  Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson
manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is
important to be able to quantize it, in order to obtain more precise results
than through classical mechanics. An available method is the deformation
quantization, which consists in constructing a star-product on the algebra of
formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of
star-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$.
The aim of this article is to determine this Hochschild cohomology in the case
of singular curves of the plane -- so we rediscover, by a different way, a
result proved by Fronsdal and make it more precise -- and in the case of Klein
surfaces. The use of a complex suggested by Kontsevich and the help of
Gr\"obner bases allow us to solve the problem.
