  We calculate the Euler characteristics of all of the Teichmuller curves in
the moduli space of genus two Riemann surfaces which are generated by
holomorphic one-forms with a single double zero. These curves can all be
embedded in Hilbert modular surfaces and our main result is that the Euler
characteristic of a Teichmuller curve is proportional to the Euler
characteristic of the Hilbert modular surface on which it lies.
  The idea is to use techniques from algebraic geometry to calculate the
fundamental classes of these Teichmuller curves in certain compactifications of
the Hilbert modular surfaces. This is done by defining meromorphic sections of
line bundles over Hilbert modular surfaces which vanish along these Teichmuller
curves.
  We apply these results to calculate the Siegel-Veech constants for counting
closed billiards paths in certain L-shaped polygons. We also calculate the
Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic,
SL_2(R)-invariant measure on the moduli space of Abelian differentials in genus
two (previously calculated in unpublished work of Kontsevich and Zorich).
