Intersection of Algebraic Curves P & Q at p - Proof

[tommyj]

Hi I am pretty stuck on a proof so any help would be great:

Let P and Q be two projective curves, and let p belong to both of them. Show that the intersection number of P and Q at p is equal to one iff the tangent lines to p of P and Q are distinct

NB-we have defined intersection numbers in terms of the resultant, and i also do not take algebra this term so all of the results in terms of ideals and such on the internet are of no use to me

thanks

i should also probably say that we are working in P^2

 

[mathwonk]

I'm a little puzzled. Isn't the resultant a global definition of intersection number rather than a local one? Check out the book of William Fulton on algebraic curves, available free on his website. Or accept this copy.