Proj

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

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

$$(\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}.$$

$$(\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}.$$

Making use of the natural Hermitian structue on $\mathbb {C} ^{n+1}$$\mathbb {C} ^{n+1}$, i.eof the norm $|x|=\sum _{i=0}^{n}|x_{i}|^{2}$$|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 $\mathbb {P} ^{n}$$\mathbb {P} ^{n}$ with the quotient $S^{2n+1}/S^{1}$$S^{2n+1}/S^{1}$ where $S^{k}$$S^{k}$ denotes the $k$$k$-dimensional sphere. In particular, $\mathbb {P} ^{n}$$\mathbb {P} ^{n}$ is compact for the complex topology. The space

$$\mathbb {P} ^{n}=\{{\text{lines in }}\mathbb {C} ^{n+1}\}=(\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}=S^{2n+1}/S^{1}$$

$$\mathbb {P} ^{n}=\{{\text{lines in }}\mathbb {C} ^{n+1}\}=(\mathbb {C} ^{n+1}\setminus \{0\})/\mathbb {C} ^{*}=S^{2n+1}/S^{1}$$

is called the "projective space of dimension $n$$n$".

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

$$R=\bigoplus _{d\geq 0}R_{d}.$$

$$R=\bigoplus _{d\geq 0}R_{d}.$$

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

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

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

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

$$R/I=\bigoplus _{d\geq 0}R_{d}/(I\cap R_{d}).$$

$$R/I=\bigoplus _{d\geq 0}R_{d}/(I\cap R_{d}).$$

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

$$\{p_{1}=\ldots =p_{k}=0{\text{ in }}\mathbb {P} ^{n}\}=\operatorname {Proj} \,\mathbb {C} /(p_{1},\ldots ,p_{k}).$$

$$\{p_{1}=\ldots =p_{k}=0{\text{ in }}\mathbb {P} ^{n}\}=\operatorname {Proj} \,\mathbb {C} /(p_{1},\ldots ,p_{k}).$$

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

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

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

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

$${\mathcal {O}}=\{(v,[x])\in \mathbb {C} ^{n+1}\times \mathbb {P} ^{n}\ \vert \ v\in [x]\}.$$

$${\mathcal {O}}=\{(v,[x])\in \mathbb {C} ^{n+1}\times \mathbb {P} ^{n}\ \vert \ v\in [x]\}.$$

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

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

As the coordinates $x_{i}$$x_i$ are linear functionals on the lines $\mathbb {C} x$$\mathbb {C} x$, the natural interpretation of the $x_{i}$$x_i$'s is as sections of the dual line bundle of ${\mathcal {O}}$${\mathcal {O}}$, called ${\mathcal {O}}$${\mathcal {O}}$. More generally, degree $d$$d$ homogeneous polynomials naturally define sections of the line bundle obtained by the $d$$d$-th tensor product of the dual of ${\mathcal {O}}$${\mathcal {O}}$, i.e. of ${\mathcal {O}}$${\mathcal {O}}$. This line bundle is called ${\mathcal {O}}$${\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 $d$$d$ curve in $\mathbb {P} ^{2}$$\mathbb {P} ^{2}$ is of genus $g={\frac {1}{2}}(d-1)(d-2)$$g={\frac {1}{2}}(d-1)(d-2)$, by degenerating the degree $d$$d$ curve to a union of $d$$d$ lines.

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