## I.2: Projective Varieties $\star$

:::{.remark}
\envlist

- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. 
  \[
\kxn_1 &= \gens{x_1, x_2,\cdots, x_n}\\
\kxn_2 &= \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}
  .\]
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::