# Proj

For the complex topology, any connected component of an affine variety is either a single point or non-compact. Indeed, if ${\displaystyle X}$ is a connected component of an affine variety, the functions ${\displaystyle x_{i}}$ can't have a maximum on ${\displaystyle X}$ unless they are constant so if ${\displaystyle X}$ is compact then each ${\displaystyle x_{i}}$ is equal to some constant ${\displaystyle a_{i}}$ on ${\displaystyle X}$, i.e. ${\displaystyle X}$ is the point ${\displaystyle (a_{1},\ldots ,a_{n})}$. This non-compactness of affine varieties is a motivation to introduce projective spaces and projective varieties.

Let ${\displaystyle \mathbb {P} ^{n}}$ be the space of complex lines in ${\displaystyle \mathbb {C} ^{n+1}}$. We denote ${\displaystyle x=(x_{0},\ldots ,x_{n})}$ the points of ${\displaystyle \mathbb {C} ^{n+1}}$. As a line is determined by some direction vector, well-defined up to scaling by non-zero scalars, ${\displaystyle \mathbb {P} ^{n}}$ can be identified with the quotient

${\displaystyle (\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}.}$
Making use of the natural Hermitian structue on ${\displaystyle \mathbb {C} ^{n+1}}$, i.eof the norm ${\displaystyle |x|=\sum _{i=0}^{n}|x_{i}|^{2}}$, we see that a line is well-defined by a unitary vector up to scaling by some scalar of unit norm. This gives an identification of ${\displaystyle \mathbb {P} ^{n}}$ with the quotient ${\displaystyle S^{2n+1}/S^{1}}$ where ${\displaystyle S^{k}}$ denotes the ${\displaystyle k}$-dimensional sphere. In particular, ${\displaystyle \mathbb {P} ^{n}}$ is compact for the complex topology. The space
${\displaystyle \mathbb {P} ^{n}=\{{\text{lines in }}\mathbb {C} ^{n+1}\}=(\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}=S^{2n+1}/S^{1}}$
is called the "projective space of dimension ${\displaystyle n}$".

In order to describe the projective space in terms of algebra, it is enough to work on ${\displaystyle \mathbb {C} ^{n+1}}$ in a ${\displaystyle \mathbb {C} ^{*}}$-equivariant way. More precisely, we consider the action of ${\displaystyle \mathbb {C} ^{*}}$ on ${\displaystyle \mathbb {C} [x_{0},\ldots ,x_{n}]}$ with ${\displaystyle \lambda \in \mathbb {C} ^{*}}$ acting on a degree ${\displaystyle d}$ homogeneous polynomial by multiplication by ${\displaystyle \lambda ^{d}}$. Let us denote ${\displaystyle R=\mathbb {C} [x_{0},\ldots ,x_{n}]}$ and ${\displaystyle R_{d}}$ the finite dimensional vector space of degree ${\displaystyle d}$ homogeneous polynomials. As any polynomial can be uniquely decomposed in homogeneous pieces, we have the direct sum decomposition

${\displaystyle R=\bigoplus _{d\geq 0}R_{d}.}$
This decomposition is the eigenspace decomposition for the ${\displaystyle \mathbb {C} ^{*}}$-action on ${\displaystyle R}$: ${\displaystyle R_{d}}$ is the eigenspace of the action of ${\displaystyle \lambda \in \mathbb {C} ^{*}}$ for the eigenvalue ${\displaystyle \lambda ^{d}}$. The space ${\displaystyle R_{d}}$ is called the "weight space of weight ${\displaystyle d}$" of the ${\displaystyle \mathbb {C} ^{*}}$-action on ${\displaystyle R}$.

"'Exercise"': We say that a finite dimensional representation ${\displaystyle W}$ of ${\displaystyle \mathbb {C} ^{*}}$ is "algebraic" if the corresponding group homomorphism ${\displaystyle \mathbb {C} ^{*}\longrightarrow GL(W)}$ is algebraic. Show that the data of a ${\displaystyle \mathbb {Z} }$-graded vector space ${\displaystyle V=\bigoplus _{d\in \mathbb {Z} }V_{d}}$ with finite-dimensional graded components ${\displaystyle V_{d}}$ is equivalent to the data of a vector space ${\displaystyle V}$ with a ${\displaystyle \mathbb {C} ^{*}}$-action such that every ${\displaystyle v\in V}$ is contained in a finite-dimensional ${\displaystyle \mathbb {C} ^{*}}$-invariant subspace on which the ${\displaystyle \mathbb {C} ^{*}}$-action is algebraic.

Let ${\displaystyle f}$ be a degree ${\displaystyle d}$ homogeneous polynomial. Then, for every ${\displaystyle x\in \mathbb {C} ^{n+1}}$, we have ${\displaystyle f(x)=0}$ if and only if ${\displaystyle f(\lambda x)=\lambda ^{d}f(x)=0}$. So it makes sense to say that ${\displaystyle f}$ vanishes at some point ${\displaystyle [x]\in \mathbb {P} ^{n}}$. An ideal ${\displaystyle I}$ of ${\displaystyle R=\bigoplus _{d\geq 0}R_{d}}$ is called "homogeneous" if ${\displaystyle I=\bigoplus _{d\geq 0}(I\cap R_{d})}$ or equivalently if ${\displaystyle I}$ is generated by homogeneous polynomials. It makes sense to say that all the functions ${\displaystyle f}$ in a given homogeneous ideal ${\displaystyle I}$ of ${\displaystyle R}$ vanish at some point ${\displaystyle [x]\in \mathbb {P} ^{n}}$ and so every homogeneous ideal ${\displaystyle I}$ of ${\displaystyle R}$ defines a subvariety of ${\displaystyle \mathbb {P} ^{n}}$. In other words, the subvarieties of ${\displaystyle \mathbb {C} ^{n+1}}$ defined by the homogeneous ideals of ${\displaystyle R}$ are precisely the ${\displaystyle \mathbb {C} ^{*}}$-invariant subvarieties of ${\displaystyle \mathbb {C} ^{n+1}}$ and so define subvarieties of ${\displaystyle \mathbb {P} ^{n}}$. The ${\displaystyle \mathbb {C} ^{*}}$-invariant subvariety of ${\displaystyle \mathbb {C} ^{n+1}}$ associated to a subvariety of ${\displaystyle \mathbb {P} ^{n}}$ is called the "affine cone" of this subvariety. A subvariety of ${\displaystyle \mathbb {P} ^{n}}$ is called a "projective variety".

Let ${\displaystyle I}$ be a homogeneous ideal of ${\displaystyle R}$. Then the quotient ring ${\displaystyle R/I}$ is also graded:

${\displaystyle R/I=\bigoplus _{d\geq 0}R_{d}/(I\cap R_{d}).}$
We define the "homogeneous spectrum" of the graded ring ${\displaystyle R/I}$, denoted ${\displaystyle \operatorname {Proj} \,R/I}$, as being the set of homogeneous ideals of ${\displaystyle R/I}$ which are maximal amongst homogeneous ideals, except the so called "irrelevant ideal"${\displaystyle (R/I)^{+}=\bigoplus _{d\geq 1}R_{d}/(I\cap R_{d})}$, which is the ideal of the origin in the ${\displaystyle \mathbb {C} ^{*}}$-invariant subvariety of ${\displaystyle \mathbb {C} ^{n+1}}$ defined by ${\displaystyle I}$, and thus corresponds to the empty set after passing to projective space. It is possible to show that ${\displaystyle \operatorname {Proj} \,R/I}$ can be naturally identified with the set of points of the projective variety defined by ${\displaystyle I}$. In other words, if ${\displaystyle p_{1},\ldots ,p_{k}}$ are homogeneous polynomials in ${\displaystyle \mathbb {C} }$, we have
${\displaystyle \{p_{1}=\ldots =p_{k}=0{\text{ in }}\mathbb {P} ^{n}\}=\operatorname {Proj} \,\mathbb {C} /(p_{1},\ldots ,p_{k}).}$

"'Example"': Let ${\displaystyle C}$ be the projective curve in ${\displaystyle \mathbb {P} ^{2}}$ defined by the degree 3 homogeneous polynomial ${\displaystyle zy^{2}-x(x^{2}-z^{2})}$. Its affine cone (with real coordinates) is a union of lines through the origin. Consider its intersection with the plane ${\displaystyle \{z=1\}}$. We can identify ${\displaystyle C\cap \{z=1\}}$ with the plane affine curve ${\displaystyle Y^{2}=X(X^{2}-1)}$ via the map ${\displaystyle [x:y:z]\mapsto (x/z,y/z)}$. We think of the remaining points on ${\displaystyle C}$ as "at infinity" with respect to this affine set.

"'Exercise"': Rewrite the table of dualities by replacing ${\displaystyle \operatorname {Spec} \,}$ by ${\displaystyle \operatorname {Proj} \,}$, ideals by homogeneous ideals, rings by graded rings, ${\displaystyle \mathbb {C} ^{n}}$ by ${\displaystyle \mathbb {P} ^{n}}$...

"'Exercise"': Work out the details of ${\displaystyle \mathbb {P} ^{n}=\mathbb {C} ^{n}\cup \mathbb {P} ^{n-1}}$. Pass back and forth between homogeneous polynomials in ${\displaystyle \mathbb {P} ^{n}}$ and polynomials in ${\displaystyle \mathbb {C} ^{n}}$.

A degree ${\displaystyle d}$ homogeneous polynomial is a function on ${\displaystyle \mathbb {C} ^{n+1}}$ but is not a function on ${\displaystyle \mathbb {P} ^{n}}$. A natural question is: what is the interpretation of the degree ${\displaystyle d}$ homogeneous polynomials in terms of the projective space ${\displaystyle \mathbb {P} ^{n}}$? For ${\displaystyle [x]\in \mathbb {P} ^{n}}$, a degree ${\displaystyle d}$ homogeneous polynomial defines a function on the line ${\displaystyle \mathbb {C} x}$ which is of degree ${\displaystyle d}$. For example, if ${\displaystyle d=1}$, the coordinates ${\displaystyle x_{i}}$ are linear functionals on the line ${\displaystyle \mathbb {C} x}$. We call ${\displaystyle {\mathcal {O}}_{x}(-1)}$ the line ${\displaystyle \mathbb {C} x\subset \mathbb {C} ^{n+1}}$ and ${\displaystyle {\mathcal {O}}}$ the union of these lines, i.e.

${\displaystyle {\mathcal {O}}=\{(v,[x])\in \mathbb {C} ^{n+1}\times \mathbb {P} ^{n}\ \vert \ v\in [x]\}.}$
As the fibres of the natural map from ${\displaystyle {\mathcal {O}}}$ to ${\displaystyle \mathbb {P} ^{n}}$, ${\displaystyle (v,[x])\mapsto [x]}$, are copies of ${\displaystyle \mathbb {C} }$, i.e. lines, we see that ${\displaystyle {\mathcal {O}}}$ is a "line bundle" on ${\displaystyle \mathbb {P} ^{n}}$. As the fibre over a point of ${\displaystyle \mathbb {P} ^{n}}$, i.e. over a line in ${\displaystyle \mathbb {C} ^{n+1}}$, is precisely this line, ${\displaystyle {\mathcal {O}}}$ is called the "tautological" line bundle on ${\displaystyle \mathbb {P} ^{n}}$.

Let us consider the natural map from ${\displaystyle {\mathcal {O}}}$ to ${\displaystyle \mathbb {C} ^{n+1}}$, ${\displaystyle \pi \colon (v,[x])\mapsto v}$. The fibre of ${\displaystyle \pi }$ over ${\displaystyle v\in \mathbb {C} ^{n+1}}$ is the set of lines passing through ${\displaystyle v}$. If ${\displaystyle v}$ is non-zero, there is a unique such line and so ${\displaystyle \pi }$ is one-to-one above ${\displaystyle \mathbb {C} ^{n+1}\setminus \{0\}}$. If ${\displaystyle v=0}$, any line passes through zero and so the fibre ${\displaystyle \pi ^{-1}(0)}$ is ${\displaystyle \mathbb {P} ^{n}}$. This map ${\displaystyle \pi \colon {\mathcal {O}}\rightarrow \mathbb {C} ^{n+1}}$ is called the "blow-up of the origin in ${\displaystyle \mathbb {C} ^{n+1}}$": one goes from ${\displaystyle \mathbb {C} ^{n+1}}$ to ${\displaystyle {\mathcal {O}}}$ by separating the various lines intersecting each other at the origin and this effectively replaces the origin of ${\displaystyle \mathbb {C} ^{n+1}}$ by the projective space ${\displaystyle \mathbb {P} ^{n}}$.

As the coordinates ${\displaystyle x_{i}}$ are linear functionals on the lines ${\displaystyle \mathbb {C} x}$, the natural interpretation of the ${\displaystyle x_{i}}$'s is as sections of the dual line bundle of ${\displaystyle {\mathcal {O}}}$, called ${\displaystyle {\mathcal {O}}}$. More generally, degree ${\displaystyle d}$ homogeneous polynomials naturally define sections of the line bundle obtained by the ${\displaystyle d}$-th tensor product of the dual of ${\displaystyle {\mathcal {O}}}$, i.e. of ${\displaystyle {\mathcal {O}}}$. This line bundle is called ${\displaystyle {\mathcal {O}}}$.

"'Exercise"': What is the interpretation of the quotient of an affine variety by the action of a finite group in terms of coordinate rings?

"'Exercise"': Prove the degree-genus formula, a smooth degree ${\displaystyle d}$ curve in ${\displaystyle \mathbb {P} ^{2}}$ is of genus ${\displaystyle g={\frac {1}{2}}(d-1)(d-2)}$, by degenerating the degree ${\displaystyle d}$ curve to a union of ${\displaystyle d}$ lines.

"'Exercise"': Let ${\displaystyle X}$ be a "reasonable" topological space. Show that the closed subsets of ${\displaystyle X}$ are exactly the subsets of the form ${\displaystyle \{f_{1}=\ldots =f_{k}=0\}}$ where the ${\displaystyle f_{i}}$ are continuous functions on ${\displaystyle X}$. Now invent the Zariski topology by replacing continuous functions by polynomials.