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May 4, 2022 at 7:45 comment added Evgeny Shinder In the case when $X$ is singular, we do not expect to move divisors off singular points, e.g. in a quadric cone $xy = z^2$ lines making up the cone all pass through the singular point $0$ and there is no way to move them off it -- in the sense of linear equivalence, and probably in the sense you are asking as well.
May 4, 2022 at 3:24 history edited user127776 CC BY-SA 4.0
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May 3, 2022 at 13:18 comment added abx No in general: if you blow up a point in a (smooth) surface, the exceptional curve cannot move.
May 3, 2022 at 13:04 history edited user127776 CC BY-SA 4.0
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May 3, 2022 at 2:52 history asked user127776 CC BY-SA 4.0