0
$\begingroup$

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).

$\endgroup$
  • 1
    $\begingroup$ 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). $\endgroup$ – tomasz Feb 1 '15 at 17:40
  • $\begingroup$ Can you explain it further? I am mixed up. @tomasz $\endgroup$ – user175343 Feb 1 '15 at 18:30
  • $\begingroup$ I can't, not really, not now anyway. $\endgroup$ – tomasz Feb 1 '15 at 18:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy