The Fano threefold $X$ of index $2$, degree $5$ and Picard number $1$ is known to be a general codimension $3$ linear section of the $Pl\ddot{u}cker$ embedding of Gr(2,5).

My first question: what does 'general' mean specifically in the above definition?

If we use $p_{ij}$ where $1\leq i<j\leq 5$ to denote the coordinates of $\mathbb{P}(\bigwedge^2 k^5)$, then Gr(2,5) is defined by the ideal generated by: $$p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}$$ $$p_{12}p_{35}-p_{13}p_{25}+p_{15}p_{23}$$ $$p_{13}p_{45}-p_{14}p_{35}+p_{15}p_{34}$$ $$p_{12}p_{45}-p_{14}p_{25}+p_{15}p_{24}$$ $$p_{23}p_{45}-p_{24}p_{35}+p_{25}p_{34}$$ If we look at the short exact sequence $$0\to I_X(2)\to \mathcal{O}_{\mathbb{P}^6}(2)\to\mathcal{O}_X(2)\to 0$$ (here $\mathbb{P}^6$ is the codimension 3 linear section of $\mathbb{P}(\bigwedge^2 k^5)$ we used to cut $X$), we can obtain sequence: $$0\to H^0(I_X(2))\to H^0(\mathcal{O}_{\mathbb{P}^6}(2))\to H^0(\mathcal{O}_X(2))\to H^1(I_X(2)))\to 0$$ where the $H^0(\mathcal{O}_{\mathbb{P}^6}(2))$ is $28$ dimensional and $H^0(\mathcal{O}_X(2))$ is $23$ dimensional.

My second question: It seems natural to guess $H^1(I_X(2)))=0$ and $H^0(I_X(2))$ is then five dimensional and spanned by the above equations restricted to the codimension 3 linear section $\mathbb{P}^6$. Is this a true statement? Is there a reference for it?

Thanks for the help in advance!