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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?

Thank you for your assistance.

share|improve this question
    
$tB-C$ 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

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