In $ X = \mathbb{P}^{n}$ we have the Euler sequence: $$0 \longrightarrow \mathcal{O}_{X} \longrightarrow \mathcal{O}_{X}(1)^{\oplus (n+1)} \longrightarrow T_{X} \longrightarrow 0 $$
Let $Y \subset X$ be a smooth subvariety and $\pi: \widetilde{X} \longrightarrow X$ the morphism of blow up along of $Y$.
Is there an exact sequence in $\widetilde{X}$ analogous to the Euler sequence in $X$?
1) Is $\widetilde{X}$ an algebraic variety? Is it projective?
2) Is Bott's formula valid for $\widetilde{X}$?
Sorry if my questions are trivial to you, but I am a beginner in Algebraic Geometry.
About exact sequence:
3) What are the references where I can find exact sequences after a blow up along a smooth subvariety?
4) More precisely, I would like to learn more about exact sequences in $\widetilde{X}$ and later learn to calculate cohomologies.
Thanks in advance.