On the restrictions of the tangent bundle of the Grassmannians (Q1188091)

Let \(X\) be a projective variety and \(Q\) a vector bundle on \(X\); let \(q\) be a surjection \[ q:\bigoplus^ m{\mathcal O}_ X\to Q\to 0 \] and put \(S=\text{Ker} q\). The author studies the deformations of \(S\) obtained by moving \(q\) in the space \(A(*,Q)\) of surjections \(\bigoplus^ m{\mathcal O}_ X\to Q\). Under some cohomological assumptions \((h^ 1{\mathcal O}_ X=h^ 2(Q^*\otimes S)=0)\) the author proves that the induced map \(A(*,Q)\to V(S)=\) space of formal deformations of \(S\), is smooth. This suggests the use of \(A(*,Q)\) in the study of the stratification of \(V(S)\) induced by geometric properties of the map \(X\to\) Grassmannian defined by a spanned vector bundle \(Q\).