Given a finitely generated $k$algebra $A$ over alg. closed $k$, a family of curves of degree $d$ is defined to be a subscheme $X\subset \mathbb P^2_A$ flat over $A$ whose fibers over closed points of $A$ are planes curves of degree $d$. Why is the ideal $I_X\subset A[x,y,z]$ of $X$ necessarily generated by a single homogeneous polynomial of degree $d$ in $A[x,y,z]$, and what conditions are there on such a polynomial for the corresponding subscheme of $\mathbb P^2_A$ to be flat over $A$?
