1_2x

Projective space: \(\left\{{\mathbf{a} \coloneqq{\left[ {a_0, \cdots, a_n} \right]} {~\mathrel{\Big\vert}~}a_i \in k}\right\}/\sim\) where \(\mathbf{a} \sim \lambda \mathbf{a}\) for all \(\lambda \in k\setminus\left\{{0}\right\}\), i.e. lines in \({\mathbf{A}}^{n+1}\) passing through \(\mathbf{0}\).

Graded rings: a ring \(S\) with a decomposition \(S = \oplus _{d\geq 0} S_d\) with each \(S_d\in {\mathsf{Ab}}{\mathsf{Grp}}\) 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: \({\mathfrak{a}}\subseteq S\) is homogeneous when it’s of the form \({\mathfrak{a}}= \bigoplus _{d\geq 0} ({\mathfrak{a}}\cap S_d)\).

• \({\mathfrak{a}}\) 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 \(fg\in {\mathfrak{a}}\implies f,g\in {\mathfrak{a}}\) for \(f,g\) homogeneous.

\(k[x_1, \cdots, x_{n}]= \bigoplus _{d\geq 0} k[x_1, \cdots, x_{n}]_d\) where the degree \(d\) part is generated by monomials of total weight \(d\).

• E.g. \(k[x_1, \cdots, x_{n}]_1 = \left\langle{x_1, x_2,\cdots, x_n}\right\rangle\), \(k[x_1, \cdots, x_{n}]_2 = \left\langle{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}\right\rangle\).
• Useful fact: by stars and bars, \(\operatorname{rank}_k k[x_1, \cdots, x_{n}]_d = {d+n \choose n}\). E.g. for \((d, n) = (3, 2)\),

Arbitrary polynomials \(f\in k[x_0, \cdots, x_{n}]\) do not define functions on \({\mathbf{P}}^n\) 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 \begin{align*} \operatorname{ev}_f: {\mathbf{P}}^n &\to \left\{{0, 1}\right\} \\ p &\mapsto \begin{cases} 0 & f(p) = 0 \\ 1 & f(p) \neq 0. \end{cases} .\end{align*} So \(Z(f) \coloneqq\left\{{p\in {\mathbf{P}}^n {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\) makes sense.

Projective algebraic varieties: \(Y\) is projective iff it is an irreducible algebraic set in \({\mathbf{P}}^n\). Open subsets of \({\mathbf{P}}^n\) are quasi-projective varieties.

Homogeneous ideals of varieties: \begin{align*} I(Y) \coloneqq\left\{{f\in k[x_0, \cdots, x_{n}]^ { \mathrm{homog} } {~\mathrel{\Big\vert}~}f(p) =0 \, \forall p\in Y}\right\} .\end{align*}

Homogeneous coordinate rings: \begin{align*} S(Y) \coloneqq k[x_0, \cdots, x_{n}]/I(Y) .\end{align*}

\(Z(f)\) for \(f\) a linear homogeneous polynomial defines a hyperplane.