How is this the moduli space of quartic forms?

Question

In this paper, near the top of p. 9, Poonen says:

Let $V = \rm{Spec} \ \mathbb{Q}[a, b, c, d, e]$ be the moduli space of quartic forms, ...

What does this mean? I am not an expert in moduli spaces; is he claiming, for example, that the prime ideals $(a - 2)$ and $(a^7 - 2 b^7)$ correspond to quartic forms?

Answer 1

To say that $V$ is a moduli space of quartic forms is expressed in terms of the functor of points. That is, for a scheme $X$ an "$X$-valued point" of $V$ is by definition a morphism of schemes $X \rightarrow V$. To say that $V$ is a moduli space of quartic forms means that to give a morphism of schemes $X \rightarrow V$ is the same as to give a quartic form "on $X$".

Take for example $X = \mathrm{Spec}(\mathbb{C})$, then this is given by a ring homomorphism $\mathbb{Q}[a,b,c,d,e] \rightarrow \mathbb{C}$ which is just to pick $a,b,c,d,e$ as complex numbers. This gives a quadratic form $aX^4 + bX^3Y + cX^2Y^2 + dXY^3 + eY^4$.

Now suppose $X$ is a curve and we are given a morphism $X \rightarrow V$. Then for each closed point $x = \mathrm{Spec}(k)$ on $X$, one gets a map $\mathrm{Spec}(k) \rightarrow V$ which again is given by a ring homomorphism $\mathbb{Q}[a,b,c,d,e] \rightarrow k$. Once again, this is simply choosing the values of a,b,c,d,e this time in the field $k$. One can view this as a (continuously varying) family of quartic forms, one for each closed point of the curve $X$. This viewpoint is missing part of the picture (specifically what happens for other types of points), but is a good start.

Answer 2

A moduli space is just a space of some sort parametrizing objects of some specified sort (perhaps under an equivalence relation, say, like isomorphism).

In this case, we notice that, that a quartic form is determined precisely by $5$ numbers. Thus any point in $\mathbb Q^5 \backslash \{ 0\}=Spec \mathbb Q[a,b,c,d,e]_{(a,b,c,d,e)}$. determines a quartic form.

Thus maximal ideals in the ring $R= \mathbb Q[a,b,c,d,e]$ correspond to quartic forms. The prime ideal $(a-2)$ then correspond the closed subvariety consisting of the quartic forms where $a=2$, and the other variables vary freely. Similarly, the prime ideal $(a^7-2b^7)$ correspond to all quartic forms where the first two coeffcients satisfy $a^7=2b^7$.