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



Uniruled locus in the parameter space of hypersurfaces
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 $\sum d\_i − n$.


