This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Degree of an algebraic variety" – news · newspapers · books · scholar · JSTOR(January 2021) (Learn how and when to remove this message)
In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety
with n hyperplanes in general position.[1] For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem).
The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.
The degree of a hypersurface is equal to the total degree of its defining equation. A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces.
The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations.
^In the affine case, the general-position hypothesis implies that there is no intersection point at infinity.
and 27 Related for: Degree of an algebraic variety information
Algebraicvarieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, analgebraicvariety is defined as...
specifically in algebraic geometry, the dimension ofanalgebraicvariety may be defined in various equivalent ways. Some of these definitions are of geometric...
In the mathematical field ofalgebraic geometry, a singular point ofanalgebraicvariety V is a point P that is 'special' (so, singular), in the geometric...
In algebraic geometry, a morphism between algebraicvarieties is a function between the varieties that is given locally by polynomials. It is also called...
projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by...
in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraicvariety that is also analgebraic group...
In algebraic geometry, an affine algebraic set is the set of the common zeros over analgebraically closed field k of some family of polynomials in the...
computational algebraic geometry, as they are the easiest known way for computing the dimension and the degreeofanalgebraicvariety defined by explicit...
In algebraic geometry, the function field ofanalgebraicvariety V consists of objects that are interpreted as rational functions on V. In classical algebraic...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
algebra and algebraic geometry, where a finite length may occur only when the dimension is zero. The degreeofanalgebraicvariety is the length of the...
numbers, by using algebraic methods. The notion ofalgebraic number field relies on the concept of a field. A field consists of a set of elements together...
In algebraic geometry, a projective variety over analgebraically closed field k is a subset of some projective n-space P n {\displaystyle \mathbb {P}...
is a glossary ofalgebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For...
Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space ofdegree 0 line bundles. It is the connected component of the identity...
is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or analgebraicvarietyof dimension n − 1, which...
transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension ofanalgebraicvariety is the transcendence...
{\displaystyle k(Y)} ). We see that the degreeofanalgebraic function field is not a well-defined notion. The algebraic function fields over k form a category;...
modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include...
mathematics, a rational variety is analgebraicvariety, over a given field K, which is birationally equivalent to a projective space of some dimension over...
matroid. No good characterization ofalgebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid...
In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano 1934, 1942), is analgebraicvariety that generalizes certain aspects of complete...
under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and...
in various degreesof generality. A prototype is the varietyof complete flags in a vector space V over a field F, which is a flag variety for the special...
mathematics, analgebraic surface is analgebraicvarietyof dimension two. In the case of geometry over the field of complex numbers, analgebraic surface...
Global Information