  For g at least 2, the Thurston spine Pg is the subspace of Teichmueller space
Tg , consisting of the marked surfaces for which the set of shortest curves,
the systoles, cuts the surface into polygons. Our main result is the existence
of an infinite set A of integers such that codim Pg is a o(g/ log g), when g
varies over A.
  This proves the recent conjecture of M. Fortier Bourque.
