Generic inner projections of projective varieties and an application to the positivity of double point divisors
Transactions of the American Mathematical Society
Let X ⊆ P N be a smooth nondegenerate projective variety of dimension n ≥ 2, codimension e and degree d with the canonical line bundle ω X defined over an algebraically closed field of characteristic zero. The purpose here is to prove that the base locus of |O X (d − n − e − 1) ⊗ ω ∨ X | is at most a finite set, except in a few cases. To describe the exceptional cases, we classify (not necessarily smooth) projective varieties whose generic inner projections have exceptional divisors. As
... ivisors. As applications, we prove the (d − e)-regularity of O X , Property (N k−d+e ) for O X (k), and inequalities for the delta and sectional genera. 4603 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4604 ATSUSHI NOMA Definition (Condition (E m )). Let m be an integer with 0 < m ≤ e−1. For general points x 1 , . . . , x m ∈ X, we define E x 1 ,...,x m (X) to be the closure of E x 1 ,. ..,x m (X) is the closure of the positive-dimensional fibres of the projection π Λ,X : X \ Λ → P N −m from Λ = x 1 , . . . , x m since π −1 Λ,X (π Λ,X (z)) = x 1 , . . . , x m , z ∩ X \ Λ and dim X ∩ Λ = 0. We say Condition (E m ) holds for For convenience, we define (E 0 ) so that no X satisfies (E 0 ) i.e., the condition is false. For example, a scroll over a projective curve satisfies (E 1 ), and the Veronese surface in P 5 satisfies (E 2 ) but not (E 1 ). Here, by a scroll over a projective variety Z, we mean the birational image of the projective bundle P Z (F) for some locally free sheaf F on Z by the morphism defined by some subsystem of |O P(F ) (1)|. We make a few remarks for the definition: (1) Clearly, for an integer m with 1 ≤ m < e − 1, if X satisfies (E m ), then X satisfies (E m+1 ). (2) To assume the inequalities in (0.2) and (0.3) is equivalent to assuming the equalities there. In fact, by the general position lemma (see Lemma 1.2), dim x 1 , . . . , x m , z ∩ X ≤ 1 and dim E x 1 ,...,x m (X) ≤ n − 1 for general points x 1 , . . . , x m ∈ X and for any point z ∈ X. (3) (E m ) is equivalent to the condition for π Λ,X : X \ Λ → P N −m to have an exceptional set of codimension one. With these definitions, our first result is stated as follows.