I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my comments below, then my questions follow.
Let $C$ be the conic given by the equation $$F(x,y) = ax^2 + bxy + cy^2 + dx + ey + f = 0$$ and let $\delta$ be the determinant $$\det \begin{bmatrix} 2a & b & d \\ b & 2c & e \\ d & e & 2f \\ \end{bmatrix}$$
(a) Show that if $\delta \neq 0$ then $C$ has no singular points.
Answered here.
(b) Conversely, show that if $\delta = 0$ and $b^2 - 4ac \neq 0$ then there is a unique singular point on $C$.
Given these two conditions we can classify the curve as two intersecting lines or a single point. So there is one singular point on the curve. This is from wikipedia.
(c) Let $L$ be the line $y = \alpha x + \beta$ with $\alpha \neq 0$. Show that the intersection of $L$ and $C$ consists of either zero, one, or two points.
Substitute the equation of the line into the equation of the conic. Then we get a quadratic equation $(a + \alpha b + \alpha^2 c)x^2 + (\beta b + 2\alpha\beta c + d + \alpha e)x + (c\beta^2 + \beta e + f) = 0$. This equation has zero, one, or two solutions.
(d) Determine the conditions on the coefficients which ensure that the intersection $L \cap C$ consists of exactly one point. What is the geometric significance of these conditions?
Let $(a + \alpha b + \alpha^2 c)x^2 + (\beta b + 2\alpha\beta c + d + \alpha e)x + (c\beta^2 + \beta e + f) = a'x^2 + b'x + c'$ There is exactly one solution if the determinant $b'^2 - 4a'c'$ of the quadratic equation is 0. Geometrically, the intersections collide: the line must be tangent.
My questions:
(b) Can anyone provide a reference other than Wikipedia for this? I've searched a lot but can't find anything that actually gives a derivation.
(c,d) What is the condition for zero, one, two roots? The field is not specified so I assume these roots could be complex. Then
- we have zero roots if $a' = b' = 0$
- we have one root if $a' = 0, b' \neq 0$
- we have two roots if $a' \neq, b' \neq 0$
But the two roots could be one root of multiplicity two. Is that correct? Or should I only be thinking about the third case (over real numbers), then considering the possible roots, with complex roots left out.
(d) How do I flesh out a proof of this fact? (i.e. How would I prove that $b'^2 - 4a'c'$ implies the line is tangent?)