# Friday, April 01

> Reference for toric geometry: Fulton's Toric Varieties, Oda's *Convex bodies in algebraic geometry*.

:::{.proposition title="?"}
Claim from last time: 
\[
\cocomplex{H}(\PP^n; \OO(d)) \da \RR\Gamma(\PP^n; \OO_{\PP^n}(d)) \cong \Hc(\mcu; \OO_{\PP^n}(d))
,\]
where this isomorphism is of graded vector spaces.
We also saw
\[
\bigoplus _{d\in \ZZ} H^0(\PP^n; \OO(d)) \cong \kxn = \bigoplus _{\vector d \geq 0} k \prod x_i^{d_i}
,\]
and in top degree,
\[
\bigoplus _{d\in \ZZ} H^n(\PP^n; \OO(d)) \cong \prod x_i\inv k\adjoin{x_0\inv, \cdots, x_n\inv}
,\]
with all intermediate degrees vanishing.
There is a nondegenerate pairing 
\[
H^0(\PP^n; \OO(d)) \times H^n(\PP^n; \OO(-n-1-d)) \to k\cdot \prod x_i\inv \cong k
\]
which is concretely realized by multiplying monomials and projecting onto the span of $\prod x_i\inv$ (so setting all other monomials to zero).
This is an instance of Serre duality, but this example is in fact used in the proof.
:::

:::{.proof title="?"}
Compute $\oplus_d \Hc(\mcu; \OO(d))$ by first writing $\PP^n = \AA^n_{x_0\neq 0} \union \AA^n_{x_1\neq 0}$ and look at global sections:
\[
0 \to k[x_0^{\pm 1}, x_1, \cdots, x_n] \oplus 
k[x_0, x_1^{\pm 1}, x_2,\cdots, x_n] \oplus \cdots
\to k[x_0^{\pm 1}, x_1^{\pm 1}, x_2, \cdots ] \oplus \cdots
\to \cdots \to
\to k[x_0^{\pm 1}, x_1^{\pm 1}, \cdots, x_n^{\pm 1}] \to 0
,\]
where we choose 1 coordinate to invert at the 1st stage, 2 coordinate to invert at the 2nd stage, and so on.
Note that this is not only $\ZZ\dash$graded, but $\ZZ\cartpower{n+1}\dash$ graded by monomials.
The claim is that the contribution of a monomial $\prod x_i^{d_i}$ to cohomology will only depend on the pattern of signs, i.e. $I\da \ts{k\st d_k < 0} \subseteq [n]$.
:::

:::{.example title="?"}
Consider $I = \emptyset$, and the contribution of $\prod x_i^{d_i}$ with $d_i \geq 0$ for all $i$.
Form a simplicial complex $X$:

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![](figures/2022-04-01_10-53-14.png)

The cohomology computes $\cocomplex{H}_{\Delta}(X; \ZZ) \cong \ZZ$ since $X$ is contractible.
:::

:::{.example title="?"}
For $I = [n]$, so all $d_i < 0$, one obtains just the faces of the complex with the boundaries deleted.

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![](figures/2022-04-01_10-58-45.png)

This computes $\cocomplex{H}_{\Delta}(X, \tilde X; \ZZ) \cong \tilde\cocomplex{H}_{\Delta}(\tilde X)$ by the LES of a pair:


:::

:::{.remark}
Recall that this LES arises from
\[
0 \to C^n(\tilde X) \to C^n(X)\to C^n(X, \tilde X)\to 0
.\]
:::

:::{.example title="?"}
For $I = \ts{0}$, so $I = \ts{0}$ with $d_0 < 0$ and $d_i \geq 0$ for $i\geq 1$.

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![](figures/2022-04-01_11-02-23.png)

This computes $\cocomplex{H}_{\Delta}(X, \tilde X; \ZZ) \cong \tilde\cocomplex{H}_{\Delta}(\tilde X) = 0$.
:::

:::{.remark}
When does this trick work? 
For any pair $(X, L)$ with $L\in \Pic X$ where the sections are $\ZZ^{n+1}\dash$graded where each graded piece is dimension at most 1.
These are referred to as **multiplicity-free**.
Examples: toric varieties:

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![](figures/2022-04-01_11-16-10.png)

:::