Timeline for Connectedness of moduli space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 21, 2019 at 17:14 | vote | accept | Xuqiang QIN | ||
| Feb 21, 2019 at 15:42 | answer | added | Jason Starr | timeline score: 9 | |
| Feb 21, 2019 at 0:23 | history | edited | Xuqiang QIN | CC BY-SA 4.0 |
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| Feb 21, 2019 at 0:21 | comment | added | Xuqiang QIN | @JasonStarr That's a very interesting example, thanks! On the other hand, would the situation be different if $d\geq 1$ (that was case I was facing, should have put that in the question, sorry!)? Projectivity is of no help in dimension $0$, but can it change anything when $d\geq 1$? | |
| Feb 20, 2019 at 23:09 | comment | added | Jason Starr | In particular, if $X$ is a general quintic hypersurface in $\mathbb{P}^4$, then there are $609250$ plane conics $R$ contained in $X$ (first proved by Sheldon Katz). For each conic $R$, the linear span of $R$ is a $2$-plane $\Pi$ that contains $R$. The intersection $\Pi\cap X$ is a plane quintic curve that contains $R$ as an irreducible component. Thus, the residual to $R$ in $\Pi\cap X$ is a plane cubic $C$. So it appears that the corresponding moduli space $M$ has (at least) $609250$ isolated points. | |
| Feb 20, 2019 at 23:06 | comment | added | Jason Starr | I believe that can happen. For a smooth projective variety $X$ of dimension $3$, for every genus $1$ curve $C$ in $X$, the Hartshorne-Serre correspondence produces a locally free sheaf $\mathcal{E}$ of rank $2$ whose determinant equals $\omega^\vee_X$ together with a global section whose zero scheme equals $C$. If $X$ is a (simply connected) Calabi-Yau, the deformations of $\mathcal{E}$ are precisely the deformations of $C$. Thus, if the set of such elliptic curves for a specified curve class is finite, it appears that $M$ has several connected components. | |
| Feb 20, 2019 at 21:23 | history | asked | Xuqiang QIN | CC BY-SA 4.0 |