Timeline for Divisorial contraction to a non-normal variety
Current License: CC BY-SA 4.0
10 events
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| Dec 4, 2020 at 4:47 | comment | added | Sándor Kovács | So, in essence your question was if there are any non-normal singularities whose normalization morphism is one-to-one. As soon as you have such an object, then take the normalization and blow up something in the locus where the normalization map is not an isomorphism. A potentially more interesting question is to do this with a small morphism. That can be done, too, but you have to pick your example more carefully. | |
| Dec 3, 2020 at 20:02 | comment | added | user114666 | Thank you very much. | |
| Dec 3, 2020 at 16:34 | comment | added | Pop | OK, glad it is useful. The surface singularity I refer to is described in affine form in Georges Elencwajg's answer to this question: math.stackexchange.com/questions/143954/… | |
| Dec 3, 2020 at 16:21 | comment | added | user114666 | Yes, an example like this is exactly what I am looking for. With divisorial contraction I do not mean Mori divisorial contraction. Just a morphism mapping a divisor to something of smaller dimension. | |
| Dec 3, 2020 at 16:02 | comment | added | Pop | There is a projective surface $Y$ with an isolated non-normal singularity $p$ such that the normalisation $W \rightarrow Y$ is bijective and $W$ is nonsingular. Let $q$ be the unique point of $W$ over $p$, and let $X \rightarrow W$ be the blowup of $q$. Is this the kind of thing you are looking for? | |
| Dec 3, 2020 at 15:33 | comment | added | Jason Starr | "Yes, it boils down to construct a birational morphism ..." I still do not quite understand your definition of "contraction". In the definition in textbooks on the Minimal Model Program, usually there is a hypothesis that the natural morphism from the structure sheaf of the target to the pushforward of the structure sheaf of the domain is an isomorphism of sheaves. Under that hypothesis, using the factorization from my comment, automatically $Y$ is normal. If you are allowing a more general notion of "contraction", please clarify your definition. | |
| Dec 3, 2020 at 14:23 | comment | added | user114666 | Yes, it boils down to construct a birational morphism $g:X\rightarrow W$ contracting $D$ onto $S$, where $\nu:W\rightarrow Y$ is the normalization of $Y$, $\nu$ is $1$-to-$1$ and an isomorphism between $W\setminus S$ and $Y\setminus \nu(S)$. | |
| Dec 3, 2020 at 12:13 | history | edited | YCor | CC BY-SA 4.0 |
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| Dec 3, 2020 at 11:19 | comment | added | Jason Starr | What is your definition of "divisorial contraction"? If $X$ is normal, then the morphism $f$ always factors through the normalization of the image of $f$. How is the induced morphism from $X$ to this normalization a "different contraction" from $f$? | |
| Dec 3, 2020 at 11:14 | history | asked | user114666 | CC BY-SA 4.0 |