Cohomological characterization of hyperquadrics of odd dimensions in characteristic two

We consider characterizations of projective varieties in terms of their tangents. Mori established the characterization of projective spaces in arbitrary characteristic by ampleness of tangent bundles. 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 (Lluis, Samuel). Kleiman and 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 intersections, and prove that, a non-linear smooth complete intersection 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.