# Projective Varieties (Tuesday, September 28)

## Projective Space

:::{.definition title="Affine space"}
Let $R$ be a ring, then the **affine space of dimension $n$ over $R$** is defined as
\[
\AA^n\slice R \da (\spec S, \OO_{\spec S}) && S \da R[\elts{x}{n}]
.\]
:::

:::{.definition title="Slice schemes"}
Let $S\in \Sch$, then $X\in \Sch\slice S$ is a **scheme over $S$** iff $X$ is a scheme equipped with a morphism of schemes $X\to S$.
These form a category where morphisms $\phi$ are commuting triangles:

\begin{tikzcd}
X 
  \ar[rd, "f_X"]
  \ar[rr, "\phi"] 
& 
& 
Y
  \ar[ld, "f_Y"] 
\\
& 
S 
& 
\end{tikzcd}

:::

:::{.remark}
Since $\ZZ \in \CRing$ is initial, there exists a unique ring morphism $\ZZ\to \RR$ for any $R\in \CRing$.
Similarly, $\spec \ZZ \in\Sch$ is final, so there exist unique morphisms $\spec R\to \spec \ZZ$ for every $R$, and thus $\Sch\slice{\spec \ZZ} \cong \Sch$ just recovers the category of schemes.
:::

:::{.remark}
Recall that if $k = \bar k \in \Field$ is algebraically closed, then $\PP^n\slice k \da \AA^{n+1}\slice k \smz/\sim$ where $\vector x\sim \lambda \vector x$ for every $\lambda \in k\units$, or equivalently $\Gr_k(k^{n+1})$, the space of lines through the origin in $k^{n+1}$ (regarded as a vector space). 
Let $f\in \kxn_d^{\homog}$ be homogeneous of degree $d$, so $f(\lambda \vector x) = \lambda^d f(\vector x)$. 
Then its vanishing locus in projective space is well-defined:
\[
V_p(f) \da \ts{ \vector x = \tv{x_0: \cdots : x_n} \subseteq \PP^n\slice k \st f(\vector x) = 0 }\subseteq \PP^n\slice k
.\]
:::

## Graded Rings and Homogeneous Ideals

:::{.definition title="Graded rings"}
A ring $R\in \CRing$ is **graded** if it admits a decomposition as an abelian group $R + \bigoplus _{d\geq 0} S_d$, where $S_d$ are the degree $d$ pieces satisfying $S_a + S_b \subseteq S_{a+b}$.
:::

:::{.remark}
Note that $S_d \in \mods{S_0}$ for any $S \in \gr\CRing$ and any degree $d$.
:::

:::{.example title="?"}
$R\da \kxn^\homog$ is graded by monomial degree: 
\[
1 + (x_0) + (x_0^2 +x_1^2) \in R_0 + R_1 + R_2
.\]
:::

:::{.definition title="Homogeneous Ideals"}
Let $S\in \gr\CRing$ be a graded ring, then an ideal $I \normal S$ is **homogeneous** if

1. $I$ is generated by homogeneous elements, and
2. For all $f\in I$, all homogeneous pieces $f_i \in S_i$ such that $f = \sum_{i\leq d} f_d$ lie in $I$.

:::

:::{.example title="?"}
If $f \da 1 + x_0 + x_0^2 + x_1^2 \in I$ is an element in a homogeneous ideal, then $1, x_0, x_0^2 + x_1\in I$ as well.
:::

:::{.remark}
If $I\normal \kxn$ is a homogeneous ideal, say $I = \gens{\elts{f}{m}}$ with each $f_i$ homogeneous of uniform degree $d$, then $V_p(I)$ is a well-defined projective variety.
:::

:::{.example title="?"}
\envlist

- Take $V_p(y^2 - (x^3 + axz^2 + bz^3))$ for $a, b\in k$ and $4a^3-27b^2 \neq 0$.
  This defines an elliptic curve.

- $\PP^n\slice k = V_p(0)$.
:::

:::{.definition title="Irrelevant Ideal"}
We define $S_+ \da \bigoplus _{d\geq 1} S_d$ to be the **irrelevant ideal**.
:::

## Projective Nullstellensatz

:::{.remark}
We again define the Zariski topology on $X = V_p(I)$ whose closed sets are of the form $V_p(J)$ for $J \subseteq (\kxn/I)^{\homog}$ 
:::

:::{.theorem title="Projective Nullstellensatz"}
Let $k[X] \da \kxn/I$ be the projective coordinate ring of $X \subset \PP^n\slice k$ and $I = I(X)$.
Then there is a bijection:
\[
\correspond{
  \text{Points }x\in X
}
&\mapstofrom
\correspond{
  \text{Homogeneous }I\in \mspec S \\
  \text{with }I\not\contains S_+
} \\
x &\mapsto I(x) \da \gens{ f\in \kxn^\homog \st f(x) = 0} \\
V_p(I) &\mapsfrom I
\]
:::

:::{.remark}
Note $I$ doesn't contain $S_+$ iff $I$ is strictly contained in $S_+$.
:::

## Proj

:::{.definition title="Proj"}
Let $S\in \gr\CRing$, then define
\[
\Proj S \da \ts{p\in  \spec S \text{ homogeneous} \st p\not\contains S_+}
,\]
where $S_+ \da \bigoplus_{d\geq 1} S_d$ is the irrelevant ideal.
:::

:::{.remark}
We'll define $\OO_{\Proj S}$ next class.
:::