We consider $u(x,t)$ , a solution of $u_t =\Delta u +|u|^{p-1}u$ which blows up at some time $T\gt 0$ , where $u:{\mathbb R}^N\times [0,T)\to {\mathbb R}$ , $p\gt 1$ , and $(N-2)p \lt N+2$ . Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set $S$ is continuous and $(N-1)$ -dimensional implies that it is $C^2$ . In particular, we compute the $N-1$ principal curvatures and directions of $S$ . Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for $S$ and refined singular behavior of $u$ near $S$ are linked through a new mechanism of algebraic cancellations that we explain in detail