Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.