Projective Algebraic Sets Projective Morphisms and Cones — Examples

For ${\mathbf{P}}_k^1$, $U_0=D_{+}(X)$ has coordinate $t=Y\mathrm{/}X$ and $U_1=D_{+}(Y)$ has coordinate $s=X\mathrm{/}Y$. On $U_0\cap U_1$, both coordinates are nonzero and $s=1\mathrm{/}t$. Thus the two affine lines glue by inversion; $t=0$ is $[1:0]$ and the point outside $U_0$ is $[0:1]$.

For the parabola $A=V(y-x^2)\subseteq {\mathbf{A}}_k^2$, use homogeneous coordinates $[Z:X:Y]$. Its homogenization is $YZ-X^2$, so ${\bar{A}}^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=V_{+}(YZ-X^2)$. The $Z=1$ chart is $y=x^2$. At infinity $Z=0$, the equation gives $X=0$, leaving exactly $[0:0:1]$.

Raw homogenization of arbitrary generators always gives the projective closure.

For $Q=V_{+}(XZ-Y^2)\subseteq {\mathbf{P}}_k^2$, the $X=1$ chart has equation $z=y^2$, and the $Z=1$ chart has equation $x=y^2$. The $Y=1$ chart has equation $xz=1$. These are the standard affine pieces, related on overlaps by ratio-coordinate changes.

Assume $\mathrm{char}k\mathrm{\ne }2$. The affine cone over $V_{+}(XZ-Y^2)$ is $V(XZ-Y^2)\subseteq {\mathbf{A}}_k^3$. Its partial derivatives are $Z,-2Y,X$, all zero at $(0,0,0)$; the defining equation also vanishes there. This is the elementary hypersurface-derivative diagnostic at the vertex only, not an invocation of a general singular-locus theorem.

An arbitrary polynomial equation has a representative-independent zero locus in projective space.

For $d\ge 1$, $[X:Y]\mapsto [X^d:Y^d]$ is defined because $X^d,Y^d$ have common degree and no common projective zero. On $X\mathrm{\ne }0$ it is $t=Y\mathrm{/}X\mapsto t^d$, and on $Y\mathrm{\ne }0$ it is $s=X\mathrm{/}Y\mapsto s^d$. The formulas agree under $s=1\mathrm{/}t$.