Hello everyone, I apologize for my english.
I have a little ploblem in a count:
Let Z a (Zariski) closed set in P(C)^n with cod(Z,P(C))^n =:c. Consider the incidence variety \Gamma_c (Z):= \{ (p,\Lambda)\in Z\times Gr(c,P(C)^n) s.t. p\in\Lambda\}\subset Z\times P(C)^n and the second factor projection
f: \Gamma_c (Z) \rightarrow Gr(c,P(C)^n).
Let p\in Z smooth then proove that
df_(p,\Lambda) T_(p,\Lambda) \Gamma_c (Z) \rightarrow T_{\Lambda} Gr(c,P(C)^n)
is surjective if and only if \Theta_p (Z), i.e. the projectif tangent space to Z at p, intersects trasversally \Lambda.
Note: if Z,W\subset P(C)^n (Zariski) closed, p\in Z^{sm}\cap W^{sm} ( Z^{sm} is the set of smooth point of Z) then Z intersects trasversally W if and only if the map
T_p Z \bigoplus T_p W \rightarrow T_p P(C)^n
given by (a,b) \rightarrow a+b is surjective.
Thanks at everyone.