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I'm reading chapter IV of Robert J Walker's book 'algebraic curves'. The last section of this chapter is about Max noether's AG+BF theorem.

I am stuck on an exercise in this section. The exercise states : the tangents at the six intersections of a cubic and a conic meet the cubic again in six points on a conic.

This statement is somewhat confusing. I think it doesn't make sense if F is reducible or 'the tangents' are tangents of the conic. Also, the conic and cubic have to meet in six distinct simple points of the cubic. otherwise, the number of the residual intersections with the cubic of the tangents may be less than six points.

So if I restates the problem as i understand, then : if a irreducible cubic and a conic meet in six distinct points Pi, and also if the tangent of the cubic at Pi meet the cubic again at Qi, then the six points Qi are on a conic.

I tried to solve the problem in this way: Let F denote the cubic and G denote the conic. If H is a quartic that passing through the 12 points Pi and Qi, then these points are the 12 intersections of F and H. Also, these are all simple points of F, and G intersects F in 6 of these points (the Pi's). Thus by the theorem7.7 in the book, the remaining six points (Qi's) are on a conic.

I realized that this proof is wrong since H may have F as a factor, so that H and F have infinite intersections.

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    I'm not sure Jesus is a user, here. – Cameron Buie Nov 30 '15 at 13:16
  • Finally I proved it : Let Ti be the tangent of the cubic F at Pi. Applying the AG+BF theorem with F, G^2, H=T1T2...T6. Then H=AG^2+BF and A is nonzero since H is not a multiple of F. F and H vanish at Qi but G doesn't vanish at Qi. Therefore Qi are on the conic A. –  Nov 30 '15 at 18:16

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