Let $C_1$ and $C_2$ be two smooth curves of degrees $m$ and $n$ in $\mathbb CP^2$. By Bezout's theorem the maximal number of their intersections is $mn$. I wonder if the minimal possible number is known for all pairs $(m,n)$?

In particular for which pairs $(m,n)$ there can be exactly one point of intersection?


For all pairs. Suppose $n\geq m$. Take for $C_1$ a curve with an inflection point of order $m$, say $F=0$ with $F(X,Y,Z)=ZY^{m-1}+X^m+Z^m$. Then take $C_2$ defined by $G(X,Y,Z)F(X,Y,Z)+Z^n=0$, where $G$ is general of degree $n-m$. Then $C_1\cap C_2$ is reduced to the point $p:=(0,1,0) $. To make sure that $C_2$ is smooth, we apply Bertini's theorem to the linear system of curves given by $G(X,Y,Z)F(X,Y,Z)+\lambda Z^n=0$. The only base point is $p$, and if $G(p)\neq 0$ the curve is smooth at $p$, so we are done.

  • $\begingroup$ Lovely! ${}{}{}$ $\endgroup$ Feb 3 '19 at 22:37

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