  Chemical reaction network theory is a field of applied mathematics concerned
with modeling chemical systems, and can be used in other contexts such as in
systems biology to study cellular signaling pathways or epidemiology to study
the effect of human interaction on the spread of disease. In this paper, we
seek to understand a chemical reaction network's equilibrium points through the
lens of algebraic geometry by computing the positive part of the steady-state
variety defined by polynomial equations arising from the assumption of
mass-action kinetics. We provide a systematic classification of positive
steady-state varieties produced by 2-species, 2-reaction networks, grounded in
combinatorial and algebraic properties. While some (restricted) techniques
exist to fully understand the ideal defining the positive steady-state variety,
this computation presents a significant challenge in general. Our
classification theorems provide a simplification of previous criteria, and aim
to provide a foundation for future analysis of larger networks.
