Construction of Fano threefold of degree $5$ and its defining equations

Question

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!

Answer 1

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.

Answer 2

For operative purposes (for example if you want to work concretely with the Fano threefold), it might be easier to check smoothness rather than generality. There is this answer by Robert Bryant that might help.

Also, notice that the equations for the Grassmannian Gr(2,5) may be obtained by taking the submaximal 4-Pfaffians of a generic 5 by 5 skew matrix of linear forms. Taking a codimension three linear section is equivalent to introduce some linear relation in three out of ten independent entries. Then any standard computer algebra system (macaulay2, magma) can easily check if the choice you made produced a smooth example.