Affine, quasi-affine, projective, and quasi-projective varieties; the affine charts of $\PP^n$

(Affine, quasi-affine, projective, and quasi-projective varieties)

An affine variety is an irreducible closed subset of $\AA^n$; a quasi-affine variety is an open subset of one. A projective variety is an irreducible closed subset of $\PP^n$; a quasi-projective variety is an open subset of one. Affine, quasi-affine, and projective varieties are quasi-projective.

(Standard charts)

$U_i \da \ts{x_i \neq 0} \subseteq \PP^n$ is open, the map \[ \phi_i: U_i \to \AA^n, \qquad \tv{x_0 : \cdots : x_n} \mapsto \qty{ x_0/x_i, \ldots, \widehat{x_i/x_i}, \ldots, x_n/x_i } \] is an isomorphism of varieties, and $\PP^n = \union_{i=0}^n U_i$.

For $Y \subseteq \PP^n$ closed, $\phi_i(Y\cap U_i)$ is the affine variety cut out by the dehomogenizations $f(x_0,\ldots,1,\ldots,x_n)$, with $1$ in position $i$, for $f \in I(Y)$. For $Z\subseteq\AA^n\cong U_0$ closed, the projective closure of $Z$ is cut out by the homogenizations of all elements of $I(Z)$.

Homogenizing only a generating set can give a larger set: for the twisted cubic $Z=V(y-x^2,z-x^3)$, the homogenizations $yw-x^2$ and $zw^2-x^3$ both vanish on the line $V(w,x)$ at infinity, which is not contained in the projective closure of $Z$.

What is a quasi-affine variety? A quasi-projective variety?

How is $\PP^n$ covered by affine spaces?

Derive the rational parametrization $\AA^1 \to V(a^2 + b^2 - 1)$ and its inverse, with their domains, and projectivize both maps.

If $X$ is projective and $F$ is homogeneous of degree $d$, show that $\ts{x \in X \st F(x) \neq 0}$ is an open affine subset of $X$.

Dependencies

D-CP2MHProjective varieties and homogeneous idealsuses

What links to this

Regular functions