  We consider characterizations of projective varieties in terms of their
tangents. S. Mori established the characterization of projective spaces in
arbitrary characteristic by ampleness of tangent bundles. J. Wahl characterized
projective spaces in characteristic zero by cohomological condition of tangent
bundles; in addition, he remarked that a counter-example in characteristic two
is constructed from odd-dimensional hyperquadrics $Q_{2n-1}$ with $n > 1$. This
is caused by existence of a common point in $\mathbb{P}^{2n}$ which every
embedded tangent space to the quadric contains. In general, a projective
variety in $\mathbb{P}^N$ is said to be strange if its embedded tangent spaces
admit such a common point in $\mathbb{P}^N$. A non-linear smooth projective
curve is strange if and only if it is a conic in characteristic two (E. Lluis,
P. Samuel). S. Kleiman and R. Piene showed that a non-linear smooth
hypersurface in $\mathbb{P}^N$ is strange if and only if it is a quadric of
odd-dimension in characteristic two. In this paper, we investigate complete
intersection varieties, and prove that, a non-linear smooth complete
intersection variety in $\mathbb{P}^N$ is strange if and only if it is a
quadric in $\mathbb{P}^N$ of odd dimension in characteristic two; these
conditions are also equivalent to non-vanishing of 0-cohomology of (-1)-twist
of the tangent bundle.
