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

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 , 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. 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. [ABSTRACT FROM AUTHOR]