Seshadri constants and surfaces of minimal degree (Q1049907)

Let \(X\) be a smooth projective surface, \(L\) be a nef line bundle on \(X\), and \(P_1,\dots,P_r\) be \(r\geq 1\) points on \(X\). One defines {\textit{the \(r\)-tuple Seshadri constant of \(L\) at the set \(P_1,\dots,P_r\)}} as the real number \[ \varepsilon\,(L;P_1,\dots,P_r):=\inf_{C\cap\{P_1,\dots,P_r\}\neq\emptyset}\,\frac{L\cdot C}{\sum{\text{mult}}_{P_i}C}, \] where the infimum is taken over all irreducible curves \(C\) of \(X\) passing through at least one of the points \(P_1,\dots,P_r\). The Kleiman criterion of ampleness implies that \[ \varepsilon\,(L;P_1,\dots,P_r)\leq \root\dim X\of{\frac{L^{\dim X}}{r}}:=\alpha(L;r). \] When this inequality is strict then the Seshadri constant is actually computed by a curve and not approximated by a sequence of curves. One defines a {\textit{Seshadri curve for \(L\) at the \(r\)-tuple \(P_1,\dots,P_r\)}} any curve \(C\) such that \[ \varepsilon\,(L;P_1,\dots,P_r)=\frac{L\cdot C}{\sum{\text{mult}}_{P_i}C}. \] The \textit{Seshadri function} \[ \varepsilon_r: (P_1,\dots,P_r)\in X^r\to \varepsilon\,(L;P_1,\dots,P_r)\in\mathbb R \] is semi-continuous and reaches its maximal value \(\varepsilon\,(L;r):=\max \varepsilon_r\) at a set which is a complement of an at most countable union of Zarisky closed proper subsets of \(X\) (i.e. at a very general multiple point \((P_1,\dots,P_r)\in X^r\)). In the case \(r=1\) one knows that if \(\varepsilon\,(L;1)\ll \alpha(L;1)\) then there exists a non-trivial fibration of \(X\) over a curve whose fibers are Seshadri curves for \(L\) [see \textit{M. Nakamaye}, J. Reine Angew. Math. 564, 205--214 (2003; Zbl 1052.14038) and \textit{T. Szemberg} and \textit{H. Tutaj-Gasińska}, Ann. Acad. Paedagog. Crac. 23, Stud. Math. 4, 225--229 (2004; Zbl 1131.14012)]. In the case \(r\geq 2\), \textit{W. Syzdek} and \textit{T. Szemberg} [``Seshadri fibrations on algebraic surfaces'', \url{arXiv:0709.2592} [math.AG], to appear in Math. Nachr.] proved a similar result, i.e. that if \(L\) is big and nef and \(\varepsilon\,(L;1)< \sqrt{\frac{r-1}{r}}\cdot\alpha(L;1)\) then there exists a fibration \(f:X\to B\) over a curve \(B\) such that for very general \((P_1,\dots,P_r)\in X^r\) and any \(i\in\{P_1,\dots,P_r\}\) the fiber over \(f(P_i)\) is a Seshadri curve of \(L\). In the paper under review the authors complete previous analysis proving that when \(r\geq 2\), if \(L\) is big and nef and \(\varepsilon\,(L;1)= \sqrt{\frac{r-1}{r}}\cdot\alpha(L;1)\), and if \(X\) is not fibred by Seshadri curves for \(L\), then either \(r=2\), \(X=\mathbf P^2\) and \(L=\mathcal O_{\mathbf P^2}(1)\), or \(X\) is a surface of minimal degree in \(\mathbf P^r\) and \(L=\mathcal O_{\mathbf P^r}(1)\). A similar result for \(r=1\) was already obtained by the same authors in the quoted paper [see Theorem 3.2 in \textit{W. Syzdek} and \textit{T. Szemberg}, loc. cit.].