# The central fiber of this family of surfaces?

I have a question on a description of a central fiber of the following family of surfaces.

Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the double cover $$T\rightarrow \mathbb{P}^2\times \mathbb{C}$$ along $(tB-C^2)\subset \mathbb{P}^2\times \mathbb{C}$, where $t$ is the parameter of $\mathbb{C}$. We think of $T$ as a one-parameter family $T\rightarrow \mathbb{C}$ of surfaces $T_t$, which doubly covers $\mathbb{P}^2$ branching along a curve of degree $2d$.

The total space $T$ has double point singularities above $2n^2$ points of $(B\cap C,0) \in \mathbb{P}^2\times \mathbb{C}$. Taking the small resolution, we obtain a nonsingular space $$\tilde{T}\rightarrow \mathbb{C},$$ which provides a degeneration of $T_t\cong \tilde{T}_t$ to $\tilde{T}_0$.

A paper I am reading claims that $\tilde{T}_0 \cong \mathbb{P}^2 \cup_C\tilde{\mathbb{P}}^2$, where $\tilde{\mathbb{P}}^2$ is the blow-up of $\mathbb{P}^2$ along $B\cap C$. How can I see this description?

$tB-C$ is an inhomogeneous polynomial. How do you deal with that? – Will Sawin Jun 1 '13 at 12:54
It should be $tB-C^2$. Thank you for pointing out the typo. – user2013 Jun 1 '13 at 13:10