Computing the equations of a variety

Abstract:

Let $X \subset {\mathbb {P}^n}$ be a projective variety or subscheme, and let $\mathcal {F}$ be an invertible sheaf on $X$. A set of global sections of $\mathcal {F}$ determines a map from a Zariski open subset of $X$ to ${\mathbb {P}^r}$. The purpose of this paper is to find, given $X$ and $\mathcal {F}$, the homogeneous ideal defining the image in ${\mathbb {P}^r}$ of this rational map. We present algorithms to compute the ideal of the image. These algorithms can be implemented using only the computation of Gröbner bases and syzygies, and they have been implemented in our computer algebra system Macaulay. Our methods generalize to include the case when $X$ is an arbitrary projective scheme and $\mathcal {F}$ is generically invertible.