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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 $A$?

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A useful theorem says that the vanishing set of a single polynomial in a flat family is another flat family when that polynomial does not vanish on an entire irreducible component in any of the fibers. In your case, since the fibers are all irreducible, the polynomial is not allowed to vanish modulo any proper ideal of $A$, so the ideal generated by the coefficients must be the unit ideal. – Will Sawin Jun 28 '12 at 15:54
Do you have a reference for this theorem? Also, when you say the fibers are irreducible, are you referring to the trivial family $\mathbb P^2_A$? THanks – HNuer Jun 28 '12 at 16:19
It's one of the first theorems in the flat morphisms section of chapter 1 of Milne's Etale Cohomology, which I don't have with me right now. Yes. – Will Sawin Jun 28 '12 at 16:41

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