Symbolic powers of monomial curves

Abstract

The answer is positive for ideal-theoretic local complete intersection (e.g., smooth) curves [S], [MK], and for arbitrary curves if char(k) is non-zero [C-N]. Problem 1 remains open for “most” singular curves over (say) C. In fact, we look at an algebraic problem related to Problem 1. Let P be a prime ideal in a commutative noetherian ring A. Denote by S(P) the direct sum ora0 P”) of the symbolic powers of P (see Section 0 for definitions), which is canonically a graded A-algebra. It is called the symbolic Rees algebra of P (or of X, if X is the variety defined by P in Spec A).