In their famous book, Faltings and Chai constructed, among other things, minimal and toroidal compactification of the Siegel moduli space. The reason for the use of the word toroidal is clear, since toroidal geometry is used to define such compactification, it is not clear to me the use of the word minimal. So the question is:


in which sense the minimal compactification is minimal?



Minimal compactification
Given the symmetric space $D$ of non-compact type for the real semi-simple Lie group $G$, there are $2^r-1$ Satake compactifications $\bar{D}$ for $D$ up to homeomorphism, where $r$ is the real rank of $G$. 


These compactifications correspond to the non-empty subsets of a set $S$ of simple roots, and as such they form a semi-lattice: if $S\_1 \subset S\_2$, then the identity of $D$ extends to a continuous mapping $\bar{D}\_{S\_2} \to \bar{D}\_{S\_1}$. 


The *Satake-Baily–Borel compactification* is one of the minimal (in the semi-lattice sense) Satake compactifications in the case where $D$ is Hermitian, i.e., has a $G$-invariant complex structure, see [this entry](http://www.encyclopediaofmath.org/index.php/Satake_compactification) in the Encyclopedia of Mathematics. 


According to [Faltings-Chai, *Degeneration of Abelian Varieties*, p. 136], the construction of the minimal compactification of $\mathcal{A}\_g$ mimics the construction of the the Satake-Baily-Borel compactification of a symmetric space, and this is why the word "minimal" is used. 


