I'm trying to work out an example of a proper family of elliptic curves in order to get a better grip on the subject. By a proper family of elliptic curves, I mean a proper flat morphism $X\to S$ such that each fiber is an elliptic curve. However, the example staring me in the face seems not to be salvageable. I would like to understand what exactly is standing in the way of this example working out, as well as to be pointed in the direction of some hands on examples of (nontrivial) families of elliptic curves.

The sort of example I feel immediately drawn to is the family $X_t:y^2=x^3+t$ parameterized by $t$ (I understand that $X_0$ is not an elliptic curve, but we should be able to simply remove that fiber). To be explicit with notation, let $X=\operatorname{Spec} k[x,y,t]/(y^2-x^3-t)$ sitting over $S=\operatorname{Spec} k[t]$. The fibers here are not elliptic curves (they're affine), which I would like to solve by moving to the closure $\overline X$ of $X$ in $\mathbb P^3$. Now we are faced with the problem that any morphism $\overline X\to S$ must be constant, as $\overline X$ is projective and $S$ is affine, so we need to extend to a family $\overline X\to \mathbb P^1$. In order for this morphism to be proper, it must be surjective, but it is far from apparent how to map any points to $\infty$ and still have the desired family of curves. It is here that I am unsure of how to proceed, although this family really feels like it should work out. What is going on here?

  • $\begingroup$ This isn't an answer to the question you're asking, but I have a suggestion in the meantime. You're jumping from the base being a point to the base being the affine or projective line - that's a big jump! Why not start in between by making your base a DVR? Then a family of elliptic curves will basically be an elliptic curve over the fraction field of your DVR which has a smooth model over the DVR, so that you compatibly get a smooth elliptic curve over the residue field via reduction. $\endgroup$ – Stahl Jul 26 '17 at 19:27
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    $\begingroup$ I found your suggestion quite helpful to think through, Stahl. I'll have to keep such examples in mind in the future. Also, I have rectified my original problem: if one instead looks at $y^2z=x^3+tz^3$ as a subvariety of $\mathbb P^2\times\mathbb A^1$, where $x,y,z$ are projective coordinates and $t$ the affine coordinate, then everything works out as expected. $\endgroup$ – Brandon Jul 26 '17 at 20:54
  • $\begingroup$ The terminology of your question is a bit confusing: usually a "proper family" would mean that the base $S$ is proper. $\endgroup$ – Nefertiti Jul 27 '17 at 9:10

One of the standard ways of doing this is to start with two general smooth cubics in $\mathbb{P}^2$, given by say $F=0, G=0$. Then we can take the pencil $\lambda F+\mu G=0$, a family of cubics. This gives a rational map $\mathbb{P}^2\to \mathbb{P}^1$ which becomes a morphism after blowing up the nine points of intersections of $F=0=G$ to get a smooth surface $X$. Thus we get a projective morphism $X\to\mathbb{P}^1$ and the general fiber is an elliptic curve.

  • $\begingroup$ Details and properties of this construction are spelled out in this paper: arxiv.org/pdf/0907.0298 , which I remember being quite nice. $\endgroup$ – user347489 Jul 26 '17 at 22:06

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