Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces

We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety $X^{n}\subset \mathbb{P}^{n+e}$ of degree $d$meets with a linear subspace $L$ of dimension $\beta$ at finite points, then $\operatorname{length} {(X\cap L)}\le d-e+\beta$ as a finite scheme. A linear subspace $L$ for which the above length attains maximal possible value is called an extremal secant subspace and such $L$ for which $\operatorname{length}{(X\cap L)}= d-e+\beta -1$ is called a next to extremal secant subspace.

In this paper, we show that if a smooth variety $X$ of degree $d\ge 6$ has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus $g\le 1$. This generalizes the results of Gruson, Lazarsfeld and Peskine (1983) for curves and the work of M-A. Bertin (2002) who classified smooth higher dimensional varieties with extremal secant lines. This is also motivated and closely related to establishing an upper bound for the Castelnuovo-Mumford regularity and giving a classification of the varieties on the boundary.