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

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