# Friday, August 25

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Regard $V \subseteq \CC^n$ as an affine variety and $V \subseteq \PP^n$ as a projective variety.
Note $\cstar\actson \PP^n$ by $t(x_0,\cdots, x_n) = (tx_0,\cdots, tx_n)$; $0$ is the only fixed point, so the action is free and $\PP^n\cong (\CC^{n+1}\smz)/\cstar$.
Projective varieties: the vanishing locus of finitely many homogeneous polynomials in $\CC[x_0,\cdots, x_n]$, regarding $x_0,\cdots, x_n$ as homogeneous coordinates in $\PP^n$.
Recall a homogeneous polynomial of degree $d$ is $f$ such that $f(tx_0,\cdots, tx_n) = t^d f(x_0, \cdots, x_n)$.
For such a $V$, define the homogeneous coordinate ring as $\CC[V] \da \CC[x_0,\cdots, x_n]/I(V)$ where $I(V)$ is the ideal generated by all homogeneous polynomials vanishing on $V$.
Here one has $V = \Proj \CC[V]$.
These polynomials define an affine variety in $\CC^{n+1}$ denoted $\tilde V$ where $\CC[\tilde V] \da \CC[V]$, i.e. just reinterpreting homogeneous polynomials as ordinary polynomials.
Call $\tilde V$ the affine cone over $V$, which is invariant under $\cstar \actson \CC^{n+1}$ and $(\tilde V \smz)/\cstar = V$ recovers $V$.
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:::{.remark}
Set $U_i \da \PP^n\smts{x_i=0} \cong \CC^n$ under the division by $x_i$ map.
The sets $\ts{x_i = 0}$ are all hyperplanes, and $\PP^n\cong \Union_{i=0}^n U_i$.
For $n=1$, $U_0$ is the complement of a point, as is $U_1$.
Similarly for $n=2$, each $U_i$ is the complement of a line.
Since $V\intersect U_i \subseteq U_i = \CC^n$, each $V \intersect U_i$ is affine, and we can write any projective variety as $V = \Union_{i=1}^n (V \intersect U_i)$ a union of affine varieties.
So any projective variety is covered by affine varieties of the same dimension.
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:::{.remark}
Claim: $\PP^n$ is a toric variety.
Note $\Intersect_{i=1}^n U_i = (\cstar)^n = \PP^n\sm \Union_{i=0}^n H_i$ where $H_i$ are the coordinate hyperplanes $H_i = V(x_i)$.
Note $\Intersect U_i = \ts{[x_0: \cdots :x_n] \st x_i\neq 0 \, \, \forall i} = (\cstar)^{n+1}/\cstar$.
The diagonal action $(\cstar)^{n+1} \actson (\cstar)^{n+1}$ is trivial; here $(\cstar)^n = (\cstar)^{n+1}/\cstar \actson \PP^n = (\CC^{n+1}\smz)/\cstar$.
Conclusion: $\PP^n$ is a projective toric variety.
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:::{.remark}
Weighted projective spaces: for $q_i\in \ZZ_{\geq 0}$, define an action
\[
\phi: \cstar \actson \cstar^{n+1} \qquad t(x_0,\cdots, x_n) \da (t^{q_0} x_0, \cdots, t^{q_n} x_n)
\]
and $\PP^n(q_0, \cdots q_n) = (\CC^{n+1}\smz)/\phi$, quotienting by this action.
Homogeneous polynomials of these weights define subvarieties of weighted projective spaces.
E.g. take $\PP(1,1,2)$ with coordinates $[x_0:x_1:x_2]$ with weights $1,1,2$ respectively.
There is an embedding 
\[
\PP(1,1,2) \embeds \PP^3 \qquad [x_0: x_1: x_2] \mapsto [x_0^2: x_1^2: x_0 x_1: x_2]
.\]
One can realize the image as $\CC[y_0, y_1, y_2, y_3] / \gens{y_2^2 - y_0 y_1}$.
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