# Monday, August 28

:::{.remark}
Goal: construct *projective normal* varieties using lattice polytopes.
Note that this is not the same as a *normal projective* -- a projective variety $X$ is **projective normal** iff the affine cone over $X$ is normal.
Recall that for a projective variety $V$ with homogeneous coordinate ring $\CC[V] = S/I(V)$ where $S \da \CC[x_0, \cdots, x_n]$, the corresponding projective variety is $V = \Proj \CC[V]$ and the affine cone over it is $\hat V \da \mspec \CC[V]$.
We can realize $\Proj \CC[V] \da \hat{V}\smz/\cstar$.
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:::{.remark}
Consider $\hat V \da \Proj \CC[x,y,z]$ and $V = \mspec \CC[x,y,z] = \CC^3 = \AA^3$.
One can check $V = \AA^3\smz/\cstar = \PP^2$.
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:::{.remark}
A lattice polytope in $M_\RR$ is any polytope whose vertices are contained in $M$.
Here we define a polytope as the convex hull of a finite set of points, or the intersection of finitely many half-spaces. Note that these two definitions are not generally equivalent -- an intersection of half-spaces can be unbounded.
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:::{.remark}
1-dimensional polytopes are closed intervals $[a, b]$.
Some special polytopes: $\Delta_n \da \Conv(0, e_1,\cdots, e_n)$, the $n\dash$simplex.
E.g. $\Delta_1 = [0, 1]$.
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:::{.remark}
On the symplectic side, polytopes will be the images of moment maps.
For example, $\PP^1\cong S^2$ with a normalized height function yields a moment map with image $\Delta_1$.
Note that the fibers are either points or circles.
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:::{.remark}
Let $P \subseteq \RR^n$ be a lattice polytope; define the affine cone $\hat V$ in the following way: place $P$ at height 1 in $\ZZ^{n+1}$ and cone it to the origin, calling the result $\Cone(P)$.
Letting $S$ be the monoid generated by lattice points in $\Cone(P)$, define $\hat{V} \da \spec \CC[S]$ and $V \da \Proj \CC[S]$.

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:::{.remark}
Note that by construction, $S$ is saturated, and thus $\hat{V}$ is normal and $V$ is projectively normal.

We'll say $P\sim P'$ is they are related by an element of $\SL_n(\ZZ)$, and they yield the same toric variety.
Scaling $P$ changes the projective embedding into $\PP^N$.
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:::{.example title="?"}
Let $[0, 1] = \Delta_1$, then $S \cong \NN^2$ and $\hat V = \spec \CC[\NN^2] = \AA^2$ and $V = \PP^1$.
There are two explicit generators of $S$, namely $(0, 1)$ and $(1, 1)$.
However, considering $P = 2\Delta_1 = [0, 2]$, one needs three generators: $(0, 1), (2, 1)$, and additionally $(1, 1)$.
Setting $x = \chi^{(0, 1)}, y = \chi^{(1, 1)}, z = \chi^{(2, 1)}$, we have $xz=y^2$, and thus $\hat V = \spec \CC[S] = \spec \CC[x,y,z]/\gens{xz-y^2}$.
By a classical theorem, a conic in $\PP^2$ is a rational $\PP^1$, so $V = \Proj \CC[x,y,z]/\gens{xz-y^2} \cong \PP^1$.
So scaling yields $\PP^1$ embedded in a different projective space with different equations.
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:::{.remark}
Note that scaling the three sides of a triangle by lengths $a,b,c$ yields a weighted projective space $\PP(a,b,c)$.
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:::{.exercise title="?"}
Let $P \da \Conv(0, 2e_1, e_2)$ and show $V = \PP(1,1,2)$.
Write equations for $V$.
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