Everyone knows that there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$

Question

This is a statement in Fulton's Introduction to intersection theory: ''there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$ '' He implied that this is clear by duality, for example. Anyone could Please give me a more lucid explanation?

Answer 1

The term duality here refers to the dual projective plane $(\Bbb{P}^2)^\ast$, definined as the set of lines of $\Bbb{P}^2$.

Explicitly, a line $ax+by+cz=0$ in $\Bbb{P}^2$ becomes the point $(a:b:c)$ in $(\Bbb{P}^2)^\ast$.

How do you dualize a smooth curve $C$ in $\Bbb{P}^2$ then? You just define the dual curve $C^\ast$ to be the set of points of $(\Bbb{P}^2)^\ast$ corresponding the tangent lines to $C$.

Notice that $C^\ast$ may be singular: for example if $C$ admits a bitangent line. Also, the degree will not be preserved in general.

As Ted Shriffin suggests in the comments, however, the dual of a conic will be a conic. Moreover, it will be smooth ! (try to see why).

Finally, the statement

there is only one nonsingular conic tangent to five general lines in $\Bbb{P}^2$

is equivalent to its dual statement:

there is only one nonsingular conic passing through five general points in $(\Bbb{P}^2)^\ast$

If you identify $(\Bbb{P}^2)^\ast$ with a standard projective plane, the dual statement is just the classical assertion that there exists a single smooth conic passing through 5 general points in projective plane. Are you fine with this assertion? Notice that an idea for proving it is contained in KReiser's comment above.

Answer 2

As a complement let me emphasize that a more precise and more elementary statement, avoiding the complication of the term "general line", is:

Given five lines in $\mathbb P^2$ , there exists a unique smooth conic tangent to these five lines if and only if no three of these lines have a common point.