mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a hyperplane section of a subset ''X'' of

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

P^''n'' is the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of ''X'' with some

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

''H''. In other words, we look at the subset ''X''_''H'' of those elements ''x'' of ''X'' that satisfy the single linear condition ''L'' = 0 defining ''H'' as a

linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...

. Here ''L'' or ''H'' can range over the dual projective space of non-zero

linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...

s in the

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

, up to

scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...

. From a geometrical point of view, the most interesting case is when ''X'' is an algebraic subvariety; for more general cases, in

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, some analogue of the Radon transform applies. In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, assuming therefore that ''X'' is ''V'', a subvariety not lying completely in any ''H'', the hyperplane sections are

algebraic set Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

s with

irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irred ...

s all of dimension dim(''V'') − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the

Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the ...

and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the

Lefschetz pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V. Description A ''pencil'' is a particular kind of linear system of div ...

References

* Algebraic geometry {{algebraic-geometry-stub