Consider the moduli stack M_1 parametrizing smooth (locally) projective genus one curves C. If C is a genus one curve over a field k, then there exists a minimal integer d > 0 such that C has an ample invertible sheaf of degree d. It turnsout there is no bound for the integer d, and it follows that M_1 does not have a presentation by a finite type scheme over Z.
Consider the moduli stack M_1(d) parametrizing pairs (C, L) where C is a smooth projective genus 1 curve, and L is an ample invertible sheaf of degree d. This does have a presentation by a finite type scheme over Z. For example when d = 3 we see that the space U = P(\Gamma(P^2, O(3))) – \Delta maps smoothly and surjectively onto M_1(d). Moreover, we have M_1(3) = [U/GL_3] (edit Oct 14, 2011: changed PGL_3 into GL_3).
Now, what’s interesting is that U is an open subscheme of a projective space. You can do the “same thing” for M_1(5) by writing every degree 5 genus one curve in P^4 as the zeros of the 2×2 pfaffians of a skew symmetric 5×5 matrix of linear forms on P^4. You can also do this for M_1(4) by writing a degree 4 genus 1 curve in P^3 as the intersection of 2 quadrics. You can also do something similar for M_1(2).
These stacks came up in a conversation with Manjul Bhargava in my office last week, and so did the following question: Can the same be done for any d > 5?
On the level of algebraic stacks, a more general question would be: Can we find obstructions to being able to write an algebraic stack M as a quotient stack [U/G] where U is an open subspace of P(V) where V is a linear representation of G? Cohomological? Intersection theory? I am hoping there may be some things once can say that avoid appealing to a classification of representations of G’s with small dimensional orbit spaces.