Pome46169CD2ACA2Lemma 12.32.1.46169CD2ACA2 Over $\mathbb C\times \mathbb P^1$, the graded algebra of fiber-homogeneous polynomial functions on the pullback of $W$ is
where all the variables have degree one and, in the finite ruling chart,
The lattice fixes $h$ and $w$, multiplies $c_-$ by $b_\lambda ^{-1}$ and $c_+$ by $b_\lambda $, and sends $u$ to $u+m(q_0h+Kw)$.
Proof. Set $s_1=y$ and $s_2=ty+\rho w$ in the finite chart. Since $\rho \ne 0$, these are linear coordinates that can replace $y,w$, with $w=(s_2-ts_1)/\rho $. In the infinite chart put
The transition is $(s_{1,\infty },s_{2,\infty })=t(s_1,s_2)$, so $s_1,s_2$ are the fiber coordinates of $\mathcal O_{\mathbb P^1}(-1)^{\oplus 2}$, while $u$ is the coordinate of a trivial line bundle.
By holomorphy in the infinite chart, a polynomial that is homogeneous of degree $j$ in $s_1,s_2$ has coefficients of degree at most $j$ in $t$. Distributing the exponent of $t$ among the $j$ fiber factors, every admissible monomial is a product of $s_1,s_2,ts_1,ts_2$. These four generators satisfy the determinant relation
and there are no further relations. Indeed, counting the $j+1$ monomials in $s_1,s_2$ and the $j+1$ admissible powers of $t$, the space of functions of degree $j$ has dimension $(j+1)^2$. The polynomial algebra in four variables modulo this nonzero quadratic relation has the same dimension
in degree $j$, where the second term is zero for $j<2$. The natural surjective graded map is therefore an isomorphism in every degree. Including $u$ adjoins a polynomial variable. The argument is uniform in $z$, and all coefficient functions are entire.
Identify the four generators as $c_-=s_1$, $h=ts_1$, $h+\rho w=s_2$ and $c_+=ts_2$; the determinant relation becomes the stated quadratic relation. These functions are polynomial also in the infinite chart, where
The lattice formulas follow by substitution in the finite chart. $\square$
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