# Projective algebraic curves-affine curves

At the projective algebraic curves there are similar identities to affine curves.

Intersection points of projective algebraic curves.

The meanings

• order of point of the curve $F$
• intersection multiplicity of algebraic

are defined similarily.

If $z \neq 0$ we correspond $P=[x, y, 1]$ to the affine $[x, y, 1]$ and we calculate at it the order and the intersection multiplicity.

If $z=0$ then we apply the appropriate dehomogenization so that the Line $z=0$ becomes finite (for example we set $y=0$ and we "send" $y$ to infinity).

Can you explain to me the case $z=0$ :

If $z=0$ then we apply the appropriate dehomogenization so that the Line $z=0$ becomes finite (for example we set $y=0$ and we "send" $y$ to infinity).

• I'm not an expert when it comes to geometry, projective geometry even less so, but I think you need to look at the fact that the projective space is locally affine: $P^2=\{[x,y,z]\mid x\neq 0\}\cup \{[x,y,z]\mid y\neq 0\}\cup \{[x,y,z]\mid z\neq 0\}$ and the three patches are (Zariski) open in $P^2$ and (as algebraic varieties) isomorphic to $A^2$, and the order of a point and intersection multiplicity are local phenomena (more or less the same way you do it for smooth manifolds). – tomasz Feb 1 '15 at 17:40
• Can you explain it further? I am mixed up. @tomasz – user175343 Feb 1 '15 at 18:30
• I can't, not really, not now anyway. – tomasz Feb 1 '15 at 18:31