A ug 1 99 6 Algebraic cuts

Let X be a projective variety with a linearized action of an algebraic group G. If X is smooth, and the ground field is C, then the geometric invariant theory quotient X//G can be identified with a quotient constructed using symplectic geometry, the “reduced space” Xr. This result, due to Mumford, Guillemin and Sternberg, connects geometric invariant theory and symplectic geometry. Suppose now that G = T is a torus, which for simplicity we will take to have dimension 1. In [L], Lerman introduced a construction called symplectic cutting, which constructs a manifold Xc related to X, with a T -action, and embeds Xr as a component of the fixed point locus X T c . The complement of Xr in Xc can be identified with an open submanifold of X, so the other components of X c are certain components of X T . The components of X c are linked by the localization theorem in equivariant cohomology. Thus, from knowledge of X one can (via the cut space) deduce results about Xr. For example, Lerman uses cutting to prove a residue formula due to Kalkman, which is closely connected to the localization theorem of Jeffrey-KirwanWitten ([G-K]). The purpose of this paper is to present an algebraic version of Lerman’s construction, called algebraic cutting, which is valid over arbitrary ground fields and for possibly singular schemes. This is useful in studying Xr from the point of view of algebraic geometry, using the equivariant intersection theory developed in [E-G] in place of equivariant cohomology. For example, Lerman’s proof of Kalkman’s formula becomes valid for smooth schemes over