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Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$.The global Seshadri constant $\varepsilon(L)$ of $L$ is defined as the infimumof Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is aninteresting question to ask if $\varepsilon(L)$ is a rational number for anypair $(X, L)$. We study this question when $X$ is a blow up of $\mathbb{P}^2$at $r \ge 0$ very general points and $L$ is an ample line bundle on $X$. Foreach $r$ we define a $\textit{submaximality threshold}$ which governs therationality or irrationality of $\varepsilon(L)$. We state a conjecture whichstrengthens the SHGH Conjecture and assuming that this conjecture is true wedetermine the submaximality threshold.

Rationality of Seshadri constants on general blow ups of $\mathbb{P}^2$