Hi, these are three questions regarding extendability of holomorphic vector fields on complex projective space to its blow up along a subvariety. Let $\mathbb{P}^n$ be the complex projective space, with homogeneous coordinates of a point $p\in\mathbb{P}^n$ $$p=[z_0:\ldots:z_n]$$

1) suppose we blow up the point $p_0=[1:0:\ldots:0]$, which are the global holomorphic vector fields of $\mathbb{P}^n$ that extend to the blow up?

I made some computations in the affine chart $\lbrace z_0\neq0 \rbrace$ and modulo (fatal) errors i obtain that my vector fields are of the form $$l_0\frac{\partial}{\partial z_0}+\sum_{i=1}^n c_iz_i\frac{\partial}{\partial z_i}$$ with $l_0\in\mathbb{C}[z_0,\ldots,z_n]$ homogeneous of degree 1 and $c_i\in\mathbb{C}$. So vector fields that extend to the blow up span a vector subspace of dimension $2n$ of $H^0(T\mathbb{P}^n)$. Am i right?

2)In a similar fashion i find that if i blow up a k-codimensional variety of type $$\lbrace z_k=\ldots=z_n=0 \rbrace$$ with $k>1$ my vector fields span a vector subspace of dimension $n(k+2)$ of $H^0(T\mathbb{P}^n)$. Again, am i right?

3) If i blow up a smooth curve of sufficiently high degree/genus do i still have holomorphic vector fields that extend to the blow up?

Thank you in advance.