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Let $n\geq3$ be a fixed positive integer, consider the parameter space $|\mathcal{O}_{\mathbb{P}^{n}}(d)|$ of degree $d$ hypersurfaces ($d\geq n+1$) in $\mathbb{P}^{n}$. Let $Z_{unir}\subset |\mathcal{O}_{\mathbb{P}^{n}}(d)|$ be the subspace parameterizing uniruled hypersurfaces, do we know some lower bounds on $\mathrm{codim}(Z_{unir})$? (For example, do we know $\mathrm{codim}(Z_{unir})\geq n+1$?)

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  • $\begingroup$ Such a bound must certainly involve $d$ — the codimension is 0 for $d\leq n$. $\endgroup$
    – abx
    Commented May 21, 2020 at 3:52
  • $\begingroup$ Yes, thanks for clarification, say $d\geq n+1$ $\endgroup$
    – user39380
    Commented May 21, 2020 at 4:11
  • $\begingroup$ In characteristic zero, an obvious lower bound is one, as for $d \ge n+1$ uniruled hypersurfaces must be singular. $\endgroup$ Commented May 22, 2020 at 21:58

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One can find such bounds in [Corollary 4.2, "Rational curves on complete intersections in positive characteristic" by E. Riedl and M. Woolf]: The space of multidegree $\underline{d}$ complete intersections in $\mathbb{P}^n$ containing a rational curve has codimension at least $\sum d_i −2n +2$. The space of uniruled hypersurfaces has codimension at least $\sum d_i − n$.

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