Hilbert curves of polarized varieties

Let $X$ be a normal Gorenstein complex projective variety. We introduce the Hilbert variety $V_X$ associated to the Hilbert polynomial $\chi(x_1 L_1,\ldots, x_\rho L_\rho)$, where $ L_1,\ldots, L_\rho$ is a basis of $\Pic X$, $\rho$ being the Picard number of $X$, and $x_1,\ldots,x_\rho$ are complex variables. After studying general properties of $V_X$ we specialize to the Hilbert curve of a polarized variety $(X,L)$, namely the plane curve of degree $\dim(X)$ associated to $\chi(xK_X+yL)$. Special emphasis is given to the case of polarized $3$-folds