Generic inner projections of projective varieties and an application to the positivity of double point divisors

Let $X\subseteq \mathbb {P}^{N}$ be a smooth nondegenerate projective variety of dimension $n\geq 2$, codimension $e$ and degree $d$ with the canonical line bundle $\omega _{X}$ defined over an algebraically closed field of characteristic zero. The purpose here is to prove that the base locus of $|\mathcal {O}_{X}(d-n-e-1)\otimes \omega _{X}^{\vee }|$ 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 applications, we prove the $(d-e)$-regularity of $\mathcal {O}_{X}$, Property $(N_{k-d+e})$ for $\mathcal {O}_{X}(k)$, and inequalities for the delta and sectional genera.