A special subvariety is a subvariety belonging to a distinguished arithmetic class in an unlikely-intersection problem. The class depends on the ambient variety. In an abelian variety, the special subvarieties are precisely torsion cosets of abelian subvarieties. In a Siegel moduli variety, the term refers to Shimura-special subvarieties, including their Hecke translates, characterized by additional rational Hodge-tensor conditions on the represented abelian varieties. These two classes are different, even though both occur in formulations of the Zilber-Pink conjecture.
The special closure of an irreducible subvariety is the smallest special subvariety containing it. In the abelian setting, if 
is the special closure of 
and 
is special, an irreducible component 
of 
is atypical when
 |
The Zilber-Pink conjecture predicts finiteness of maximal such components in its abelian formulation. For a Hodge-generic curve
in 
,
meeting a special curve gives a negative expected dimension,
so even an isolated intersection point is atypical.
Curves in 
." Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22,
1705-1745, 2021.