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Pfister and Steenbrink studied punctual Hilbert schemes for irreducible curve singularities. In particular, they investigated the structure of special punctual Hilbert schemes for certain monomial curve singularities. In this paper, we study the punctual Hilbert schemes of all degrees for the curve singularities of types $E_6$ and $E_8$. For our analysis, we introduce computational algorithms to decompose a punctual Hilbert schemes into affine cells. We also use the theory of Gr\"obner basis and known results about the compactified Jacobian of singular curves to prove our main theorems.

The punctual Hilbert schemes for the curve singularities of types $E_6$ and $E_8$