Critical \(k\)-very ampleness for abelian surfaces (Q272916)

A line bundle \(L\) on a variety \(V\) is called \(k\)-very ample if the natural map \(\Gamma(V,L) \to \Gamma(V/I_X)\) is surjective for the ideal sheaf \(I_X\) of any \(0\)-dimensional subscheme \(X\) of length \(\leq k+1\) in \(V\). One can define the integer valued function \(\phi(L)\) on the ample cone \(\mathrm{Amp}(X)\) to be the maximum value of \(k\) such that \(L\) is \(k\)-very ample but not \((k+1)\)-very ample.NEWLINENEWLINEThis article gives a polynomial formula of \(\phi(L^n)\) in terms of \(n\) in the case of a polarized abelian surface \((S,L)\) with Picard rank \(1\). The result is \(\phi(L^n) = 2(n-1)d-2\) with \(d:=c_1(L)^2/2\).NEWLINENEWLINEThe first observation to derive the formula is the equivalence of \(k\)-very ampleness and the weak index theorem with respect to the Fourier-Mukai transform with the Poincaré bundle as the kernel. The second point is that under certain situation, the weak index theorem is verified by Bridgeland stability, and one can use Fourier-Mukai transforms to find line bundles which enjoy the weak index theorem.NEWLINENEWLINEThis article gives a good application of Fourier-Mukai transforms and Bridgeland stability conditions. Although some knowledge on stability conditions for abelian surfaces is assumed, it is written quite clearly and briefly, and readers will find it very helpful to have experience on moduli spaces of stable objects for Bridgeland stability conditions.