Projective varieties and homogeneous ideals

(Projective variety)

A subset of $\PP^n$ is closed if it is $V(T)$ for a set $T$ of homogeneous elements of $S \da k[x_0,\ldots,x_n]$. A projective variety is an irreducible closed subset of $\PP^n$; a quasi-projective variety is an open subset of one.

Let $p\in\PP^n$ have representative $v\in k^{n+1}\sm\ts{0}$, and let $f\in S$ be homogeneous of degree $d$. Then $f(\lambda v)=\lambda^d f(v)$ for $\lambda\in k^\times$, so the condition $f(v)=0$ does not depend on the representative $v$, although the value $f(v)$ does when $d>0$. For $f=x_0+x_1^2$ in $k[x_0,x_1]$ and $v=(-1,1)$, $f(v)=0$ but $f(2v)=2\ne0$ when $\operatorname{char}k\ne2$. Homogenizing with respect to a new variable $Z$ gives projective closures: $V(x^2 + y^2 - 1)$ becomes $V(X^2 + Y^2 - Z^2)$, and $V(y - x^3)$ becomes $V(YZ^2 - X^3)$.

The irrelevant ideal $S_+ = (x_0,\ldots,x_n)$ is homogeneous and radical, and $V(S_+) = \emptyset=V(S)$. The projective Nullstellensatz: for a homogeneous ideal $J$, $V(J) = \emptyset$ exactly when $\sqrt{J} \supseteq S_+$, and otherwise $I(V(J)) = \sqrt{J}$. Hence $V$ and $I$ are inclusion-reversing bijections, inverse to each other, between closed subsets of $\PP^n$ and homogeneous radical ideals of $S$ other than $S_+$ [@Har10a, Exercise I.2.4].

What is a projective variety?

Why must the defining polynomials be homogeneous?

Which homogeneous ideals correspond to which closed subsets of $\PP^n$?

Over an infinite field, show that $f$ is homogeneous of degree $d$ if and only if $f(\lambda \mathbf{x}) = \lambda^d f(\mathbf{x})$ for all $\lambda$.

Find the projective closures of $V(x^2 + y^2 - 1)$ and $V(y - x^3)$ in $\PP^2$.

Describe the line at infinity in $\PP^2$, and find the intersection of the projective closures of $V(y)$ and $V(y - tx - 1)$ when $t = 0$.

Show that a real conic $V(f) \subseteq \RR^2$ with infinitely many real points is a circle if and only if the projective closure of $V(f)$ passes through the circular points $[1 : \pm i : 0]$.

Show that conics in $\PP^2$ are parametrized by $\PP^5$, and that the conics through a fixed point form a hyperplane $H_p$.

Show that over an infinite field the conics through $5$ fixed points form a single point or an infinite set, and that there is a unique conic through $5$ points in linear general position.

Show that there is a unique circle through any $3$ non-collinear points of $\RR^2$.

Describe $\PGL_{n+1}(k)$ as a quotient, and show that for two sets of $n+2$ points in linear general position in $\PP^n$ there is a unique element of $\PGL_{n+1}(k)$ carrying one to the other.

Show that the $(k+1)$-dimensional linear subspaces of $\PP^n$ containing a fixed $k$-dimensional subspace $W$ form a $\PP^{n-k-1}$.

For $\PP^n = \PP(V)$, show that $\PP(\dualof{V})$ parametrizes the hyperplanes of $\PP^n$.

Show that there is a unique rational normal curve through $n+3$ points in linear general position in $\PP^n$.

Show that $\dim(V_1 \cap V_2) \geq \dim V_1 + \dim V_2 - n$ for linear subspaces $V_1, V_2 \subseteq \PP^n$.

Dependencies

PR-7OT2ZThe ideal-variety correspondenceuses

Backlinks

D-VARQAPAffine, quasi-affine, projective, and quasi-projective varieties; the affine charts of $\PP^n$uses
PR-EFW6BRegular functions on a projective variety are constantuses
D-L6ERWThe Hilbert polynomial and its coefficientsuses
PR-VA4S3The Segre embedding, and why $\PP^m \times \PP^n$ is projectiveuses

What links to this

Affine or projective