  Let V be an even dimensional vector space with a non degenerate quadratic
form. We denote by X the variety of maximal isotropic subspaces in V (in fact
one of its two connected components). In this paper, we prove the
irreducibility of the scheme of degree d morphism f:C->X as soon as d is bigger
than 1/2dim(V)-1.
  When dim(V)=10 and d=6, this result was used by A. Iliev and D. Markushevich
in math.AG/0403122 to prove the irreducibility of the moduli space
M_{X_12}(2,1,6) where X_{12} is the Fano threefold of index 1 and degree 12.
