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I haven't been able to find a proof of the following fact, which I have seen mentioned a few times: two non-singular curves have multiplicity intersection greater than 1 at a point P if and only if both curves have the same tangent at P.

I also would like to know if that result can be generalized for greater multiplicities (i.e. two non-singular curves have multiplicity intersection n at a point P if and only if there is a line such that both curves intersect it at P with multiplicity n), or replacing lines with curves of a higher degree, such as conics (i.e. two non-singular curves have multiplicity intersection n at a point P if and only if there is a conic such that both curves intersect it at P with multiplicity n).

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  • $\begingroup$ For your first point, a reference is Fulton, Algebraic Curves, Section 3.3 (in particular, "(5)" on page 37 is exactly what you want). $\endgroup$ – Nils Matthes Nov 12 '13 at 10:26
  • $\begingroup$ Thanks, that was what I needed, but I'm still wondering if the result can be generalized. $\endgroup$ – Andres Nov 14 '13 at 3:25
  • $\begingroup$ It is not true that two curves $C_1,C_2$ intersect at a point $P$ with multiplicity $n$, if and only if there is a line intersecting $C_1$ and $C_2$ with multiplicity $n$. Consider the cubic $y-x^3$ and the parabola $y-x^2$. They intersect at the origin with multiplicity $>1$, and the line $L$ given by $y=0$ is the only line, which has intersection multiplicity $>1$ with $C_1,C_2$ there. But $I(P,C_1 \cap L)=3 \neq 2= I(P,C_2 \cap L)$. I also have to admit that I don't quite see how that statement would be a generalization of the one in your first paragraph. $\endgroup$ – Nils Matthes Nov 15 '13 at 6:26

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