Timeline for Kodaira dimension of spaces of rational curves in hypersurfaces
Current License: CC BY-SA 4.0
6 events
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| May 26, 2023 at 14:03 | comment | added | Jason Starr | I double-checked, and it is a surface of general type. For the "Hilbert scheme" model, the parameter space of plane conics contained in a general quartic threefold is smooth (this is the main issue), and the canonical bundle is straightforward to compute by adjunction (the parameter space is the zero locus of a regular section of a rank-9 vector bundle on a $\mathbb{P}^5$-bundle over the Grassmannian of linear $2$-planes in $\mathbb{P}^4$). | |
| May 26, 2023 at 13:34 | comment | added | Jason Starr | Yes, I believe the Kontsevich space is a surface of general type in that case. This case was studied by Gert Welters in his book "Abel-Jacobi Isogenies for Certain Types of Fano Threefolds", if memory serves (it is quite analogous to the general type surface parameterizing lines in a cubic threefold, studied by Clemens -- Griffiths). | |
| May 26, 2023 at 12:14 | comment | added | Puzzled | Thanks a lot. When $n=d=4$ and $a = 2$ is the Kontsevich space a surface of general type? | |
| May 26, 2023 at 10:42 | comment | added | Jason Starr | If you want negative Kodaira dimension / uniruledness just for small values of $a$, more is known. For $a=1$, I believe the result goes back to Altman-Kleiman: $(d+1)d/2 \leq n$. For higher values of $a$, I have some results in scattered papers, such as arxiv.org/abs/math/0305432. Based on the formula for the virtual canonical bundle on the Kontsevich space due to de Jong and myself, we have quite good predictions for when the moduli space is general type or uniruled: arxiv.org/abs/math/0602642. The issue is the singularities. | |
| May 26, 2023 at 10:38 | comment | added | Jason Starr | Yes, the Kodaira dimensions is negative for all $a>0$ if $d^2 \leq n+1$; in fact the Kontsevich spaces are even rationally connected. This was proved by Joe Harris and myself with a worse inequality, $d^2 \leq n/2 + C$, then I improved the inequality to $d^2 \leq n+C$, then de Jong and I improved the inequality to $d^2 \leq n+1$. Note, for applications to existence of rational points, the "expected" inequality is $d^2\leq n$, just as in the Tsen-Lang theorem. If $d^2\leq n$, there exists a "very twisting family of lines", which reproves the Tsen-Lang theorem. | |
| May 26, 2023 at 9:30 | history | asked | Puzzled | CC BY-SA 4.0 |