of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces. Projections Let be a linear subspace; i.e., for some linearly independent linear functionals si. Then the projection from E is the (well-defined) morphism The geometric description of this map is as follows: We view so that it is disjoint from E. Then, for any , where denotes the smallest linear space containing E and x (called the join of E and x.) where are the homogeneous coordinates on For any closed subscheme disjoint from E, the restriction is a finite morphism. Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Start with some projective variety If the projection from a point not on X gives Moreover, is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map This result is the projective analog of Noether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.) The same procedure can be used to show the following slightly more precise result: given a projective variety X over a perfect field, there is a finite birational morphism from X to a hypersurface H in In particular, if X is normal, then it is the normalization of H. Duality and linear system While a projective n-space parameterizes the lines in an affine n-space, the dual of it parametrizes the hyperplanes on the projective space, as follows. Fix a field k. By , we mean a projective n-space equipped with the construction: , a hyperplane on where is an L-point of for a field extension L of k and For each L, the construction is a bijection between the set of L-points of and the set of hyperplanes on . Because of this, the dual projective space is said to be the moduli space of hyperplanes on . A line in is called a pencil: it is a family of hyperplanes on parametrized by . If V is a finite-dimensional vector space over k, then, for the same reason as above, is the space of hyperplanes on . An important case is when V consists of sections of a line bundle. Namely, let X be an algebraic variety, L a line bundle on X and a vector subspace of finite positive dimension. Then there is a map: determined by the linear system V,