Seshadri constants on principally polarized abelian surfaces with real multiplication (Q2118189)

In the paper under review, the authors study the Seshadri constants on principally polarized abelian surfaces with real multiplication. Recall that for an ample line bundle \(L\) on a smooth projective variety \(X\), the Seshadri constant of \(L\) at a point \(x \in X\) is by definition the real number \[\varepsilon(L,x) = \mathrm{inf} \bigg\{ \frac{LC}{\mathrm{mult}_{x}(C)} \, : \, C \text{ irreducible curve through } x \bigg\}.\] If this invariant is independent of the chosen point \(x\), one writes \(\varepsilon(L)\). Let us recall that on complex abelian surfaces with Picard number \(\rho=1\) the Seshadri constants are fully understood. The main aim of the paper under review is to understand the Seshadri function \[\varepsilon \, : \, \mathrm{Amp}(X) \rightarrow \mathbb{R}, \quad L \mapsto \varepsilon(L)\] which associates to each ample line bundle its Seshadri constant. Let us introduce the following definition. Definition. Let \(I \subset \mathbb{R}\) be an interval. A function \(f: I \rightarrow \mathbb{R}\) is called broken linear if it is continuous and there is a non-empty and nowhere dense subset \(M \subset I\) such that the following holds: i) Around every point of \(I\setminus M\) there is an open interval contained in \(I\setminus M\) on which \(f\) is linear. ii) If \(I_{1}\), \(I_{2}\) are maximal open subintervals of \(I\) on which \(f\) is linear, then \(I_{1}, I_{2}\) are contained in \(I\setminus M\), and \(I_{1}, I_{2}\) are not adjacent to each other. The first result of the paper under review can be formulated as follows. Theorem A. Let \(X\) be a complex principally polarized abelian surface whose endomorphism ring is isomorphic to \(\mathbb{Z}[\sqrt{e}]\) for some non-square integer \(e>0\). Then the Seshadri function of \(X\) is broken linear. Another result presents an interesting structure associated with the Seshadri function. Theorem B. There exists a decomposition of the ample cone into infinitely many subcones \(\mathcal{C}_{k}\) for \(k \in \mathbb{Z}\) such that the group \(G\) of isometries of \(\mathrm{NS}(X)\) that leave the Seshadri function on \(\mathrm{Amp}(X)\) invariant acts transitively on the set of subcones. In particular, the values of the Seshadri function on any subcone of the subcones \(\mathcal{C}_{k}\) completely determine the Seshadri function on the entire ample cone.