Smooth varieties of almost minimal degree

In this article we study non-linearly normal smooth projective varieties X ⊂ P^r of deg (X) = codim (X, P^r) + 2. We first give geometric characterizations for X (Theorem 1.1). Indeed X is the image of an isomorphic projection of smooth varieties over(X, ̃) ⊂ P^{r + 1} of minimal degree. Also if over(X, ̃) is not the Veronese surface, then there exists a smooth rational normal scroll Y ⊂ P^r which contains X as a divisor linearly equivalent to H + 2 F where H is the hyperplane section of Y and F is a fiber of the projection morphism π : Y → P¹. By using these characterizations, (1) we determine all the possible types of Y from the type of over(X, ̃) (Theorem 1.2), and (2) we investigate the relation between the Betti diagram of X and the type of Y (Theorem 1.3). In particular, we clarify the relation between the number of generators of the homogeneous ideal of X and the type of Y. As an application, we construct non-linearly normal examples where the converse to Theorem 1.1 in [D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restriction linear syzygies: Algebra and geometry, Compos. Math. 141 (2005) 1460-1478] fails to hold (Remark 2).