  The aim is to give a geometric characterization of the finite generation of
the Cox ring of anticanonical rational surfaces. This characterization is
encoded in the finite generation of the effective monoid. Furthermore, we prove
that in the case of a smooth projective rational surface having a negative
multiple of its canonical divisor with only two linearly independent global
sections (e.g., an elliptic rational surface), the finite generation is
equivalent to the fact that there are only a finite number of smooth projective
rational curves of self-intersection -1. The ground field is assumed to be
algebraically closed of arbitrary characteristic.
