Numerically deciding the arithmetically Cohen–Macaulayness of a projective scheme

Abstract

In numerical algebraic geometry, a witness point set W is a key object for performing numerical computations on a projective scheme X of pure dimension d>0 defined over C. If X is arithmetically Cohen–Macaulay, W can also be used to obtain information about X, such as the initial degree of the ideal generated by X and its Castelnuovo–Mumford regularity. Due to this relationship, we develop a new numerical algebraic geometric test for deciding if X is arithmetically Cohen–Macaulay using points which lie (approximately) on a general curve section C of X. For any curve, we also compute other information such as the arithmetic genus and index of regularity. Several examples are presented showing the effectiveness of this method, even when the ideal of X is unknown.