A modest extension of Reider’s Theorem on ample divisors on a surface

The derived category is a useful tool for studying classical problems about algebraic surfaces. For example, a wall-crossing argument for moduli of derived objects was used by Arend Bayer to give a new proof of Lazarsfeld’s theorem on the Brill-Noether generality of curves on a K3 surface with Picard number one. This was recently extended by Farkas, Feyzbakhsh and Rojas to the Picard rank two case. Here, we use a non-wall-crossing argument to give inequalities of the same form as those of Reider’s theorem to obtain information about the equations that cut out the surface. This is joint work with my students Jonathon Fleck, Liebo Pan and Joseph Sullivan.