Jet schemes, in terms of Buium.
Let \(K\) be a function field in one variable over constant field \(k\).
\(K_a, K_s\) its algebraic and separable closures.
\(X/K\) an genus \(g\ge 2\) curve.
Function field Mordell conjecture says \(X(K)\) is finite, unless \(X\) is isotrivial, i.e. \(K_a\)-isomorphic to a curve over \(k\).
This is proved by Manin in characteristic \(0\).
Buium and Voloch proved an effective version of Lang’s conjecture in characteristic \(p\).
Lemma (V, Lemma 1). Vanishing of KS class of \(X\) is equivalent to \(X\) defined over \(K^p\).
Theorem (BV). \(K\) be a function field in one variable char \(p>0\). Let \(X\) be a smooth projective curve of genus \(g\ge 2\) over \(K\) embedded into Jacobian \(J\).
Assume \(X\) has non-zero Kodaira-Spencer class (equivalently, \(X\) is not defined over \(K^p\)).
If \(\Gamma\le J(K_s)\) such that \(|\Gamma/p\Gamma|<\infty\), then
\[|X\cap \Gamma|\le |\Gamma/p\Gamma| \cdot (3p)^g(8g-2)g!\]
We know that, the vector bundle, or locally free sheaf \(TX\) is the dual of the sheaf of differentials \(\Omega^1_{X/k}\). This means
\[TX = {\mathcal{H}om}_{\mathcal{O}_X}(\Omega^1_{X/k}, \mathcal{O}_X)\]
For any vector bundle \(\mathcal{V}\), the scheme structure of it can be built from the symmetric algebra of its dual.
\[ \mathcal{V}= \mathrm{Spec} \left( \mathrm{Sym}^\bullet (\mathcal{V}^\vee) \right) \]
Thus the tangent bundle \(TX\) can be constructed as a scheme from the symmetric algebra of the sheaf of differentials \(\Omega^1_{X/k}\).
\[ TX = \mathrm{Spec} \left( \mathrm{Sym}^\bullet (\Omega^1_{X/k}) \right) \]
B1Fix a derivation \(\delta=\frac{\partial }{\partial t}\) on \(K\), \(t\in K\) is a separable transcendence basis of \(K/k\).
Define the first jet scheme along \(\delta\) as
\[J^1_\delta(X) = \mathrm{Spec} \left( \mathrm{Sym}^\bullet (\Omega^1_{X/k}/I_\delta) \right) \]
where \(I_\delta = \langle \mathrm{d}f - \delta(f) | f\in \mathcal{O}_X\rangle\).
Lemma. If \(X/K\) is a smooth projective curve \(g_X\ge 2\), with non-zero KS class, then \(J^1_\delta(X)\) is an affine surface.
Let’s think about the linear forms on this space.
\[\Omega^1_{X/k}/I_\delta =\langle \mathrm{d}x_i \rangle /\langle \mathrm{d}x_i - \delta(x_i)\rangle \]