transversal surfaces

Question

I have to prove that the surfaces $E=\{([0:x_{1}:x_{2}],[y_{1}:y_{2}])\in\mathbb{P}\mathbb{C}^{2}\times\mathbb{P}\mathbb{C}\}$ and $V=\{([x_{1}:0:x_{2}],[0:y_{2}])\in \mathbb{P}\mathbb{C}^{2} \times \mathbb{P}\mathbb{C}\}$ are transversal in $P=([0:0:1],[0:1])$, how can I do ?

Answer 1

I don't know the algebraic-geometric interpretation of transversality, the following is a solution in terms of differential topology, or differential geometry (both of which seem to me more appropriate tags to this question, by the way).

First note that at any $(p,q)\in\mathbb{C}\mathbb{P}^2\times\mathbb{C}\mathbb{P}^1$ the tangent space is the direct sum $T_p\mathbb{C}\mathbb{P}^2\oplus T_q\mathbb{C}\mathbb{P}^1$. We want to show that the sum of the tangent spaces to $E,V$ at the specified point is the whole tangent space of the ambient manifold, and it suffices to show it for each component separately. However, it is obvious that $TE$ covers the whole $T\mathbb{C}\mathbb{P}^1$ component, thus we can focus on the $T\mathbb{C}\mathbb{P}^2$ component, ignoring the $\mathbb{C}\mathbb{P}^1$ coordinate.

So let $E',V\subset\mathbb{C}\mathbb{P}^2$ be given by $$E'=\{[0:x_1:x_2]\},\quad V=\{[x_1:0:x_2]\}, $$ and let $p=[0:0:1]\in\mathbb{C}\mathbb{P}^2$. It is easy to find specific tangent vectors $u_{E1},u_{E2}\in T_p E',u_{V1},u_{V2}\in T_p V'$ which are linearly independent, and this will end the proof, since the dimension of $\mathbb{C}\mathbb{P}^2$ as a real manifold is $4$.

For specific examples of such vectors one can consider the paths $\alpha,\beta:(-\epsilon,\epsilon)\to E'$ given by $$\alpha(t)=[0:t:1],\quad\beta(t)=[0:it:1],$$ then take $u_{E1}=\dot{\alpha}(0),u_{E2}=\dot{\beta(0)}$, and repeat the process with similar paths on the other submanifold $V'$.