On a notion of toric special linear systems

TL;DR

This paper investigates special linear systems on complete toric varieties with an invariant point, introducing a new expected dimension concept and analyzing degenerations induced by toric degenerations.

Contribution

It characterizes toric varieties with dense automorphism orbits and proposes a novel expected dimension for linear systems considering toric subvarieties.

Findings

Characterization of such toric varieties via their fans

Introduction of a new expected dimension for linear systems

Analysis of degenerations induced by toric degenerations

Abstract

In this note we study linear systems on complete toric varieties $X$ with an invariant point, whose orbit under the action of the automorphism group of $X$ contains the dense torus $T$ of $X$. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.