9
$\begingroup$

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?

$\endgroup$
16
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Lovely! ${}{}{}$ $\endgroup$ Feb 3 '19 at 22:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.