Quadratic equations defining the $10$-dimensional spinor variety.

Question

Let $S$ be the $10$-dimensional Spinor variety parametrizing one of the two families of $4$-dimensional linear subspaces of the non-singular quadric in $\mathbb{P}^{9}$. I have read that there exist $10$ linearly independent quadratic forms (and no more) $$ F_{1},\ldots,F_{10}\in K[X_{0},\ldots,X_{15}] $$ such that $$ S\simeq V(F_{1},\ldots,F_{10})\subseteq \mathbb{P}^{15}. $$ Nevertheless, I have not been able to prove it, nor to find a book where it is proven. Any hint would be appreciated.

Answer 1

This follows easily from representation theory of group $G = Spin(10)$ (the simply connected group of type $D_5$). The spinor variety is the homogeneous space $G/P$, where $P$ is the maximal parabolic group corresponding to the simple root $i_5$ (for the standard enumeration of vertices). The 16-dimensional spinor representation --- the space of global sections of $O_S(1)$ --- is an irreducible G-representation with the highest weight $\omega_5$ (the fundamental weight corresponding to $i_5$). Consequently, the space of quadratic equations of $S$ is the kernel of the natural map $$ Ker(Sym^2 V_{\omega_5} \to V_{2\omega_5}). $$ Representation theory allows to identify its kernel with the irreducible representation $V_{\omega_1}$. Its dimension is 10 (this is the standard representation of the quotient $SO(10)$ of $Spin(10)$), hence there are 10 quadrics through $S$.

Finally, any compact homogeneous space is an intersection of quadrics, hence $S$ is cut out by these equations.

Answer 2

You can consult the following article:

W. Lichtenstein "A system of quadrics describing the orbit of the highest weight vector", Proc. AMS 84 (1982), no. 4, 605–608, where it is proved that homogeneous varieties in their minimal embedding have ideal generated by quadratic forms. You can deduce that they are ten going to lines section and finding a canonical curve. A more subtle interesting fact is that S can be scheme theoretically defined only by 9 quadratic equations.