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?
[Math] Family of curves (in algebraic geometry)
algebraic-curvesalgebraic-geometry
Related Solutions
What Shafarevic calls a local parameter is often called a uniformizing parameter at $P$, and is also the same thing as a uniformizer of the local ring of $C$ at $P$.
The point is that if $P$ is a smooth point on a curve, then the local ring at $P$ (i.e. the ring of rational functions on $C$ which are regular at $P$) is a DVR, and hence its maximal ideal is principal; a generator of this ideal is called a uniformizer.
If $t$ is a uniformizer/local parameter/uniformizing parameter at $P$, and if $u$ is any other rational function, then if we write $u = t^k v$ where $v(P) \neq 0$ (i.e. $v$ is a unit in the local ring), then $k$ is the order of vanishing of $u$ at $P$. In particular, $u$ vanishes to order one if and only if it is equal to $t$ times a unit in the local ring, if and only if it is also a generator of the maximal ideal of the local ring at $P$, if and only if it is also a uniformizer. Thus Shafarevic and Wikipedia are reconciled.
One is supposed to think of $t$ as being a "local coordinate at $P$." In the complex analytic picture you would choose a small disk around $P$, and consider the coordinate $z$ on this disk; this a local coordinate around the smooth point $P$. This analogy is very tight: indeed, it is not hard to show (when the ground field is the complex numbers) that a rational function $t$ is a local parameter at $P$ if and only $t(P) = 0$, and if there is a small neighbourhood of $P$ (in the complex topology) which is mapped isomorphically to a disk around $0$ by $t$, i.e. if and only if $t$ restricts to a local coordinate on a neighbourhood of $P$.
Finally, this concept is ubiquitous. The fact that the local ring at a point on a smooth algebraic curve is a DVR is fundamental in the algebraic approach to the theory of algebraic curves; see e.g. section 6 of Chapter I of Hartshorne.
Let $K / k$ be finite extension of fields of degree $d$. Let $X$ be smooth geometrically connected projective curve over $K$ of genus $g$. The morphism $\mathrm{Spec} K \to \mathrm{Spec} k$ is projective (Exercise 3.3.22 of Liu's Algebraic geometry and arithmetic curves), therefore $X$ is projective over $k$. Then $$ p_{a,k}(X) = 1- \chi_k(X) = 1 - d \cdot \chi_K(X) = 1 - d (1-g). $$ You can take $g = 0$.
EDIT. More concretely, let $k$ be a field, let $f \in k[t]$ be a monic irreducible polynomial of degree $d$, let $\alpha \in \bar{k}$ be a root of $f$ and let $K = k(\alpha)$. If $f(t) = \sum_{i=0}^d a_i t^i$, then $\mathrm{Spec} K$ is isomorphic to $$ \mathrm{Proj} k[x_0,x_1] / (\sum_{i=0}^d a_i x_0^i x_1^{d-i} ). $$ Then $$ X = \mathbb{P}^1_K = \mathbb{P}^1_k \times_k \mathrm{Spec} K = \mathbb{P}^1_k \times_k \mathrm{Proj} k[x_0,x_1] / (\sum_{i=0}^d a_i x_0^i x_1^{d-i} ) $$ is a closed subscheme of $\mathbb{P}^1_k \times_k \mathbb{P}^1_k$, hence it is a closed subscheme of $\mathbb{P}^3_k$ by Segre embedding, i.e. $$ X = \mathrm{Proj} k[z_0, z_1, z_2, z_3] / (z_0 z_3 - z_1 z_2, \sum_{i=0}^d a_i z_0^i z_1^{d-i}, \sum_{i=0}^d a_i z_2^i z_3^{d-i}).$$
Best Answer
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.