Exercise 1.1.(c) in Hartshorne's Deformation Theory:

Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(x,y,z)$ of degree $d$ in the coordinate ring $S=k[x,y,z]$.

(c) For any finitely generated $k$-algebra $A$, we define a family of curves of degree $d$ in $\mathbb P^2$ over $A$ to be a closed subscheme $X\subseteq\mathbb P^2_A$, flat over $A$, whose fibers above closed points of $\mathrm{Spec}\,A$ are curves in $\mathbb P^2$. Show that the ideal $I_X\subseteq A[x,y,z]$ is generated by a single homogeneous polynomial $f$ of degree $d$ in $A[x,y,z]$.

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 $d$.