  The goal of this work is to consider some geometric properties of
geometrically irreducible subschemes on degenerations of separably rationally
connected varieties. It is known that these geometrically irreducible
subschemes exist when the ground field has characteristic zero or contains an
algebraically closed subfield. In this note, we show that the dimension of this
geometrically irreducible subscheme has a lower bound by the Fano index of the
generic fiber.
