Some basic facts about the Gushel-Mukai varieties?

Question

I'm a beginner in the area of algebraic geometry and reading on the paper Arxiv:1510.05448 about Gushel-Mukai varieties. I have some easy questions:

Let $V_5$ be the vaector space of dimension $5$ and $K$ be a vector space. Then we get the projective cone $C_KGr(2,V_5)\subset\mathbb{P}(\bigwedge^2V_5\oplus K)$ of $Gr(2,V_5)\subset\mathbb{P}(\bigwedge^2V_5)$ with vertex $K$, that is, the join of $Gr(2,V_5)$ with $\mathbb{P}K$.

(I) How to show that $K_{C_KGr(2,V_5)}=-(5+k)H$ where $\dim K=k$? I know nothing about the canonical divisor of the join of varieties. I only know $K_{Gr(2,V_5)}=-5H$.

(II) They also claim that there exists a resolution $$0\to\mathscr{O}(−5)\to V_5^{\vee}\otimes\mathscr{O}(−3)\to V_5\otimes\mathscr{O}(−2)\to\mathscr{O}\to\mathscr{O}_{C_KGr(2,V_5)}\to 0.$$ I don't know where it comes from? I don't think it comes from the Koszul resolution but it maybe some transform of it?

Answer 1

The general fact for (i) is that of $X \subset \mathbb{P}^n$ has $K_X = rH$ and $CX \subset \mathbb{P}^{n+1}$ is the cone over $X$ then $K_{CX} = (r-1)H_{CX}$. Indeed, consider first $$ \tilde{X} = \mathbb{P}_X(\mathcal{O}_X \oplus \mathcal{O}_X(-H)) \stackrel{p}\to X. $$ On the one hand, if $\tilde{H}$ is the relative hyperplane class of this projective bundle, then $$ K_{\tilde{X}} = p^*(K_X + H) - 2\tilde{H} = p^*(r+1)H - 2\tilde{H}. $$ On the other hand, there is a morphism $\pi \colon \tilde{X} \to CX$ that contracts the exceptional section $E \subset \tilde{X}$, and $\tilde{H}$ is the pullback of the hyperplane class $H_{CX}$ of $CX$. Finally, $E \sim \tilde{H} - p^*H$, hence $$ K_{\tilde{X}} = (r - 1)\tilde{H} - (r + 1)E, $$ and since $\mathrm{Cl}(CX)$ is the quotient of $\mathrm{Cl}(\tilde{X})$ by $E$, it follows that $K_{CX} = (r-1)H_{CX}$.

As for (ii), the general fact is that the resolution for the cone over $X$ looks like the resolution for $X$ (the morphisms are the same, they are independent on the extra variable).