On Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic
Linus Rösler
TL;DR
The paper tackles the problem of deriving effective Fujita-type freeness results in arbitrary characteristic by establishing sharp lower bounds for Seshadri constants of ample adjoint divisors. It introduces a strategy that combines proper-intersection estimates with adjunction on components, driven by the asymptotic order of vanishing, and uses a generalized Bend-and-Break framework to control curves through a fixed point. The main contributions are a concrete surface bound $\varepsilon_{\rm int}(K_S+4A;x)\geq 3/4$ and a threefold result stating that, for any $\delta>0$, all but finitely many curves through $x$ satisfy $\frac{(K_X+6A).C}{\mathrm{mult}_x C} \geq \frac{1}{2\sqrt{2}}-\delta$, implying rationality of submaximal constants in this regime. These results advance Murayama's conjecture in arbitrary characteristic by providing explicit, characteristic-free lower bounds and clarifying the structure of Seshadri-curve attainability in low dimensions.
Abstract
We develop a new approach towards obtaining lower bounds of the Seshadri constants of ample adjoint divisors on smooth projective varieties $X$ in arbitrary characteristic. Let $x\in X$ be a closed point and $A$ an ample divisor on $X$. If $X$ is a surface, we recover some known lower bounds by proving, e.g., that $\varepsilon(K_X+4A;x)\geq 3/4$. If $X$ is a threefold, we prove that for all $δ>0$ and all but finitely many curves $C$ through $x$, we have $\frac{(K_X+6A).C}{\operatorname{mult}_x C}\geq\frac{1}{2\sqrt{2}}-δ$. In particular, if $\varepsilon(K_X+6A;x)<1/(2\sqrt{2})$, then $\varepsilon(K_X+6A;x)$ is a rational number, attained by a Seshadri curve $C$.