Inductive characterizations of hyperquadrics (Q2464030)

If \(X\) is an \(n\)-dimensional variety containing an \(m\)-plane whose normal bundle is trivial, with \( m \geq n+2\), \textit{L. Ein} [Duke Math. J., 52, No. 4 , 895--907 (1985; Zbl 0603.14025)] proved that then there exists a smooth projective variety \(Y\) and a vector bundle \(E\) over \(Y\) such that \( Y \simeq \mathbb{P}(E)\) and and the \(m\)-plane is recovered as a fiber of \( X \rightarrow Y\). More recently, \textit{E. Sato} [Tohoku Math. J., 49, No. 3, 299--321 (1997; Zbl 0917.14026)] has proven that for projective smooth \(n\)-folds with the property that through every point of \(X\) there passes through an \(m\)-dimensional linear subspace if \( m \geq n/2 \), either \(X\) is is a projective bundle or \(m= n/ 2\). The author extends this result by replacing linear subspaces by quadric hypersurfaces.~His main theorem (Theorem 2) states that if \( m > [ n/2] +1 \) then \( N= n+1\) and \(X\) is a smooth hyperquadric. A more general result, without the assumption that the quadric subspaces pass all through one fixed point is proven by \textit{Y. Kachi} et al. [Segre's reflexivity and an inductive characterization of hyperquadrics, Mem. Am. Math. Soc. 160, No. 763 ( 2002; Zbl 1039.14016)]. The paper is organized as follows. In section two, the author recalls the definition of the secant defect and for a smooth irreducible non-degenerate projective variety to be conically connected, the variety of minimal rational tangents and a local quadratic entry locus manifold of type \(\delta\) (LQEL -manifold, for short).~In section three, he states the precise definition for a variety to be swept out by m-dimensional hyperquadrics passing through \(z \in X\). In section three he proves theorem 2 assuming that \(X\) is smooth non-degenerate.~He also proves in proposition 3 which states that if \( N \geq 3n/2 \) and \( m = [n/2] +1 \) with \( n \geq 3\) then \(X\) is projectively isomorphic to one of: a) The Segre threefold \( \mathbb{P}_1 \times \mathbb{P}_2 \subset \mathbb{P}_5 \). b) The Plücker embedding \(G(1,4) \subset \mathbb{P}_9 \). c) The 10-dimensional spinor variety \(S_{10} \subset \mathbb{P}_{15} \). d) a general hyperplane section of b) or c). In section four, the author introduces new conditions for the existance of LQEL-manifolds with secant defects. In theorem 3, he proves that if \(X\) is an \(n\)-dimensional LQEL manifold and \( \delta \) is greater than a certain bound given in terms of \(n\) then \( N= n+1\) and \(X\) is a quadric hypersurface. In corollary 3, he proves that if \(X \subset \mathbb{P}_N \) is a LQEL manifold not a quadric hypersurface then \( \delta \leq {n+ 8 \over 3}\) and \( N \geq {5(n-1) \over 3}\). Furthermore \( \delta = {n+ 8 \over 3}\) if and only if \(X\) is projectively equivalent to a hypersurface given explicitely in a list of the corollary.