The widely accepted by experts, but still conjectural,
"geometric explanation" of curious vanishing properties of the
moduli space
M_{g,k} of smooth genus g algebraic curves with punctures
is the existence of its stratification by a certain number of
affine strata or the
existence of a cover of M_{g,k} by a certain number of
open affine sets.

Recently, the author jointly with S. Grushevsky proposed an
alternative approach for geometrical explanation of the
vanishing properties of M_{g,k} motivated by certain
constructions of the Whitham perturbation theory of integrable
systems. These constructions has already found their
applications in topological quantum field theories (WDVV
equations) and N=2 supersymmetric gauge theories. In the talk
I'll present the key ideas of this approach and its latest
applications: the proof of Arbarello's conjecture and a new
upper bound for the dimension of complete complex subvarieties
in the moduli space of stable curve of compact type.