Intersection of Algebraic Curves P & Q at p - Proof

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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 both curves are distinct. This conclusion is based on the definition of intersection numbers in terms of the resultant, as discussed in the context of projective geometry. The discussion references William Fulton's book on algebraic curves for further insights into the topic.

PREREQUISITES
  • Understanding of projective geometry, specifically P^2
  • Familiarity with the concept of intersection numbers in algebraic geometry
  • Knowledge of the resultant and its application in determining intersection properties
  • Basic concepts of tangent lines in the context of curves
NEXT STEPS
  • Study William Fulton's book on algebraic curves for a comprehensive understanding of intersection theory
  • Learn about the properties and applications of the resultant in algebraic geometry
  • Explore the concept of tangent lines to curves in projective space
  • Investigate local versus global definitions of intersection numbers in algebraic geometry
USEFUL FOR

Mathematicians, algebraic geometers, and students studying projective curves and intersection theory will benefit from this discussion.

tommyj
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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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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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