Numerically deciding the arithmetically Cohen-Macaulayness of a projective scheme

In numerical algebraic geometry, a witness point set W is a key object for performing nu-merical 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 reg-ularity. 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. Key words and phrases. Numerical algebraic geometry, witness set, arithmetically Cohen-Macaulay, Castelnuovo-Mumford regularity, arithmetic genus 2010 Mathematics Subject Classification. Primary 65H10; Secondary 14Q05, 14Q15.