Hilbert curves of polarized varieties , I ∗

Contents 1 Conventions and basic notation 2 2 Hilbert variety: the general framework 3 3 The Hilbert curve 8 4 Cubic Hilbert curves 12 5 Image of the Hilbert curve in P 3 17 6 Fibrations and singular points of the Hilbert curve at infinity 19 7 Serre-invariant curves 23 Abstract Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V X associated to the Hilbert polynomial χ(x 1 L 1 ,. .. , x ρ L ρ), where L 1 ,. .. , L ρ is a basis of Pic(X), ρ being the Picard number of X, and x 1 ,. .. , x ρ 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 χ(xK X + yL). Special emphasis is given to the case of polarized 3-folds.