Rationality of Seshadri constants and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture (Q323718)

In the paper under review, the authors show an exciting relation between the SHGH Conjecture and rationality of one-point Seshadri constants for blow ups of the projective plane.NEWLINENEWLINELet us recall that if \(X\) is a smooth projective variety and \(L\) is a nef line bundle on \(X\), then the Seshadri constant of \(L\) at a point \(x \in X\) is the real number defined as NEWLINE\[NEWLINE\varepsilon(L,x) = \inf_{C} \frac{C.L}{\mathrm{mult}_{x}C},NEWLINE\]NEWLINE where the infimum is taken over all irreducible curves passing through \(x\). In a similar manner one can introduce the multi-point Seshadri constant, i.e., for \(x_{1}, \dots, x_{s}\) pairwise distinct points in \(X\) we define NEWLINE\[NEWLINE\varepsilon(L,s) = \inf_{C} \bigg\{ \frac{L.C}{\sum_{i=1}^{s} \mathrm{mult}_{x}(C)}: \{x_{1}, \dots, x_{s}\} \cap C \neq \emptyset \bigg\}.NEWLINE\]NEWLINE The celebrated Nagata Conjecture tells us that for \(s\geq 9\) the multi-point Seshadri constant of the hyperplane bundle \(\mathcal{O}_{\mathbb{P}^{2}}(1)\) on the projective plane satisfies NEWLINE\[NEWLINE\varepsilon(\mathcal{O}_{\mathbb{P}^{2}}(1), s) = \frac{1}{\sqrt{s}}.NEWLINE\]NEWLINE Another interesting conjecture is due to Segre, Harbourne, Gimigliano, and Hirschowitz, the so-called SHGH conjecture.NEWLINENEWLINE{SHGH Conjecture.} Let \(X\) be the blow up of the projective plane in \(s\) very general points with the exceptional divisors \(E_{1}, \dots, E_{s}\). We denote by \(H\) the pull-back of the hyperplane section. Let the integers \(d, m_{1} \geq \dots \geq m_{s} \geq -1\) with \(d \geq m_{1} + m_{2} + m_{3}\) are given. Then the line bundle NEWLINE\[NEWLINEdH - \sum_{i=1}^{s} m_{i}E_{i}NEWLINE\]NEWLINE is non-special.NEWLINENEWLINEIt is worth pointing out that the SHGH Conjecture implies the Nagata Conjecture.NEWLINENEWLINENow we are ready to formulate the main result of the paper.NEWLINENEWLINE{Main Theorem.} Let \(s\geq 9\) be an integer for which the SHGH Conjecture holds true. ThenNEWLINENEWLINEa) either there exist points \(P_{1}, \dots,P_{s}\), a line bundle \(L\) on the blow up \(X_{s}\) of the projective plane along \(P_{1}, \dots, P_{s}\) and a point \(p \in X_{s}\) such that NEWLINE\[NEWLINE\varepsilon(L;p) \mathrm{ is irrational},NEWLINE\]NEWLINENEWLINENEWLINEb) or the SHGH Conjecture fails for \(s+1\)NEWLINENEWLINECorollary. If all one-point Seshadri constants on the blow-up of \(\mathbb{P}^{2}\) in nine very general points are rational, then the SHGH conjecture fails for ten points.NEWLINENEWLINEThe paper is nicely written and (unfortunately) there are some typos.