p)6=p??'2(p) =p? '= P(f)??? ????? ?? ?? ????? ??f??? ?????? ???? ?? ?????? ??? ?? ????? ??f? c d ????'2=a2+bc b(a+d) c(a+d)bc+d2 ????? ??M??? ?????? a+d= 0? ['(q);p;'(p);q]? ?? ? ? ['(q);p;'(p);q] = ['2(q);'(p);p;'(q)] = ['(q);p;'(p);'2(q)] ?? ??????? ??????? ?? ?? ?? ????? ??????? ??? ?? ????? ?? ????K??? ?? ????? ???? ??????a;b? ????? ???? ???? ?????p2D? ??? ??????a;b;p;'(p) a???1??b???0? ?? ?[a;b;p;'(p)] ='(p)=p?? ????? ?????'???
0??1???? ??????? ???? ????? ?????'(z) =z????2K? ?????
a??a0??b??b0? ?????? ??'????? ?? ???? ????? ????M?????? ??? ?????? ?????? ??????? 1 0 ????? ??? ????? ??? ??? ???'? ??????b='(a)??c='(b)? ?? ?b6=a? ?????? ?? ??????'1(b) =b??'1(b) =a? ?????? ????d='(c)????
2 1(a) = 2(d) =b; 2 1(b) = 2(c) =c; 2 1(c) = 2(b) =d
???? 2 1='????? ?? ?????? ?????? ??? ??? ?????? R?? R1
8A2R;9N2N;8nN;junj A??un=1
?????? ??? ?????([xk:yk])??RP1???? ????1= [1 : 0]??jxk=ykj ! 1
SL(2;R)=R? ?? ?????? ???a b
c d
2PGL(2;R)
1 0 01 P t=1t 0 1 lim k!1Pkt(z) =1 E =cossin sincos zcos+ sin=z2sin+zcos (z2+ 1)sin= 0 H e2 0 0e2 ?? >0? ?? ? lim k!+1Hk(z) =1 8z6= 0 lim k!1Hk(z) = 08z6=1 P 1t 0 1 1 0 =1 0 1t 0 1 0 1t =1 0 +0 1t 1t 0 1 ??1 1 0 1 ???? ??????? ????1? ???? ?? ???f? ???? ??? ?????? ?????? ??????? 1t 0 1 ?? ?????'6= Id?? ?t6= 0?quotesdbs_dbs43.pdfusesText_43
GEOMETRIE PROJECTIVE - Université Paris-Saclay