High Quality Content by WIKIPEDIA articles!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 n ? 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem.

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