  Let $X$ be a projective variety with a torus action, which for simplicity we
assume to have dimension 1. If $X$ is a smooth complex variety, then the
geometric invariant theory quotient $X//G$ can be identifed with the symplectic
reduction $X_r$. Lerman introduced a construction (valid for symplectic
manifolds) called symplectic cutting, which constructs a manifold $X_c$, such
that $X_c$ is the union of $X_r$ and an open subset $X_{>0} \subset X$.
Moreover, there is a natural torus action on $X_c$ such that $X_r$ is a
component of the fixed locus. Using localization for equivariant cohomology,
this construction can be used to study of $X_r$.
  In this note, we give an algebraic version of this construction valid for
projective but possibly singular varieties defined over arbitrary fields. This
construction is useful for studying $X_r$ from the point of view of algebraic
geometry, using the equivariant intersection theory developed by the authors.
At the end of the paper we briefly give an adaptation of Lerman's proof of the
Kalkman residue formula and use it to give some formulas for characteristic
numbers of quotients by a torus.
