Timeline for Do subvarieties naturally map to the hilbert scheme of points?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2018 at 22:12 | comment | added | jacob | Ok, so youre suggesting that the map doesn't extend to $X^{[n]}$. But is there some slightly different space it DOES extend to? | |
| Jun 3, 2018 at 5:57 | comment | added | Will Sawin | So the only such ideals are the ideals generated by homogeneous polynomials of degree $d$, which have codimension ${m+d-1 \choose m} - {m + d-3 \choose m}$. So at minimum you have to solve some weird Diophantine equation to find an example. | |
| Jun 3, 2018 at 5:55 | comment | added | Will Sawin | If you let $V$ be the vanishing locus of a nondegenerate homogeneous quadratic form $f$ in $m$ variables, it has an isolated singularity at the origin. If for some $k$ we can extend the map to that point then it must define an $O_m$-invariant ideal of codimension ${m + k -2 \choose m-1}$ inside the quotient $\mathbb C[x_1,\dots,x_m]/f$. As a representation of $O_m$ this splits into the spherical harmonic representations which are irreducible and nonisomorphic. | |
| Jun 3, 2018 at 5:10 | comment | added | jacob | Yeah, I think one has to take a large enough $k$ dependent on $V$. Namely, I think $k$ has to at least be big enough such that at every point of $v$, the defining ideal for $V$ does not lie entirely in the $k$'th infinitesimal neighborhood of $v$. | |
| Jun 3, 2018 at 4:53 | comment | added | Will Sawin | For $X$ a smooth variety, the moduli space of points of $X$ plus a $d$-dimensional subspace of their tangent space is proper - a Grassmanian bundle on $X$. So any such moduli space of local germs in the case $k=1$ must either (1) not include this space as a subvariety, (2) not map the smooth points of $V$ into this subvariety, (3) map all the points of $V$ into this subvariety, or (4) not be separated. One can rule out (3) by an explicit example, and then I have a hard time seeing how any of the others could be viable. | |
| Jun 3, 2018 at 3:39 | history | asked | jacob | CC BY-SA 4.0 |