Relative projective-line cohomology and apolarity

Work over a sufficiently small affine $U=\mathrm{Spec}A$ that trivializes $E$ and $M_U$ in [F1]. Put $V=A^2$, use the line convention $\mathbb{P}(V)$, and write $L{\mathrm{\mid }}_{E_U}=\mathcal{O}(n)\otimes {\pi }^{\ast }{M}_U$. The relative Euler sequence, or its direct two-chart differential calculation, gives $K_{\pi }{\mathrm{\mid }}_{E_U}=\mathcal{O}(-2)\otimes {\pi }^{\ast }(\det V{)}^{\vee }\mathrm{.}$ Indeed ${\mathrm{\Omega }}_{\mathbb{P}(V)\mathrm{/}U}^1$ is the determinant of the kernel of $V^{\vee }\otimes \mathcal{O}(-1)\to \mathcal{O}$, and the displayed formula has central $\lambda I_V$-weight zero, as a relative canonical bundle must. Hence $L\otimes K_{\pi }^{\otimes (n+1)}=\mathcal{O}(-n-2)\otimes {\pi }^{\ast }(M_U\otimes (\det V{)}^{-n-1})\mathrm{.}$

Apply [F2] to these two twists and [F4] to identify the resulting modules with higher direct images on $U$. For $n\ge 0$ this gives ${\pi }_{\ast }L{\mathrm{\mid }}_U\cong {\mathrm{Sym}}^n(V^{\vee })\otimes M_U,R^q{\pi }_{\ast }L{\mathrm{\mid }}_U=0(q>0),$ and $R^q{\pi }_{\ast }(L{K}_{\pi }^{n+1}){\mathrm{\mid }}_U=0$ for $q\mathrm{\ne }1$. For $n=-1$ the two twists are both $\mathcal{O}(-1)\otimes M_U$ and [F2] makes every direct image zero. The calculations hold after every affine restriction because their Čech matrices are defined over $A$ and tensor with the new base ring.

For $n\ge 0$, the monomial bases of [F2] give a perfect pairing over $A$: a degree-$n$ monomial $x_0^a{x}_1^{n-a}$ pairs to $1$ with the Laurent class $x_0^{-a-1}{x}_1^{-(n-a)-1}$ and to $0$ with the other basis classes. This is the same ordered residue normalization as [F3] after every field specialization. It identifies $H^1(\mathcal{O}(-n-2))$ with ${\mathrm{Sym}}^n(V^{\vee }{)}^{\vee }\otimes \det V={\mathrm{Sym}}^{n}V\otimes \det V$ as a $GL(V)$-representation: the determinant factor records how the ordered Laurent residue changes under a coordinate matrix. After the determinant twist of step 1.1, $R^1{\pi }_{\ast }(L{K}_{\pi }^{n+1}){\mathrm{\mid }}_U\cong {\mathrm{Sym}}^{n}V\otimes (\det V{)}^{-n}\otimes M_U\mathrm{.}$ The canonical wedge pairing in rank two gives a $GL(V)$-equivariant isomorphism $V^{\vee }\cong V\otimes (\det V{)}^{-1}$; its $n$th symmetric power identifies the right side with ${\mathrm{Sym}}^n(V^{\vee })\otimes M_U={\pi }_{\ast }L{\mathrm{\mid }}_U$. Choose the scalar of this map so the dual pair of ordered monomials has residue $1$; the wedge and residue formulas then determine the same nonzero invariant normalization on every chart.

On an overlap two projective trivializations differ by a $PG{L}_2$-matrix. After an affine refinement lift it to $g\in G{L}_2$; replacing $g$ by $\lambda g$ changes the action on ${\mathrm{Sym}}^n{V}^{\vee }$ by ${\lambda }^{-n}$ and the action on ${\mathrm{Sym}}^{n}V\otimes (\det V{)}^{-n}$ by ${\lambda }^n{\lambda }^{-2n}={\lambda }^{-n}$. The transition of $M_U$ is the same on both sides because both functors are linear in $L$. Thus the $G{L}_2$-equivariance of step 3.1 and equal central weights show that the local maps agree on overlaps independently of the lift, and they glue to $a_L$. The Čech constructions, wedge map and descent use only restriction and tensor operations, so $a_L$ is compatible with base restriction and coordinate changes. The explicit projective-line Čech calculation of [F2] supplies the required cohomology comparison, and the completed local normal form [F1] is used at steps 1.1–2.1. AC is inherited through [F1], [F2] and [F4].