  Let f be a polinomial with coefficients in a finite field F. Let $\Psi : F
\to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds
for the exponential sums $\sum_{x\in F^n} \Psi (Tr_{F/F_p} (f(x)))$ in some
cases where the highest degree form of f defines a singular projective
hypersurface X (e.g. when X is an arrangement of lines in P^2). The bound
involves the Milnor numbers of the singularities of X. The proof goes via the
classical cohomological interpretation of this exponential sums through
Grothendieck's trace formula.
