Let \mathbb K denote an algebraically closed field. We study the relation between an ideal I\subseteq\mathbb K[x_1,\ldots,x_n] and its cross sections I_\alpha=I+\langle x_1-\alpha\rangle . In particular, we study under what conditions I can be recovered from the set I_S=\{(\alpha,I_\alpha)\,:\,\alpha\in S\} with S\subseteq\mathbb K . For instance, we show that an ideal I=\bigcap_iQ_i , where Q_i is primary and Q_i\cap\mathbb K[x_1]=\{0\} , is uniquely determined by I_S when |S|=\infty . Moreover, there exists a function B(\delta,n) such that, if I is generated by polynomials of degree at most \delta , then I is uniquely determined by I_S when |S|\geq B(\delta,n) . If I is also known to be principal, the reconstruction can be made when |S|\geq 2\delta , and in this case, we prove that the bound is sharp.