Hilbert curves of polarized varieties

Let XX be a normal Gorenstein complex projective variety. We introduce the Hilbert variety VXVX associated to the Hilbert polynomial χ(x1L1+⋯+xρLρ)χ(x1L1+⋯+xρLρ), where L1,…,LρL1,…,Lρ is a basis of Pic(X), ρρ being the Picard number of XX, and x1,…,xρx1,…,xρ are complex variables. After studying general properties of VXVX we specialize to the Hilbert curve of a polarized variety (X,L)(X,L), namely the plane curve of degree dim(X)dim(X) associated to χ(xKX+yL)χ(xKX+yL). Special emphasis is given to the case of polarized threefolds.