Quivers and equations a la Pl\"ucker for the Hilbert scheme

Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove that a similar statement holds when one replaces linear subspaces with algebraic subschemes of the projective space. We exhibit equations of degree 1 and 2 that define schematically the Hilbert schemes $\mathbf{Hilb}^{p}_{\mathbb P^n}$ for all (possibly nonconstant) Hilbert polynomials $p$. The equations are reminiscent of the Pl\"ucker relations on the Grassmannians: they are built formally with permutations on indexes on the Pl\"ucker coordinates. Our method relies on a new construction of the Hilbert scheme as a quotient of a scheme of quiver representations.