Family of curves (in algebraic geometry) How can I view, in algebraic geometry, a family of curves over a base curve?
For instance, can the family
$y^2 = x(x-1)(x-t)$
be viewed as a family over
$\mathbb{P}^1$ ?
How can I understand this rigorously? Can someone point me to a reference?
 A: A family $X \to S$ is just a flat morphism of varieties. The "members" of the family are the fibers $X_s$ above the points $s \in S$. Intuitively, the flatness of the morphism says that the members of the family look alike. 
This is in Hartshorne, the section on flat morphisms. For an introduction without the language of sheaf cohomology, try Eisenbud and Harris's The Geometry of Schemes. I don't know any reference where you can learn about families in the context of varieties and not schemes, but since flatness is the key condition underlying a family, you really need scheme-theoretic language anyway.
In your example, we have a morphism to $\mathbb{A}^1$ which just takes $(x, y, t)$ satisfying your equations and returns $t$. This morphism is flat because the ring of functions $k[t]$ on $\mathbb{A}^1$ is a Dedekind domain and so a module is flat over it iff it is $k[t]$-torsion-free.
When you say it is a family over $\mathbb{P}^1$, I assume you mean some compactification of your surface admits a flat map to $\mathbb{P}^1$ extending this one. One natural compactification is to view your surface as a subvariety of $\mathbb{P}^2 \times \mathbb{P}^1$ where the variables in the first are $x, y, z$ and the latter are $s, t$. So the bihomogeneous polynomial of your surface is then
$$
y^2zs = x^3s - x^2zt -x^2zs + xz^2t
$$ 
(To compute this, I expanded everything out, then made it homogeneous separately in $x, y, z$ and $t, s$) and the map is just the restriction of the projection to $\mathbb{P}^1$; on the open subvariety where $s, z\neq 0$, this is your old family. You'll need to prove this is flat over the new point $\infty$ we have added to the $t$-line. 
