Show that the set of plane curves of degree $d$ is in one-to-one correspondence with $\mathbb{C}\mathbb{P}^{d(d+3)/2}.$

Question

Show that the set of plane curves of degree $d$ is in one-to-one correspondence with $\mathbb{C}\mathbb{P}^{d(d+3)/2}.$

A plane curve is a degree $d$ hypersurface in $\mathbb{C}\mathbb{P}^2.$ I can see the answer for specific values of $d,$ but I don't know how to generalize it.

If $d = 1,$ then the plane curves look like $a_0x_0 + a_1x_1 + a_2x_2 = 0,$ which corresponds to $\mathbb{C}\mathbb{P}^2,$ and $1*(1 + 3)/2 = 2.$

If $d = 2,$ then the plane curves look like $a_0x_0^2 + a_1x_1^2 + a_2x_2^2 + a_3x_0x_1 + a_4x_0x_2 + a_5x_1x_2= 0,$ which corresponds to $\mathbb{C}\mathbb{P}^5,$ and $2*(2 + 3)/2 = 5.$

This feels like a combinatorics issue, but I do not see how it falls into place.

Answer 1

Hint: A plane curve of degree $d$ is cut out by a polynomial of degree $d$ in $3$ variables. What object do the collection of all such polynomials form? When do two of these polynomials correspond to the same curve?

Bigger hint:

The polynomials form a vector space. Two polynomials determine the same curve iff they're nonzero scalar multiples of eachother, so we want the projective space of this vector space. Can you determine the dimension of the vector space of such polynomials?

Biggest hint:

By stars and bars, the dimension of the space of polynomials of degree $d$ in three variables is $\binom{d+2}{2}$. Can you verify the expected relation between this number and the dimension of the projective space you've been given?