A Frobenius variant of Seshadri constants
Let \(X\) be a projective variety of dimension \(n\) over an algebraically closed field of \(\text{char\,}p> 0\) and let \(L\) be an ample line bundle over \(X\). The authors define and study a new version of Seshadri constants for \(L\). They prove that ower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. Specifically, for any smooth point \(x\in X\), let \(s_F(L^m; x)\) be the largest integer \(e\) such that the restriction map \[ H^0(X, L^m)\to H^0(X, L^m\otimes O_X/{\mathfrak m}^{[p^e]}_x) \] is surjective, where \({\mathfrak m}^{[p^e]}_x\) is the ideal locally generated by the \(p^e\)-powers of the generators of the maximal ideal \({\mathfrak m}_x\) of the local ring of \(X\) at \(x\). Now, the authors define the Frobenius-Seshadri constant by: \[ \varepsilon_F(L; x):= \sup_{m\geq 1} {p^{s_F(L^m; x)}- 1\over m}. \] The main result of the paper asserts that if \(X\) is smooth and (a) if \(\varepsilon_F(L; x)> 1\), then \(\omega_X\otimes L\) is globally generated at \(x\), and (b) if \(\varepsilon_F(L; x)> 2\) for each \(x\in X\), then \(\omega_X\otimes L\) is very ample. Let \(\varepsilon(L; x)\) be the (classical) Seshadri constant introduced by Damailly. Then, it is easy to see that \(\varepsilon_F(L; x)\geq\varepsilon(L; x)/n\). Thus, from the above result, one deduces that (a') if \(\varepsilon(L; x)> n\), then \(\omega_X\otimes L\) is globally generated at \(x\), and (b') if \(\varepsilon(L; x)> 2n\) for each \(x\in X\), then \(\omega_X\otimes L\) is very ample. The characteristic zero analogues of (a') and (b') were obtained easily as consequences of the Kawamata-Viehweg vanishing theorem. The authors compare the two Seshadri and Frobenius-Seshadri constants in the case of torus-invariant points on smooth toric varieties in terms of the polytope combinatorics.
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