Though
generators for
ordinary powers of I are well understood when I is given in terms of its generators as I = (f_1, \ldots, f_k), it is still very difficult in many cases to determine the generators of symbolic powers of I. But in the
geometric setting, there is a clear geometric interpretation in the case when I is a
radical ideal over an
algebraically closed field of
characteristic zero. If X is an
irreducible variety whose ideal of vanishing is I, then the
differential power of I consists of all the
functions in R that vanish to order ≥
n on X, i.e. :I^{\langle n\rangle} := \{ f \in R \mid f \text{ vanishes to order} \geq n \text{ on all of }X \}. Or equivalently, if \mathbf{m}_p is the
maximal ideal for a point p\in X, I^{\langle n\rangle}=\bigcap _{p \in X} \mathbf{m}_p^n.
Theorem (Nagata, Zariski) Let I be a prime ideal in a
polynomial ring K[x_1, \ldots , x_N ] over an algebraically closed field. Then :I^{(m)}=I^{\langle m\rangle} This result can be extended to any
radical ideal. This formulation is very useful because, in
characteristic zero, we can compute the differential powers in terms of generators as: :I^{\langle m\rangle}=\left\langle f \mid \frac{\partial^\mathbf{a}f}{\partial x^\mathbf{a}}\in I \text{ for all }\mathbf{a}\in \mathbb{N} ^N \text{ where }|\mathbf{a}|=\sum_{i=1}^N a_i\leq m-1 \right\rangle For another formulation, we can consider the case when the base
ring is a
polynomial ring over a
field. In this case, we can interpret the
n-th symbolic power as the
sheaf of all function
germs over X = \operatorname{Spec}(R) \text{ vanishing to order} \geq n \text{ at } Z=V(I) In fact, if X is a
smooth variety over a
perfect field, then : I^ {(n)} = \{f \in R \mid f \in \mathbf{m}^ n \text{ for every closed point }\mathbf{m} \in Z\} == Containments ==