Invariant polynomials and Mukai's models of moduli spaces of curves and K3 surfaces

Beginning in the 1980s, Mukai introduced birational models of some moduli spaces of curves and some moduli spaces of K3 surfaces. They are defined as geometric invariant theory quotients. Little is known about the boundaries of these spaces. We describe an approach to efficiently evaluate certain invariant polynomials associated to these GIT problems. This allows us to show that several singular curves and surfaces are GIT semistable in Mukai’s models. In the appendix, we give a combinatorial formula for an ${\mathrm{SL}}_n$-invariant in terms of the Gelfand-Tsetlin basis.