Let $X \hookrightarrow \mathbb{P}^N$ be a compact sub manifold of dimension $n$. Let $\mathbb{P}(d, N)$ denote the projectivization of degree $d$ homogeneous polynomial on $\mathbb{P}^N$. Each projective subspace $U \subset \mathbb{P}(d, N)$ defines a family $\{X_P\}_{[P] \in \mathbb{P}(d, N)}$ of hypersurfaces in $X$. Let $\check{U}$ be the projectivized dual space of $U$.
We define the modification of $X$ determined by the linear system $\{X_P\}_{P \in U}$ to be the variety
$$\hat{X} = \{(x,H) \in X \times \check{U} : P(x) =0, \;\forall P \in U \;s.t.\; H(P)=0\} $$
The base locus is defined to be $B=B_U = \cap_{P \in U} X_P$.
Question: Consider the projection of $\pi_X: \hat{X} \to X$, the fiber over $X\backslash B$ consists of exactly one-point. It is similar in spirit of the construction of blow-up of a point in a plane. So what is the relation between $\hat{X}$ and $\text{Bl}_B(X)$?
PS: I am reading about complex Morse theory from Liviu Nicolaescu's online note. http://www3.nd.edu/~lnicolae/Morse2nd.pdf The definitions are taken from page 202.