Construction
We begin with a closed embedding \(X\hookrightarrow Y\) and end with a flat morphism \(\pi:M\to \mathbb P^1\) which fits into a diagram
where (1) \(M_t \cong Y\) for \(t\neq \infty\) and (2) \(M_\infty\) is the sum of two disjoint effective Cartier divisors \(\mathbb P(C\oplus 1) + \Bl_X Y\), where \(C = C_XY\) is the normal cone of \(X\) in \(Y\). This is the construction of \(M\) and an outline of the proof that it possesses these two properties.
- Define \(M = \Bl_{X\times \{\infty\}}(Y\times \mathbb P^1)\).
- Observe that the normal cone to \(X\times \{\infty\}\) in \(Y\times \mathbb P^1\) is \(C\oplus 1\), so the exceptional divisor of this blowup is \(\mathbb P(C\oplus 1)\).
- Observe that we have a sequence of embeddings
\[X = X\times\{\infty\}\hookrightarrow X\times \mathbb P^1 \hookrightarrow Y\times \mathbb P^1,\] so by the lemma on chains of closed embeddings, the blowup \(\Bl_{X\times \{\infty\}}(X\times \mathbb P^1)\) embeds into \(M\). Since \(X\times\{\infty\}\) is a Cartier divisor in \(X\times \mathbb P^1\), \(\Bl_{X\times \{\infty\}}(X\times \mathbb P^1)\cong X\times \mathbb P^1\) and we therefore have an embedding \[X\times \mathbb P^1 \hookrightarrow M.\]
- Similarly, \(X = X\times \{\infty\}\hookrightarrow Y = Y\times \{\infty\} \hookrightarrow Y\times \mathbb P^1\) is a sequence of closed embeddings, yielding an embedding \(\Bl_XY\hookrightarrow M.\)
- For any scheme \(S\) flat over \(T\) and any closed subscheme \(Z\subset S\), the composition \(\Bl_ZS\to T\) of the blowdown map to S with the map to \(T\) is also flat. Hence \(M \to \mathbb P^1\) is flat since the projection map \(Y\times \mathbb P^1 \to \mathbb P^1\) is flat.
- The fiber \(M_t\) over any \(t\neq \infty\) is isomorphic to \(Y\times \{t\}\)
- Since \(\mathbb P(C\oplus 1)\) and \(\Bl_XY\) both globally embed in \(M\), we can check the structure of \(M_\infty\) locally to verify condition (2). That proof is below.
Proof
Assume \(Y = \Spec A\) and \(I\subset A\) is the ideal defining \(X\). We study \(M \) near \(\{\infty\}\) by choosing the chart \(\mathbb A^1 = \mathbb P^1 \setminus \{0\}\) for \(\mathbb P^1\) so that \[Y\times \mathbb A^1 = \Spec (A\otimes k[T]) = \Spec A[T].\] The ideal defining \(X\times \{0\}\) in \(Y\times \mathbb A^1\) is then \((I, T)\), so the blowup of \(Y\times \mathbb A^1\) along \(X\times \{0\}\) is \(\Proj(S^\bullet)\) with \[S^n = (I,T)^n.\] The scheme \(\Proj(S^\bullet)\) is covered by affines opens \(\Spec S^\bullet_{(a)}\) where \[S^\bullet_{(a)} = \{s/a^n ~ \mid ~ s\in S^n\}\] and \(a\) runs through a set of generators for the ideal \((I, T)\) in \(A[T]\). For \(a\in I\), the exceptional divisor \(\Proj(C\oplus 1)\) is defined in \(\Spec S^\bullet_{(a)}\) by the equation \(a/1\), thinking of \(a\) as a degree \(0\) element in of \(A \subset S^0 = A[T]\). The divisor \(\Bl_XY\) is defined by \(T/a\) in \(\Spec S^\bullet_{(a)}\). The fiber \(M_\infty\) of \(M\to \mathbb P^1\) is defined by \((T)\) in \(\Proj S^\bullet\), and since \(T = \frac{a}{1} \frac{T}{a}\), \[M_\infty = \mathbb P(C\oplus 1) + \Bl_XY.\]