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Let $X$ be a smooth projective variety (over the complex numbers) of dimension at least $2$, $B$ a finite set of closed points. Consider the closed subscheme $E:=B \times X + \Delta \subset X \times X$, where $\Delta$ is the diagonal of $X \times X$. Denote by $\tilde{X}$ the blow-up of $X \times X$ along $E$. Consider the induced projection map $p_2:\tilde{X} \to X \times X \xrightarrow{\mbox{pr}_2} X$, where $\mbox{pr}_2:X \times X \to X$ is the projection onto the second coordinate. Denote by $X':=p_2^{-1}(X \backslash B)$. Then,

1) Is the restriction of $p_2$, $p_2|_{X'}: X' \to X\backslash B$ a flat morphism?

2) If 1) is true then is the fiber over any $t \in X\backslash B$, denoted $\tilde{X}_t$ the blow up of $X$ along $B \cup t$?

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  • $\begingroup$ Doesn't this reduce to the formal-local case, which since $X$ is smooth over $\mathbb C$, means you only need understand the case of $X$ a vector space? $\endgroup$
    – Allen Knutson
    Commented Aug 14, 2016 at 2:37
  • $\begingroup$ Yes to both questions, because both $E$ and $X \times X$ are flat over $X \setminus B$. $\endgroup$
    – Sasha
    Commented Aug 17, 2016 at 22:06

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