Sets of points and their conductor

Let A be the coordinate ring of set X of s distinct K-rational points P1,…,Ps in PKn. The integral closure of A in its total ring of fractions has the form A¯=⊕i=1sK[ti] where K[ti] is isomorphic to the coordinate ring of Pi. The conductor of A in A¯ is the biggest ideal CX in A that coincides with its extension to A¯. Considered as an ideal in A¯, it is of the form CX=(t1d1,t2d2,…,tsds), where di is the least degree of a hypersurface of PKn which passes through all the points of X except Pi. The number di is called the conductor degree of Pi in X [J. London Math. Soc (2) 24 (1981) 85-96]. Given a set X of s distinct K-rational points in PKn, we determine the ideals of points of X which have the same conductor degree in X (Theorem 14). For a set of points X in PK2, we find a lower bound for dimK(A/CX), which depends only on the degree matrix of X (Theorem 23), and we show that this lower bound is sharp.