Grassmannian.info

Projective duality

Dual varieties for Grassmannians of rank at most 8, embedded by the ample generator of $\operatorname{Pic}(X)$. Self-dual means $X\cong X^\vee$ projectively; a dash denotes an empty dual.

Types: ABCDEFG

No matching Grassmannians.

Dual defect

A general tangent hyperplane is tangent along $\mathbb P^\delta$, where $\delta=\operatorname{codim}(X^\vee)-1$. The defective cases with nonempty dual are $\Gr(2,2m+1)$ for $m\geq2$ ($\delta=2$) and the spinor tenfold ($\delta=4$). The self-dual cases are quadrics, $\Gr(2,5)$ and the spinor tenfold. [Tevelev, Theorems 1.12, 9.19 and 10.22; Examples 2.20–2.21 (arXiv version)]

Degrees

Computed with Oscar's IntersectionTheory from the first jet bundle: \begin{equation} \deg X^\vee=\int_X c_{d-\delta}\bigl(J_1(\mathcal O_X(1))\bigr)H^\delta, \qquad d=\dim X,\quad H=c_1(\mathcal O_X(1)). \end{equation} [Tevelev, Theorems 7.1–7.2 (arXiv version)]

References

1. J. M. Landsberg; L. Manivel. The projective geometry of Freudenthal's magic square. J. Algebra 239 (2001), 477–512. arXiv:math/9908039.
2. Evgueni Tevelev. Projective Duality and Homogeneous Spaces. Encyclopaedia of Mathematical Sciences 133, Springer (2005). arXiv:math/0112028.
3. Kuznetsov, A. G. Hyperplane sections and derived categories. Izv. Math. 70 (2006), 447–547. MR2238172.