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.