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+ ... +x_r L_r), where L_1, ... , L_r is a basis of Pic X, r being the Picard number of X, and x_1, ... ,x_r 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.