Seshadri functions on abelian surfaces (Q2679882)

In this nicely written paper the author presents an algorithm which tells us how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. 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 the real number \[ \varepsilon(X,L;x)= \mathrm{inf} \bigg\{\frac{LC}{\mathrm{mult}_{x}(C)}: \, C \text{ is irreducible passing through } x \bigg\}. \] The first part of the paper is devoted to the aforementioned algorithm, which allows to compute the Seshadri constant and all submaximal Seshadri curves of any given nef line bundle on an abelian surface. The methods presented by the author show that the Seshadri constant can be computed entirely from Pell's bounds and elliptic curves, both of which can be computed from the intersection matrix. This phenomenon is to some extent surprising. As a nice corollary, the author shows the following. Corollary. Let \(A\) be a complex abelian surface and let \(G \subset\mathrm{Aut}(\mathrm{NS}(A))\) be the subgroup of isometries with respect to the intersection product that leave the nef cone invariant. Then the Seshadri function on \(A\) is invariant under \(G\), i.e., we have that the Seshadri constant of \(L\) is equal to the Seshadri constant for \(\phi(L)\), where \(\phi \in G\) and \(L \in\mathrm{Nef}(A)\). Furthermore, \(G\) gives rise to a decomposition of the ample cone into subcones such that \(G\) acts transitively on the set of subcones and, therefore, the Seshadri function is completely determined by the values of any such subcone. Another interesting observation is that the Seshadri function is locally piecewise linear around line bundles \(L\) with \(\varepsilon(X,L;x) < \sqrt{L^{2}}\). In this context, the author delivers an interesting proposition. Proposition. Let \(A\) be an abelian surface such that the Seshadri function is piecewise linear. Then \(A\) is either a simple abelian surface with the Picard number one or is a non-simple abelian surface with Picard number two.