# Incidence variety

• Sep 17th 2009, 01:29 AM
otrebla
Incidence variety
Hello everyone, I apologize for my english.
I have a little ploblem in a count:
Let Z a (Zariski) closed set in $\displaystyle P(C)^n$ with $\displaystyle cod(Z,P(C))^n =:c$. Consider the incidence variety $\displaystyle \Gamma_c (Z):= \{ (p,\Lambda)\in Z\times Gr(c,P(C)^n) s.t. p\in\Lambda\}\subset Z\times P(C)^n$ and the second factor projection
$\displaystyle f: \Gamma_c (Z) \rightarrow Gr(c,P(C)^n)$.
Let $\displaystyle p\in Z$ smooth then proove that
$\displaystyle df_(p,\Lambda) T_(p,\Lambda) \Gamma_c (Z) \rightarrow T_{\Lambda} Gr(c,P(C)^n)$
is surjective if and only if $\displaystyle \Theta_p (Z)$, i.e. the projectif tangent space to Z at p, intersects trasversally $\displaystyle \Lambda$.
Note: if $\displaystyle Z,W\subset P(C)^n$ (Zariski) closed, $\displaystyle p\in Z^{sm}\cap W^{sm}$ ( $\displaystyle Z^{sm}$ is the set of smooth point of Z) then Z intersects trasversally W if and only if the map
$\displaystyle T_p Z \bigoplus T_p W \rightarrow T_p P(C)^n$
given by $\displaystyle (a,b) \rightarrow a+b$ is surjective.
Thanks at everyone.