Worst Unstable Points of a Hilbert Scheme

In this paper, we describe the worst unstable points of a Hilbert scheme for some special Hilbert polynomials and ambient spaces using Murai’s work on Gotzmann monomial sets. We investigate the geometry of the projective schemes represented by worst unstable Hilbert points and see that in certain cases that they fail to be $K$-stable or attain maximal regularity.