# Global sections of a Vojta divisor: a lemma for Faltings' theorem and Vojta's inequality

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!