A flat fat-point family and its base changes

The family has no points on $Y=0$: there its equation becomes $X^2=0$ but $X$ is invertible. On $Y\mathrm{\ne }0$ its algebra is $A[x]\mathrm{/}(x^2-t)$, free over $A$ with basis $1,x$ by division by the monic polynomial. It is finite flat of rank two, and its closed immersion is finitely presented. Every fibre has length two and therefore constant Hilbert polynomial $2$ by [F1]. At $t=0$ it is a double point. In characteristic different from two, nonzero fibres are either two distinct rational points or a quadratic field point. In characteristic two a nonzero fibre can also be a double point: at $t=1$, $x^2-1=(x-1{)}^2$. Total length is always two.

For an arbitrary $A$-algebra $B$, its algebra pulls back to $B[x]\mathrm{/}(x^2-t_B)$, still free on $1,x$. Thus the family remains finitely presented and flat after any base change. By [F1], the corresponding map to the Hilbert scheme is the composite of $\mathrm{Spec}B\to \mathrm{Spec}A$ with the original classifying map, and its scheme theoretic pulled-back family is exactly this algebra. In particular the assertion applies when $B$ has nilpotents.