On the parity conjecture for Hilbert schemes of points on threefolds

Let Hilb^d(ℙ^3) be the Hilbert scheme of d points in ℙ^3, and let T_z denote the tangent space to a point z ∈Hilb^d(ℙ^3). Okounkov and Pandharipande have conjectured that T_z and d have the same parity for every z. For points z parametrizing monomial ideals, the conjecture was proved by Maulik, Nekrasov, Okounkov, and Pandharipande. In this paper, we settle the conjecture for points z parametrizing homogeneous ideals. In fact, we state a generalization of the problem to Quot schemes of ℙ^3, and we settle it for points parametrizing graded modules.