  Let K denote an algebraically closed field. We study the relation between an
ideal I in K[x1,...,xn] and its cross sections I_a=I+<x1-a>. In particular, we
study under what conditions I can be recovered from the set I_S={(a,I_a):a in
S} with S a subset of K. For instance, we show that an ideal I=cap_i Q_i, where
Q_i is primary and Q_i cap K[x1]={0}, is uniquely determined by I_S when S is
infinite. Moreover, there exists a function B(d,n) such that, if I is generated
by polynomials of degree at most d, then I is uniquely determined by I_S when
|S|>=B(d,n). If I is also known to be principal, the reconstruction can be done
when |S|>=2d, and in this case, we prove that the bound is sharp.
