  We refine the asymptotic behavior of solutions to the semilinear heat
equation with Sobolev subcritical power nonlinearity which blow up in some
finite time at a blow-up point where the (supposed to be generic) profile
holds. In order to obtain this refinement, we have to abandon the explicit
profile function as a first order approximation, and take a non explicit
function as a first order description of the singular behavior. This non
explicit function is in fact a special solution which we construct, obeying
some refined prescribed behavior. The construction relies on the reduction of
the problem to a finite dimensional one and the use of a topological argument
based on index theory to conclude. Surprisingly, the new non explicit profiles
which we construct make a family with finite degrees of freedom, namely
$\frac{(N+1)N}{2}$ if $N$ is the dimension of the space.
