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Let $X\subset\mathbb{P}^n$ be a general hypersurface of degree $d\leq n$, and $\overline{\mathcal{M}}_{0,0}(X,a)$ the Kontsevich space of degree $a$ rational curves in $X$.

Does there exist an explicit relation between $n,d,a$ implying that $\overline{\mathcal{M}}_{0,0}(X,a)$ has negative Kodaira dimension?

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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented May 26, 2023 at 10:38
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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented May 26, 2023 at 10:42
  • $\begingroup$ Thanks a lot. When $n=d=4$ and $a = 2$ is the Kontsevich space a surface of general type? $\endgroup$
    – Puzzled
    Commented May 26, 2023 at 12:14
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    $\begingroup$ 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). $\endgroup$
    – Jason Starr
    Commented May 26, 2023 at 13:34
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    $\begingroup$ 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 $\mathbb{P}^4$). $\endgroup$
    – Jason Starr
    Commented May 26, 2023 at 14:03

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