Algebra and Geometry under Projections and Applications

Let X be a reduced closed scheme in P and let IX be its saturated defining ideal. As a generalization of property Np due to Green-Larzarsfeld, we say that X satisfies property N2,p if IX has only the simplest linear syzygies up to p-th step. When p = 1, then IX is generated by quadrics. Recently, many interesting geometric results have been proved for projective varieties with property N2,p (see [4], [7], [8] [13]). In this paper, we obtain higher normality, syzygetic structures and geometric properties of any isomorphic or birational projections of varieties satisfying N2,p by using the mapping cone and the vector bundle technique. In fact, this kind of uniform results can also be refined as we move the center of the projection in an ambient space. Furthermore, property N2,p can be simply generalized to property Nd,p, d ≥ 2. For a variety X satisfying the condition Nd,p, first of all we can show by projection method that a linear section X ∩ L is d-regular (as a scheme) if dim(X ∩ L) = 0 and 1 ≤ dimL ≤ p (see also Theorem 1.1 in [7]). Thus, a projective variety with property N2,p has no (p + 2)-secant p-plane. Second, we give not only other algebraic and geometric structures for projected varieties but also the structure of multiple loci for projections of varieties with the condition Nd,p. In addition, many interesting examples are provided.