PomeC3F367EB396ELemma 12.33.1.C3F367EB396E Over $\mathbb C\times \mathbb P^1$, the graded algebra of fiber-homogeneous polynomial functions on the pullback of $W$ is
where all five variables have degree one. The lattice fixes $h$ and $w$, sends $c_-$ to $b_\lambda ^{-1}c_-$ and $c_+$ to $b_\lambda c_+$, and translates $u$ by $m(q_0h+Kw)$.
Proof. As in the proof of lemma 12.31.1, holomorphy in the infinite ruling chart bounds the degree in $t$ of the coefficient of $y^ j$ by $2j$; the variable $w$ does not change between the charts. Distributing the exponent $a$ among $j$ integers in $\{ 0,1,2\} $, every monomial $t^ ay^ j$ with $0\leq a\leq 2j$ is a product of $y$, $ty$ and $t^2y$. These generators satisfy $y(t^2y)=(ty)^2$.
In degree $j$, the space of functions of $y$ and $t$ has dimension $2j+1$. The polynomial algebra in three variables modulo the nonzero quadratic relation has the same dimension
where the second term is zero for $j<2$, so the natural surjective graded map is an isomorphism. Adjoining the two trivial fiber coordinates $u,w$ gives the stated algebra, with entire coefficients in $z$. In the other chart,
so all generators are holomorphic there. Substitution gives their lattice transformations. $\square$
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