  We consider the capillary-gravity water wave equation in two dimensions. We
assume that the fluid is inviscid, incompressible, irrotational and the air
density is zero. We construct an energy functional and prove a local
wellposedness result without assuming the Taylor sign condition. When the
surface tension $\sigma$ is zero, the energy reduces to a lower order version
of the energy obtained by Kinsey and Wu [23] and allows angled crest
interfaces. For positive surface tension, the energy does not allow angled
crest interfaces but admits initial data with large curvature of the order of
$\sigma^{-\frac{1}{3} + \epsilon} $ for any $\epsilon >0$.
