# Can a polynomial of degree 2 vanish on three different lines?

Suppose $p(x,y,z)$ is a homogeneous polynomial of degree 2. My question is:

Can I have three distinct lines $L_1,L_2,L_3$ in the projective space $\mathbb{P}^2$ such that $p$ vanishes on every point of the three lines?

My intuition says that this is not possible, but I'm just starting to learn about projective geometry so I do not know how to prove this. Any help?

• Are you sure you want $p$ to be a homogeneous polynomial in 3 variables? Projective 3-space has 4 co-ordinates. If 3 variables, it vanishes along infinitely many lines. – Mohan Oct 15 '15 at 22:44
• Of course, I meant $\mathbb{P}^2$, thanks! – u1571372 Oct 15 '15 at 22:47
• If $p$ vanishes along a line defined by $l(x,y,z)=0$, then, $l$ divides $p$. – Mohan Oct 15 '15 at 22:52
• Nullstellensatz? – Hoot Oct 15 '15 at 23:01
• @Hoot, how can Nullstellensatz be applied here? – u1571372 Oct 15 '15 at 23:02

If $l(x,y,z)=0$ defines a line and $p(x,y,z)$ vanishes on it, then changing variables, you may assume that $l=x$. Thus the restriction of $p$ to $x=0$ is just $p(0,y,z)$. If this polynomial is not identically zero, it is clear that $p$ does not vanish at some point on the line. So, it follows that $p(0,y,z)=0$, which is same as saying $x$ divides $p$. Now changing back to the original coordinates, this means $l$ divides $p$.