I would like to understand why the set of classes of divisors corresponding to the conics of $\mathbb P^2$ passing through $3$ non-colinear points is a linear system of dimension $2$.
Thank you very much.
Edit :
Well, there are $6$ degree $2$ monomials of $k[x_0,x_1,x_2]$. We can identify all the conics with $\mathbb P^5$. Then, when one fixes a point contained in the conic, it implies that the conic belongs to a hyperplane of $\mathbb P^5$. By the same way, fixing $1\le k\le 5$ points of the conic determines a projective subspace of $\mathbb P^5$ of dimension $5-k$. So in our case, the projective subspace which parametrizes all possible conics is of dimension $2$. What is still unclear for me is how to translate it properly into the language of linear system of divisors.