Exercise 111.60.4. Let $k$ be a field. Let $X = \mathbf{P}^1_ k$ be the projective space of dimension $1$ over $k$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. For $d \in \mathbf{Z}$ denote $\mathcal{E}(d) = \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(d)$ the $d$th Serre twist of $\mathcal{E}$ and $h^ i(X, \mathcal{E}(d)) = \dim _ k H^ i(X, \mathcal{E}(d))$.

Why is there no $\mathcal{E}$ with $h^0(X, \mathcal{E}) = 5$ and $h^0(X, \mathcal{E}(1)) = 4$?

Why is there no $\mathcal{E}$ with $h^1(X, \mathcal{E}(1)) = 5$ and $h^1(X, \mathcal{E}) = 4$?

For which $a \in \mathbf{Z}$ can there exist a vector bundle $\mathcal{E}$ on $X$ with

\[ \begin{matrix} h^0(X, \mathcal{E})\phantom{(1)} = 1 & h^1(X, \mathcal{E})\phantom{(1)} = 1 \\ h^0(X, \mathcal{E}(1)) = 2 & h^1(X, \mathcal{E}(1)) = 0 \\ h^0(X, \mathcal{E}(2)) = 4 & h^1(X, \mathcal{E}(2)) = a \end{matrix} \]

Partial answers are welcomed and encouraged.

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