  We consider $u(x,t)$, a solution of $\partial_tu = \Delta u + |u|^{p-1}u$
which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to
\mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be
the blow-up set of $u$, that is the set of all blow-up points. Under suitable
nondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensional
continuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a
$\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic
behavior of $u$ near blow-up. In order to obtain such a refined behavior, 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 way we escape logarithmic scales of the variable $(T-t)$ and
reach significant small terms in the polynomial order $(T-t)^\mu$ for some $\mu
> 0$. The refined asymptotic behavior yields geometric constraints of the
blow-up set, leading to more regularity on $S$.
