A note on symbolic and ordinary powers of homogeneous ideals

Abstract

In this note we are interested in the graded modulesMk=I(k)/Ik and\(N_k : = \overline {I^k } /I^k \), whereI is a saturated ideal in the homogeneous coordinate ringS=K[x0,…,xn] of ℙn,I(k) is the symbolic power and\(\overline {I^k } \) is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙn.