Intersection of Algebraic Curves P & Q at p - Proof

In summary, the intersection number of two projective curves, P and Q, at a point p is equal to one if and only if the tangent lines to p of P and Q are distinct. This can be seen by considering the resultant, a global definition of intersection number, and checking out William Fulton's book on algebraic curves. Please note that algebraic knowledge is necessary for this proof.
  • #1
tommyj
22
0
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
 
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  • #2
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.
 

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FAQ: Intersection of Algebraic Curves P & Q at p - Proof

What is the definition of an intersection of algebraic curves P and Q at p?

An intersection of algebraic curves P and Q at p is a point where the two curves intersect on a 2-dimensional plane. This point is determined by the coordinates of p, which satisfy both the equations of P and Q.

How can the intersection of P and Q at p be proven?

The intersection of P and Q at p can be proven by substituting the coordinates of p into the equations of P and Q. If the resulting values are equal, then p is a point of intersection.

Can there be more than one intersection of P and Q at p?

Yes, there can be more than one intersection of P and Q at p. This can occur if the two curves have more than one point of intersection on the 2-dimensional plane.

How does the degree of P and Q affect the number of intersections at p?

The degree of P and Q can affect the number of intersections at p. If the degree of P and Q is higher, there is a higher likelihood of multiple intersections at p. However, it is not a guarantee and the exact number of intersections can vary.

Are there any special cases in which P and Q may not have any intersections at p?

Yes, there are special cases in which P and Q may not have any intersections at p. This can occur if the two curves are parallel or if they do not intersect at all on the 2-dimensional plane.

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