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 $\mathrm{codim}(Z_{unir})\geq n+1$?)

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

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