# What is known about the least cardinal where $\kappa$ fails to be supercompact?

Assume $$\kappa$$ is $$\lambda$$-supercompact for some $$\lambda$$ but not fully supercompact. Are there any known restrictions (or provably non-restrictions) on the least $$\delta$$ such that $$\kappa$$ is not $$\delta$$-supercompact (e.g. $$\text{cf}(\delta)\geq\kappa$$ or $$\text{lim}(\delta)$$)?

For a successor ordinal $$\alpha$$, let $$\kappa_\alpha$$ be the least $$\kappa>\alpha$$ which is $$\kappa^{+\alpha}$$-supercompact. Then $$\kappa_\alpha$$ is not $$2^{\kappa_\alpha^{+\alpha}}$$-supercompact. This is because an embedding $$j : V \to M$$ with critical point $$\kappa_\alpha$$ witnessing $$2^{\kappa_\alpha^{+\alpha}}$$-supercompactness would have that $$M$$ has a measure witnessing $$\kappa_\alpha$$ is $$\kappa_\alpha^{+\alpha}$$-supercompact. (Here, we need to invoke the theorem of Solovay that if $$\lambda$$ is regular and $$\kappa$$ is $$\lambda$$-supercompact, then $$\lambda^{<\kappa}=\lambda$$.) Since $$j(\kappa_\alpha) > \kappa_\alpha$$, there is $$\kappa<\kappa_\alpha$$ that is $$\kappa^{+\alpha}$$-supercompact, contradicting the definition of $$\kappa_\alpha$$.
On the other hand, if $$\kappa$$ is $$\kappa^{+\alpha}$$-supercompact, where $$\alpha<\kappa$$ is a limit ordinal, then $$\kappa$$ is $$\kappa^{+\alpha+1}$$-supercompact, since a $$\lambda$$-supercompactness measure for $$\kappa$$ witnesses $$\lambda^{<\kappa}$$-supercompactness. See Solovay-Reinhardt-Kanamori.
I suppose an interesting specialization of the question is, if $$\kappa$$ is $$\kappa^{+}$$-supercompact and $$2^{\kappa^+} > \kappa^{++}$$, is it $$\kappa^{++}$$-supercompact? We can preserve the $$\kappa^+$$-supercompactness and kill the $$\kappa^{++}$$-supercompactness by adding a nonreflecting stationary subset of $$\kappa^{++}$$, but I don't know how to do it without collapsing $$2^{\kappa^+}$$ to $$\kappa^{++}$$ simultaneously.
One can get fairly close bounds by observing that if $$\kappa$$ is $$\lambda$$-supercompact, then by using a supercompactness measure of least Mitchell rank, there is a $$\lambda$$-supercompactness embedding $$j:V\to M$$ for which $$\kappa$$ is not $$\lambda$$-supercompact in $$M$$. But $$\kappa$$ will be $$\theta$$-supercompact in $$M$$ whenever $$2^{\theta^{<\kappa}}\leq\lambda^{<\kappa}$$. So from the point of view from $$M$$, we have a good understanding of the exact point of failure of supercompactness of $$\kappa$$. Indeed, under GCH, these bounds are nearly tight.