In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace.
Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication.
From a geometrical point of view, the most interesting case is when X is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies.
In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1.
Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension.