Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves.

Suppose we have the line

$C_1: x+y=2$

and the circle

$C_2: x^2 + y^2 = 2$.

We see that these intersect at the point $(1,1)$. We then put the curves in homogeneous coordinates and see that they are described by

$C_1: X+Y=2Z$ and $C_2: X^2 + Y^2 = 2Z^2$ and we get the single intersection point $(1:1:1) \in \mathbb{P}^2$.

Now, it is said in the text that since the line $C_1$ is tangent to the circle $C_2$ at $(1,1)$, this intersection point "should count double". Can anyone explain the intuition behind this statement?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Intuitively, if we have two irreducible plane curves $F$ and $G$ that intersect at a point $P,$ then the intersection multiplicity $I(P, F \cap G)$ is the maximum number of intersections of $F$ and $G$ near $P$ if we wiggle $F$ and $G$ a little.

For example, take $F$ to be a curve and $G$ to be a straight line that goes through $F$ at $P$ transversally. Then there is at most single point of intersection even if we wiggle $G$ a bit, so the multiplicity of this intersection is $1.$

Now take the cubic with a cusp $F= Y^2-X^3$ and the line $G= X.$ Then $G$ intersects $F$ at $P=(0,0)$ and if we shift $G$ a little to the right, it intersects $F$ in two places so $I(P,F\cap G)=2.$

share|improve this answer
    
Thanks, that's interesting. I had never heard this idea of ``wiggling'' our curves to explain the intersection multiplicity. Do you have a reference for that? –  nigelvr May 9 at 1:07
    
@nigelvr I actually have never read this anywhere. I first learned about Intersection Multiplicity when reading Section 3.3 of Fulton's Algebraic Curves. There he introduces it as a number that we want to satisfy 7 properties, and I figured out on my own what such a number does intuitively. When I later told my supervisor about it, he seemed to think it was obvious so I think it is well known but people don't like to write down imprecise things. –  Ragib Zaman May 9 at 2:13
    
Let me give you some warnings about where to be careful. I assume you are working with Real numbers. Suppose I took $F=Y^2-X^3$ again and now $G=Y.$ If I wiggle $G$ left and right, it stays the same. If I move it up and down, it still only intersects at $1$ point. If I rotate it a little around the origin, then I can get two intersections. So is the Intersection Multiplicity $2$ ? What other ways can someone "wiggle" the curve? It turns out (as it does for many things in algebraic geometry) to get a good consistent notion of Intersection Multiplicity, one needs to consider intersections.... –  Ragib Zaman May 9 at 2:18
    
... not just in the original field (here the Real numbers) but in the algebraic closure of the field (here the complex numbers). If you do this, and wiggle $G$ to $G' = Y-\epsilon$ then you see that you get $3$ distinct intersection points, with $Y=\epsilon$ and $X$ being the $3$ cube roots of $\epsilon^2.$ So the intersection number here is actually $3.$ –  Ragib Zaman May 9 at 2:21

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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