Prof. Qing Liu
Let $X$ be a projective scheme over an affine base. We develop a technique for proving the existence of closed subschemes $H$ with various favorable properties. We offer several applications of this technique, including the existence of hypersurfaces in $X$ containing a given closed subscheme and intersecting properly a given closed set, and the existence of finite quasi-sections. This is joint work with O. Gabber and D. Lorenzini.