# Uniruled locus in the parameter space of hypersurfaces

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

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