It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles.
This is not true in families, though, not even locally on the base of the family, as it is possible to construct flat $1$-parameter families of VB that have generically the same splitting type and "jump" to a different splitting type at the special fiber. This prevents the moduli stack $\mathfrak{M}$ of all VB on $\mathbb{P}^1$ (or the corresponding functor $F$ of isomorphism classes) to be separated (in the sense of valuative criteria). Here we took the notion of isomorphism of families to mean, as usual: the families $\mathscr{E}$ and $\mathscr{E}'$ of VB on $X$ parametrized by $T$ are isomorphic if there is a line bundle $L$ on $T$ such that $\mathscr{E}\simeq\mathscr{E}'\otimes L$.
For a smooth projective curve $X$, the moduli functor $F^s$ of stable bundles is representable by a smooth separated scheme $M^s$, and the stack $\mathfrak{M}^s$ of stable bundles is a gerbe over the latter. The moduli functor $F^{ss}$ of isomorphism classes of semistable bundles has a non separated coarse moduli space with proper connected components, and collapsing $S$-equivalence classes in the sense of Seshadri fixes this: now the functor $F'^{ss}$ of such $S$-equivalence classes of semistable bundles has a coarse moduli space (each connected component of) which is a projective variety (in particular separated).
By Grothendieck's theorem, the relation of S-equivalence on $\mathbb{P}^1$ coincides with the relation of isomorphism, so we can avoid talking of $S$-equivalence in this case. Also, there are no stable VB on $\mathbb{P}^1$ (of rank $>1$).
In what follows the curve $X$ is $\mathbb{P}^1$.
Since every VB is uniquely determined by its splitting type, and there is a countable set of choices for the possible line bundles occurring as summands, each connected component of $M^{ss}$, which is a variety, has to be a reduced point $\mathrm{Spec}(\mathbb{C})$.
Recall we let $\mathfrak{M}$ be the stack in groupoids of all VB on $\mathbb{P}^1$, with isomorphisms of families as $1$-arrows, and $F$ the corresponding functor of isomorphism classes.
Q.1: How does $\mathfrak{M}$ look like?
Q.2: Does $F$ have a coarse moduli space $M$? If yes, how does it look like?
It seems possible to me that an answer to question 1 or 2 has something to do with the partial order structure on the set $\boldsymbol{\mathrm{HNP}}$ of "Harder-Narasimhan polygons" (see Shatz, The decomposition and specialization of algebraic families of vector bundles). Maybe the specialization order on (field-valued) points of $M$ (or of $F$, or of $\mathfrak{M}$) reflects somehow the structure of the lattice $\boldsymbol{\mathrm{HNP}}$?
Best Answer
Well, it is not clear to me what exactly you mean by Q.1 but here is some kind of an answer. Let's even work with $G$-bundles for any reductive group $G$ (you can take $G=GL(n)$ if you want, but that's not necessary). Consider the quotient $G((t))/G[t^{-1}]$. It has a natural scheme structure of infinite type. This scheme is isomorphic to the scheme of $G$-bundles on $\mathbb P^1$ endowed with a trivialization at the formal neighborhood of $\infty$. It is also called the thick affine Grassmannian of $G$, so we'll denote it by $Gr_G$. Now the stack of $G$-bundles on $\mathbb P^1$ is obviously the quotient $G[[t]]\backslash Gr_G$. The isomorphism classes of $G$-bundles correspond to $G[[t]]$-orbits on $Gr_G$ which are in one-to-one correspondence with dominant coweights of $G$ (this is the analog of Grothendieck's theorem for any $G$). So, for any such coweight $\lambda$ there is a stratum $\mathcal M^{\lambda}$ in $\mathcal M$ and we have $\mathcal M^{\mu}\subset \overline{\mathcal M^{\lambda}}$ iff $\lambda\leq \mu$ with respect to the standard partial order on coweights ($\lambda\leq \mu$ means that $\mu-\lambda$ is a sum of positive coroots). The geometry of $\overline{\mathcal M^{\lambda}}$ is pretty complicated (for example, it is quite singular). In fact, the closures of $G[[t]]$-orbits on $Gr_G$ are special examples of affine Schubert varieties.