Projections from Subvarieties

Let X ⊂ P be an n-dimensional connected projective submanifold of projective space. Let p : P → P denote the projection from a linear P ⊂ P . Assuming that X 6⊂ P we have the induced rational mapping ψ := pX : X → P. This article started as an attempt to understand the structure of this mapping when ψ has a lower dimensional image. In this case of necessity we have Y := X ∩ P is nonempty. The special case when Y is a point is very classical: X is a linear subspace of P . The case when q = 1 and Y = P = P was settled for surfaces by the fourth author [17] and by Ilic [12] in general. Beyond this even the special case when q ≥ 2 and Y = P is open. We have found it convenient to study a closely related question, which includes many special cases including the case when the center of the projection P is contained in X . Problem. Let Y be a proper connected k-dimensional projective submanifold of an n-dimensional projective manifold X . Assume that k > 0. Let L be a very ample line bundle on X such that L ⊗ JY is spanned by global sections, where JY denotes the ideal sheaf of Y in X . Describe the structure of (X,Y, L) under the additional assumption that the image of X under the mapping ψ associated to |L⊗ JY | is lower dimensional. Let us describe our progress on this problem. In §3 we study upper and lower bounds for the dimensions of the spaces of sections of powers tL of a very ample line bundle L on a projective manifoldX . The need for such bounds arises naturally when we consider line bundles which are multiples of a very ample line bundle. One general result Proposition (3.8) gives an upper bound for an integer t0 such that for t ≥ t0, h(tL⊗ JY ) > 0.