Let $k$ be an algebraically closed field. For a finitely generated $k$-algebra $A$, a family of curves of degree $d$ in $\mathbb P^2$ over $A$ is a closed subscheme $X\subset\mathbb P^2_A$, flat over $A$, whose fibers above closed points of $\mathrm{Spec}\,A$ are curves in $\mathbb P^2$, defined by a single homogeneous polynomial of degree $d$. Show that the ideal $I_X\subset A[x,y,z]$ is generated by a single homogeneous polynomial of degree $d$.
My attempts: the condition on fibers above closed points is equivalent to the condition that for any $\mathfrak m\in\mathrm{Specm}\,A$ the ideal $(I_X,\mathfrak m)/\mathfrak m$ is principal. Equivalently, for any $\mathfrak m$ there is $f_{\mathfrak m}\in I_X$ such that $I_X\subset (f_{\mathfrak m},\mathfrak m)$. Also, it is clearly sufficient to prove that $I_X$ contains a homogeneous element of degree $d$.