  Strong asymptotics of polynomials orthogonal on the unit circle with respect
to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad
|z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$
for $|z|=1$ and can be extended as a holomorphic and non-vanishing function to
an annulus containing the unit circle. The formulas obtained are valid
uniformly in the whole complex plane. As a consequence, we obtain some results
about the distribution of zeros of these polynomials, the behavior of their
leading and Verblunsky coefficients, as well as give an alternative proof of
the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants
for such type of weights. The main technique is the steepest descent analysis
of Deift and Zhou, based on the matrix Riemann-Hilbert characterization
proposed by Fokas, Its and Kitaev.
