Let $X$ be a quasi-projective variety over $\mathbb{C}$. Let $I$ be an ideal sheaf supported at a closed point on $X$. Is the exceptional divisor for the blow-up of $X$ along the ideal sheaf $I$, connected? Any reference will be most welcome.
2026-03-26 12:33:43.1774528423
Connectedness of exceptional divisors
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No. Consider the blowup of a nodal curve at the node. This is the same as the normalization, so the exceptional divisor is just the two isolated points lying over the node.