Résumé : Many nice Fano manifolds and K3 surfaces can be obtained as linear sections of homogeneous spaces. I will study low-codimensional sections of the spinor tenfold, that admit non-trivial moduli starting from codimension four. The corresponding family exhibits an extremely rich geometry, connected with the exceptional complex Lie algebra of type E 8, the theory of graded Lie algebras, as well as the classical Kummer quartic surfaces in three dimensional projective space.