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Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the Zariski closure of the union of linear spans of $s$-tuples of points lying on $X$. We exhibit two functions $s_0(n)\le s_1(n)$ such that $\sigma_s(Gr(2,n))$ has the expected dimension whenever $n\geq 9$ and either $s\le s_0(n)$ or $s_1(n)\le s$. Both $s_0(n)$ and $s_1(n)$ are asymptotic to $\frac{n^2}{18}$. This yields, asymptotically, the typical rank of an element of $\wedge^{3} 1pt {\mathbb C}^{n+1}$. Finally, we classify all defective $\sigma_s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case., Comment: 17 pages

We prove that the special instanton bundles of rank 2 n 2n on P 2 n + 1 ( C ) {\mathbb {P}^{2n + 1}}(\mathbb {C}) with a symplectic structure studied by Spindler and Trautmann are stable in the sense of Mumford-Takemoto. This implies that the generic special instanton bundle is stable. Moreover all instanton bundles on P 5 {\mathbb {P}^5} are stable. We get also the stability of other related vector bundles.

We define some stable vector bundles on the complex quadric hypersurface Q n {Q_n} of dimension n n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q 4 ≃ Gr ⁡ ( 1 , 3 ) {Q_4} \simeq \operatorname {Gr} (1,\,3) . We call them spinor bundles. When n = 2 k − 1 n = 2k - 1 there is one spinor bundle of rank 2 k − 1 {2^{k - 1}} . When n = 2 k n = 2k there are two spinor bundles of rank 2 k − 1 {2^{k - 1}} . Their behavior is slightly different according as n ≡ 0 ( mod 4 ) n \equiv 0\;(\bmod 4) or n ≡ 2 ( mod 4 ) n \equiv 2\;(\bmod 4) . As an application, we describe some moduli spaces of rank 3 3 vector bundles on Q 5 {Q_5} and Q 6 {Q_6} .