On Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic

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 $\epsilon (K_X+4A;x)\ge 3\mathrm{/}4$. If $X$ is a threefold, we prove that for all $\delta >0$ and all but finitely many curves $C$ through $x$, we have $\frac{(K_X+6A)\mathrm{.}C}{{\mathrm{mult}}_{x}C}\ge \frac{1}{2\sqrt{2}}-\delta$. In particular, if $\epsilon (K_X+6A;x)<1\mathrm{/}(2\sqrt{2})$, then $\epsilon (K_X+6A;x)$ is a rational number, attained by a Seshadri curve $C$.