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!



Construction of Fano threefold of degree $5$ and its defining equations
To give a 3-dimensional linear section $X$ of $Gr(2,V)$ with $\dim V = 5$ is equivalent to giving a 3-dimensional subspace $A \subset \Lambda^2V^\vee$ (the space of linear equations of $X$). Then smoothness of $X$ is equivalent to the property
$$
\mathbb{P}(A) \cap Gr(2,V^\vee) = \varnothing.
$$


For the second question note that there is a resolution 
$$
0 \to \mathcal{O}(-5) \to \mathcal{O}(-3)^{\oplus 5} \to \mathcal{O}(-2)^{\oplus 5} \to I\_X \to 0
$$
for the ideal of $X$ (obtained by restricting to $\mathbb{P}(A^\perp) = \mathbb{P}^6$ the resolution of the ideal of $Gr(2,V)$). Twisting it by $\mathcal{O}(2)$ it is easy to check the vanishing of $H^1(I\_X(2))$ and to identify $H^0(I\_X(2))$ with the space of Plucker quadrics.


