Hilbert scheme of 9 and 10 points and applications to secant varieties of pencils

We describe the irreducible components of the Hilbert scheme of d points on affine space for d=9, 10. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. As the main consequence, we establish the equality of cactus Grassmann and the secant Grassmann variety in the corresponding cases. This is a joint project with Hanieh Keneshlou and Klemen Sivic.