Timeline for induced morphism through normalization
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| Mar 21, 2020 at 0:43 | comment | added | Devlin Mallory | As to why your argument using the universal property of normalization doesn't go through, see the answer at mathoverflow.net/questions/46/…: the issue is that the universal property only allows you to lift maps $Z\to X$ up to $Y$ if they send associated points of $Z$ to those of $X$; if $\dim X\geq 2$, this won't be the case for maps $\mathbb P^1\to X$. | |
| Mar 20, 2020 at 22:34 | comment | added | Evgeny Shinder | I think the map is not surjective in general. That is, if the image of $\mathbf{P}^1$ lands in the singular locus of $Y$, there is no reason why the map can be lifted to $X$. One can find an example for surfaces, and I think one such example is a surface $x^2 u = y^2 v$ in coordinates $[x,y,z] \in \mathbf{P}^2$, $[u,v] \in \mathbf{P}^1$. Its singular locus is isomorphic to a projective line, and taking normalization yields a connected $2:1$ cover on the exceptional locus, hence there is no lifting for the identity map on the exceptional locus. | |
| Mar 20, 2020 at 19:03 | history | asked | alberth | CC BY-SA 4.0 |