The graph of monomial ideals

Abstract

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K [ x 1 ,..., x n ] . The definition involves Gröbner bases or the action of the algebraic torus ( K * ) n . We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter case, the edges correspond to generalized bistellar flips.