Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rationality of Seshadri constants on general blow ups of $\mathbb{P}^2$   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 infimum
of Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is an
interesting question to ask if $\varepsilon(L)$ is a rational number for any
pair $(X, L)$. 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 $X$. For
each $r$ we define a $\textit{submaximality threshold}$ which governs the
rationality or irrationality of $\varepsilon(L)$. We state a conjecture which
strengthens the SHGH Conjecture and assuming that this conjecture is true we
determine the submaximality threshold.

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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph