generalizing the theory we describe in this thesis will lead to a “nice” (e.g. smooth, toroidal...) compactification $M^n_{A,I}\subset B^n_{A,I}$ dominating $\overline{M}^n_{A,I}$, in such a way that $B^n_{A,I}$ itself has a meaningful geometric modular interpretation.

The case $A={{\mathbb{F}}}_q[t]$ and $I=(t)$
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In the special case of Drinfeld ${{\mathbb{F}}}_q[t]$-modules of rank $n$ and level $(t)$, the fine moduli space $M^n_{{{\mathbb{F}}}_q[t],(t)}$ is obtained via base change to $\Spec{{\mathbb{F}}}_q(t)$ from: $$\Omega^n:={{\mathbb{P}}}^{n-1}_{{{\mathbb{F}}}_q}\setminus\bigcup_{H}H,$$ where $H$ runs over the set of ${{\mathbb{F}}}_q$-hyperplanes in ${{\mathbb{P}}}^{n-1}_{{{\mathbb{F}}}_q}$. The scheme $\Omega^n$, often referred to as the *Drinfeld period domain* over ${{\mathbb{F}}}_q$, has been studied by Rapoport [@Rap], Orlik-Rapoport [@Orl] and Pink-Schieder [@P-Sch]. As in [@P-Sch], it will be useful to adopt a coordinate free formulation in order to describe the compactifications of $\Omega^n$ that we are interested in. Let $V$ be a finite dimensional vector space over ${{\mathbb{F}}}_q$ of dimension $n$. Let $P_V:=\Proj\big(\Sym V\big)$, and define $$\Omega_V:=P_V\setminus\bigcup_{H}H,$$ where $H$ runs over the set of ${{\mathbb{F}}}_q$-hyperplanes in $P_V$. Any basis of $V$ induces an isomorphism $P_V\stackrel\sim\to{{\mathbb{P}}}^{n-1}_{{{\mathbb{F}}}_q}$ sending $\OV$ to $\Omega^n$. In [@P-Sch], a smooth compactification $B_V$ of $\OV$ is constructed that differs in general from the tautological compactification $P_V$. The scheme $B_V$ is projective and defined as a closed subscheme of $\prod_{0\neq 