Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

Abstract

We consider flags E• = {X ⊃ E ⊃ {q}}, where E is an exceptional divisor defining a non-positive at infinity divisorial valuation νE of a Hirzebruch surface Fδ , q a point in E and X the surface given by νE , and determine an analogue of the Seshadri constant for pairs (νE , D), D being a big divisor on Fδ . The main result is an explicit computation of the vertices of the Newton- Okounkov bodies of pairs (E•, D) as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.