Hilbert surfaces of bipolarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety VX associated to the Hilbert polynomial χ(x1L1; : : : ; xÞLÞ), where L1; : : : ;LÞ is a basis of Pic(X), Þ being the Picard number of X, and x1; : : : ; xÞ are complex variables. After reviewing general properties of VX, we focus on the following specific topics. First, we consider the Hilbert surface of a bipolarized variety (X;L1;L2), namely, the surface of degree dim(X) in a 3-dimensional affine space, associated to χ(xKX + yL1 + zL2). Special emphasis is given to the case of 3-folds. Next, we treat the case of the Hilbert curve of a polarized 4-fold (X;L), that is, the plane quartic curve associated to χ(xKX + yL). We also study quotients of Hilbert surfaces under the Serre involution s induced by Serre duality, and we characterize surfaces in a 3-dimensional affine space which are invariant under s.