Kähler-Differentiale/Projektiver Raum/Einführung/Textabschnitt

Es gibt die exake Sequenz

0\,{\stackrel {}{\longrightarrow }}\,\Omega _{{\mathbb {P} }_{R}^{N}{|}R}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(-1)^{\oplus N+1}\,{\stackrel {}{\longrightarrow }}\,{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}\,{\stackrel {}{\longrightarrow }}\,0

und dazu dual

0\longrightarrow {\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}\longrightarrow {\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(1)^{\oplus N+1}\longrightarrow T_{{\mathbb {P} }_{R}^{N}}\longrightarrow 0.

Für die symmetrische Potenzen gilt die Beziehung

0\longrightarrow \operatorname {Sym} ^{q-1}{\left({\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(1)^{\oplus N+1}\right)}\longrightarrow \operatorname {Sym} ^{q}{\left({\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(1)^{\oplus N+1}\right)}\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}\longrightarrow 0.

Ausgeschrieben ist dies

0\longrightarrow \bigoplus _{\binom {q-1+N}{N}}{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(q-1)\longrightarrow \bigoplus _{\binom {q+N}{N}}{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(q)\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}\longrightarrow 0.

Bei ${}N\geq 2$ ist somit

{}{\begin{aligned}\dim _{K}{\left(\Gamma ({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)})\right)}&={\binom {q+N}{N}}{\binom {q+N}{N}}-{\binom {q-1+N}{N}}{\binom {q-1+N}{N}}\\&={\left({\binom {q+N}{N}}-{\binom {q-1+N}{N}}\right)}^{2}\\&={\left({\frac {(q+N)\cdots (q+1)-(q+N-1)\cdots q}{N!}}\right)}^{2}\\&={\left({\frac {(q+N-q)(q+N-1)\cdots (q+1)}{N!}}\right)}^{2}\\&={\left({\frac {(q+N-1)\cdots (q+1)}{(N-1)!}}\right)}^{2}\\&={\binom {q+N-1}{N-1}}^{2}.\end{aligned}}

Unter Berücksichtigung von Twists ist

0\longrightarrow \bigoplus _{\binom {q-1+N}{N}}{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(q+\ell -1)\longrightarrow \bigoplus _{\binom {q+N}{N}}{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(q+\ell )\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}(\ell )\longrightarrow 0.

Bei ${}N\geq 2$ ist somit

{}{\begin{aligned}\dim _{K}{\left(\Gamma {\left({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}\right)}\right)}&={\binom {q+N}{N}}{\binom {q+\ell +N}{N}}-{\binom {q-1+N}{N}}{\binom {q+\ell -1+N}{N}}\\\end{aligned}}

Bei ${}\ell <-q$ gibt es keine globalen Schnitte.

Zu einer Hyperfläche

{}Z\subseteq {\mathbb {P} }_{R}^{N}\,

vom Grad ${}k$ hat man über

0\longrightarrow {\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}(-k){\stackrel {F}{\longrightarrow }}{\mathcal {O}}_{{\mathbb {P} }_{R}^{N}}\longrightarrow {\mathcal {O}}_{Z}\longrightarrow 0

die exakte Sequenz

0\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}(-k)\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}{|}Z\longrightarrow 0

Wenn ${}N\geq 3$ ist, so ist

{}H^{1}({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}(\ell ))=0\,

für alle Twists ${}\ell$ und somit ist

{}{\begin{aligned}\dim _{K}{\left(\Gamma {\left({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}{|}Z(\ell )\right)}\right)}&=\dim _{K}{\left(\Gamma {\left({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}(\ell )\right)}\right)}-\dim _{K}{\left(\Gamma {\left({\mathbb {P} }_{R}^{N},\operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}\right)}(\ell -k)\right)}\right)}\\\end{aligned}}

Mit der Tangentialsequenz (bei ${}Z$ glatt)

0\longrightarrow T_{Z}\longrightarrow T_{{\mathbb {P} }_{R}^{N}}{|}_{Z}\longrightarrow {\mathcal {O}}_{Z}(k)\longrightarrow 0

ist

0\longrightarrow \operatorname {Sym} ^{q}{\left(T_{Z}\right)}\longrightarrow \operatorname {Sym} ^{q}{\left(T_{{\mathbb {P} }_{R}^{N}}{|}Z\right)}\longrightarrow \operatorname {Sym} ^{q-1}{\left(T_{{\mathbb {P} }_{R}^{N}}{|}Z\right)}\longrightarrow 0.