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

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 $$1\leq i to denote the coordinates of $$\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!

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.