The author shows some methods connecting descriptive set theory with definable proper forcing. He proves in (ZFC + large cardinals) the following result:
Let \(I\) be a \(\sigma \)-ideal \(\sigma \)-generated by a projective collection of closed sets. The poset \(P_I = \text{Borel}(\mathbb R) \setminus I\) is proper and adds a single real \(r_{\text{gen}}\) of an almost minimal degree: If \(V \subseteq V[s] \subseteq V[r_{\text{gen}}]\) is an intermediate model for some real \(s\), then \(V[s]\) is a Cohen extension of \(V\) or else \(V[s] = V[r_{\text{gen}}]\).