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This is E.5 of Hindry, Silverman's Deophantine Geometry. I want a global section of a Vojta divisor from some polynomials.

Let $K$ be a number field, $C$ a smooth projective $K$-curve of genus $g \ge 2 $, $K_C$ its canonical divisor, and $J$ its Jacobian variety. Then, since $\operatorname{Pic}^0C_{\overline{K}} \cong J(\overline{K})$, we can find $A \in \operatorname{Pic}C_{\overline{K}}$ such that $(2g - 2)A \sim K_C$, and taking replace $K$ by somefinite extension, we have $A \in \operatorname{Pic}C$.

Now since $\operatorname{deg}A = 1$, $A$ is ample. Thus for sufficiently large $N$, $NA$ is very ample and induces a closed immersion $\phi_{NA} : C \to \mathbb{P}^n_K$. (And we write it by $\phi_{NA}(P) = [x_0(P) : \cdots : x_n(P)]$.)

Next, since $A$ is ample on $C$, $A \times C + C \times A$ is ample on $C \times C$. Thus for sufficiently large $M$, $B := (M + 1) A \times C + (M + 1) C \times A - \Delta$ is very ample on $C \times C$. ($\Delta$ is the diagonal closed subset of $C \times C$.) Thus $B$ induces a closed immersion $\phi_B : C \times C \to \mathbb{P}^m_K$, write $\phi_B(P) = [y_0(P) : \cdots : y_m(P)]$.

Now for sufficiently large $d$, the global sections of $dB$ is generated by the monomials of degree $d$ in $y_i$, and for sufficiently large $\delta_i$, the global sectinos of $E := \delta_1 NA \times C + \delta_2 C \times NA$ are generated by the monomials of bidegree $(\delta_1, \delta_2)$ in $x_i$'s and $x'_i$'s. (we write $\phi_{NA} \times \phi_{NA} (P,Q) = [x_i(P)] \times [x'_i(Q)]$.) (These properties about generators follow from the cohomological ampleness criterion.)

We say $\Omega := E - dB$ is a Vojta divisor on $C \times C$. And I want a global section of $\Omega$ from some polynomials.

Let $s_1$ be a global section of $dB$. Then this is a homogeneous polynomial of degree $d$ in $y_i$. Further, if $s$ is a global section of $\Omega$, then $s s_1$ is a global section of $E$. Thus $ss_1$ is a bihomogeneous polynomial of bidegree $(\delta_1, \delta_2)$ in $x_i$ and $x'_j$. Thus, taking $y_i ^d$ as $s_1$ above, we have

$$ s = (\frac{F_i(x, x')}{y_i^d})|_{C \times C}, (1)$$

where $F_i$ is bihomogenous polynomial of bidegree $(\delta_1, \delta_2)$ in $x_i$ and $x'_j$, and since this is true for every $i$, we have the equation

$$ (\frac{F_i(x, x')}{y_i^d})|_{C \times C} = (\frac{F_j(x, x')}{y_j^d})|_{C \times C}. (2)$$

Thus from a global section of $\Omega$, we have a set of bihomogenous polynomials of bidegree $(\delta_1, \delta_2)$ $F = \{ F_i \}$ satisfying (2).

Now my question is: is the converse true? That is, if we have a set of of bihomogenous polynomials of bidegree $(\delta_1, \delta_2)$ $F = \{ F_i \}$ satisfying (2), is $s$, as in (1), a global section of $\Omega$?

Thank you very much!

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