Assume $X$ is a complete scheme and $B$ a closed subscheme. Is the blowup $Bl_B X$ always complete?
If $B$ is smooth, this would be clear to me. But how should I think of the case when $B$ is singular?
Another question: if $X$ is projective, is the blowup $Bl_B X$ also projective?
Thanks!
The blowup is the projective spectrum of the graded algebra $$ \bigoplus_{n \ge 0} I_B^n, $$ where $I_B$ is the ideal of $B$. Therefore, it is projective over $X$. Consequently, it is complete/projective, if $X$ is.