2025/10/19 by Batavia, Manav, Sundaram, Kesavan Mohana, Pandey, Vaibhav +1
#13A50 #13D45 #14F20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13C40 #Secondary 13A35
paper · doi:10.48550/arxiv.2510.17049
The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic m-residual intersection of an ideal generated by n indeterminates for all m≥ n and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero.