# Nine points in $\Bbb{P}^2$ determine a cubic?

I'm sure this is a silly question, but for some reason I'm stuck in it.

The $$k$$-vector space generated by the monomials $$x,y,z$$ give all possible equations for lines in $$\Bbb{P}^2$$ and has projective dimension $$3-1=2$$. So two points in $$\Bbb{P}^2$$ determine a line.

Analogously, the space of $$x^2,xy,xz, y^2,yz,z^2$$ gives the equations of conics and has projective dimension $$6-1=5$$. So five points determine a conic.

For cubics, the space is $$x^3,x^2y,x^2z,xy^2,xz^2,xyz,y^3, y^2z, yz^3, z^3$$ with projective dimension $$10-1=9$$. Then shouldn't nine points determine a cubic?

Take for example a cubic $$C$$ and three concurrent lines $$L_1,L_2,L_3$$ meeting at $$P\notin C$$, each $$L_i$$ meeting $$C$$ simply. (for example, take $$C:yz^2-(x^2-3z^2)x=0$$, $$L_1:y=0$$, $$L_2:y-z=0$$ and $$L_3:y+z=0$$).

The intersection $$C\cap L_1L_2L_3$$ consists of $$9$$ distinct points, but $$C$$ and $$L_1L_2L_3$$ are distinct cubic through them.

What am I missing?

• @saulspatz sure, thank you. Jun 3, 2021 at 19:17
• The points should be in "general positions". For example, if we have five points on the same line, then they don't uniquely determine a conic. Jun 3, 2021 at 19:22
• I think this can be viewed as conditions for solution to a linear system. If you write the equation for a cubic, you have 10 monomials, so you have 10 coefficients of the monomials. Now, for each point you want on the curve you put in the values of $x,y,z$ you get one linear equation in the coeffcients. Since you're working projective, you eliminate the $z$ and remain with 9 coefficients multiplied by a number, and a constant term. Then after 9 points you have to check the determinant of the $9x9$ matrix to see if it's non-zero.
– user932138
Jun 3, 2021 at 19:34

9 general points determine a curve, but 9 specific points might determine lots of curves. It’s similar to how a $$9\times9$$ matrix is usually invertible, but if you’re not careful about choosing your 9 rows, they could be linearly dependent and have lots of solutions.