Disconnectedness of the Hilbert Schemes of $E_6\mathrm{/}{P}_6$

In this note, we show that the Hilbert scheme ${\text{Hilb}}_{P_{d,4}(t)}(E_6\mathrm{/}{P}_6)$ associated with the Hilbert polynomial $P_{d,4}(t)$ is disconnected by determining that it has exactly two connected components. This result adapts Seong’s methods, successfully extending the disconnectedness of Grassmannians to the exceptional type $E_6$.