# Friday, September 01

:::{.remark}
Abstract variety: $V\in \Top$ with a sheaf of $\CC\dash$algebras $\OO_V$.
Every point $p\in V$ admits a neighborhood $W\ni p$ such that $(W, \ro{\OO_V}W)$ is isomorphic to an affine variety $(W, \OO_W)$.
Examples: any affine or projective variety.
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:::{.example title="?"}
Cover $\PP^1$ by two copies of $\AA^1\smz$ and glue by $z\mapsto z\inv$.
Similarly, cover $\PP^2$ by 3 copies of $\AA^2$.
Similarly, cover $\PP^n$ by $n+1$ copies of $\AA^n$.
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:::{.remark}
In dimensions $d\geq 3$, there are abstract varieties which are neither affine or projective.
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:::{.remark}
Consider the cones spanned by $e_1, e_1+e_2 \leadsto \sigma_1$ and $e_1+e_2, e_2 \leadsto \sigma_2$.
Take $\spec\CC[S_{\sigma_i}]$ to get copies of $\AA^2$.
Check that $\sigma_1\dual = \Cone(-e_1 + e_2, -e_2)$; since $\matt 0 {-1} {-1} 1$ can be right-multiplied to obtain $\id$, we have $\sigma_\dual \intersect M \cong \NN^2$.
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:::{.remark}
Gluing affine varieties: let $\tau = \sigma_1 \intersect \sigma_2$.
This yields inclusions $X_{\tau\dual} \embeds X_{{ \sigma_i\dual }$ since \( \sigma_i\dual \intersect M \subseteq \tau \intersect M \).
Letting $\Sigma$ be comprised of \( \sigma_1 = \Cone(e_1), \sigma_2 = \Cone(-e_1) \) in $\RR$ and $\tau = \ts{0}$, we have \( \sigma_i \dual \intersect M \cong \NN \) and \( \spec \CC[S_{\sigma_i}] = \AA^1 \) and \( \spec \CC[\tau \dual \intersect M] \cong \spec \CC[\ZZ] \cong \cstar \), we have $X_{\Sigma} \da \AA^1 \amalg_{\cstar}\AA^1$ where the gluing map is $z\mapstofrom z\inv$; thus this is $\PP^1$.
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:::{.remark}
The previous example with $e_1, e_1 + e_2, e_2$ results in $\Bl_1 \AA^2$.
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