I am inexperienced in algebraic geometry, all I learned from this was reading some class notes and many questions clarified with the help of this platform. I'm reading a part of a book and I found some more doubts:
"Let $X \subset \mathbb{P}^n$ be an irreducible projective variety. Consider the Gauss map $$ \gamma: X_{reg}\longrightarrow \mathbb{G}(k,n) $$ that assigns to each point $p \in X_{reg}$ the translate to the origin of the projectivized tangent hyperplane $\mathbb{P}T_p(X_{reg}).$ Let $\Gamma$ be the closure of the graph of $\gamma$ in $X \times\mathbb{G}(k,n)$. Let $\widetilde {\Gamma}$ be the normalization of $\Gamma$. We have a natural morphism $\alpha:\widetilde {\Gamma} \longrightarrow \mathbb{G}(k,n)$ induced by projection onto the second factor."
1) It is correct that $\alpha(\bigstar)=p_2 \circ \nu(\bigstar)$, where $p_2$ is projection onto the second factor and $\nu:\widetilde{\Gamma} \longrightarrow \Gamma$ is regular map birational, given by normalization ???
2) What is the advantage of working with $\alpha$ instead of directly $p_2$? What do you get by looking at $\widetilde {\Gamma}$ instead of $\Gamma$???
Thanks in advance.