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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
Hyperplane_section
Theorem in algebraic geometry
theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those
Lefschetz_hyperplane_theorem
Algebraic geometry theorem
Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields
Theorem_of_Bertini
On the existence of hyperplanes separating disjoint convex sets
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar
Hyperplane_separation_theorem
Construction in algebraic geometry
variety, and the divisors on V {\displaystyle V} are hyperplane sections. Suppose given hyperplanes H {\displaystyle H} and H ′ {\displaystyle H'} , spanning
Lefschetz_pencil
Concept in algebraic geometry
a hyperplane in P n {\displaystyle \mathbb {P} ^{n}} (because the zero set of a section of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is a hyperplane).
Ample_line_bundle
Geometrical concept
section of a solid, the cross section of an n-dimensional body in an n-dimensional space is the non-empty intersection of the body with a hyperplane (an
Cross_section_(geometry)
approach. If X is a smooth projective variety of dimension m and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called stable
Stable_vector_bundle
Problem in convex geometry
whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies
Busemann–Petty_problem
Russian-born American mathematician (1884–1972)
the École Centrale Paris. He proved theorems on the topology of hyperplane sections of algebraic varieties, which provide a basic inductive tool (these
Solomon_Lefschetz
Theorem in algebraic geometry
non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersection H ⋅ H
Hodge_index_theorem
Theory in algebraic geometry
Z ⊂ K. Weak Lefschetz axiom: For any smooth hyperplane section j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space), the maps j∗:Hi(X)→Hi(W)
Weil_cohomology_theory
American mathematician
Shiffman, Bernard; Sibony, Nessim (1981). "Average growth estimates for hyperplane sections of entire analytic sets". Math. Ann. 257 (1): 43–59. doi:10.1007/BF01450654
Bernard_Shiffman
Mapping from a Euclidean space to itself
a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension
Reflection_(mathematics)
Set of conjectures in algebraic geometry
fixed smooth hyperplane section W = H ∩ X, where X is a given smooth projective variety in the ambient projective space P N and H is a hyperplane. Then for
Standard conjectures on algebraic cycles
Standard_conjectures_on_algebraic_cycles
Vector bundle existing over a Grassmannian
dual of the hyperplane bundle or Serre's twisting sheaf O P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)} . The hyperplane bundle is the
Tautological_bundle
Type of surface in algebraic geometry
is the number of zeroes of a generic section. For a Fano surface S, a 1-form w defines also a hyperplane section {w=0} into P4 of the cubic F. The zeros
Fano_surface
Set of methods for supervised statistical learning
hyperplane. This is called a linear classifier. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane
Support_vector_machine
hypersurfaces, the generic such surface has no curve on it apart from the hyperplane section. In more modern language, the Picard group is infinite cyclic, other
Max Noether's theorem on curves
Max_Noether's_theorem_on_curves
Characteristic class in algebraic topology
\xi \in H^{2}({\mathbb {C} }P^{n})} be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent
Todd_class
H=\operatorname {Proj} (S/(I,f)).} If f is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection
Scheme-theoretic_intersection
Concept in algebraic geometry
of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety
Local_cohomology
1960–67 foundational treatise on algebraic geometry by Alexander Grothendieck
edition essentially complete; some changes made in last sections; the section on hyperplane sections made into the new Chapter V of second edition (draft
Éléments de géométrie algébrique
Éléments_de_géométrie_algébrique
Term in mathematics
of dimension n in ℂℙn+m are the intersection of hyperplane sections, we can use the Lefschetz hyperplane theorem to deduce that Hj(X)=ℤ for j<n. In addition
Complete_intersection
{Q} ).} Here η {\displaystyle \eta } is the fundamental class of a hyperplane section, f ∗ {\displaystyle f_{*}} is the direct image (pushforward) and R
Decomposition theorem of Beilinson, Bernstein and Deligne
Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
Group of Italian mathematicians who studied birational geometry (c. 1885–1935)
system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many modern techniques
Italian school of algebraic geometry
Italian_school_of_algebraic_geometry
Branch of geometry
is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete
Contact_geometry
Theorem on extension of bounded linear functionals
guise the theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem and has numerous uses in convex geometry. The theorem
Hahn–Banach_theorem
Subspace defined by a polynomial of degree 2 over a field
If q is the product of two linear forms, then X is the union of two hyperplanes. It is common to assume that n≥1 and q is irreducible, which excludes
Quadric_(algebraic_geometry)
Set of statistical processes for estimating the relationships among variables
the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical
Regression_analysis
Path of an object through spacetime
related by dudτ=dwdτ, then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve
World_line
Sphere-like surface
viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because For an ovoid O {\displaystyle {\mathcal {O}}} and a hyperplane ε {\displaystyle
Ovoid_(projective_geometry)
American mathematician
Andreotti, Aldo; Frankel, Theodore (1959). "The Lefschetz theorem on hyperplane sections". Annals of Mathematics. Second Series. 69 (3): 713–717. doi:10.2307/1970034
Theodore_Frankel
Algebraic structure in linear algebra
any subspace of dimension 1 less, i.e., of dimension n−1 is called a hyperplane. The counterpart to subspaces are quotient vector spaces. Given any subspace
Vector_space
Concept in linear algebra
a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958
Householder_transformation
Regular tessellation in 4D Euclidean space
maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such (4-dimensional) 24-cell lies in the hyperplane). Accordingly
24-cell_honeycomb
American mathematician (1916–1988)
to algebraic geometry. In 1950, he published a paper called The hyperplane sections of normal varieties, which has proved fundamental in later advances
Abraham_Seidenberg
British mathematician (born 1964)
Ph.D. from Harvard University in 1991. His dissertation, On the Hyperplane Sections of a Variety in Projective Space, was supervised by Joe Harris. McKernan
James_McKernan
Generalizations of codimension-1 subvarieties of algebraic varieties
An over k is equal to zero. Since projective space Pn over k minus a hyperplane H is isomorphic to An, it follows that the divisor class group of Pn is
Divisor_(algebraic_geometry)
Scottish mathematician
in 1979. McQuillan, Michael Liam (1999). "Holomorphic curves on hyperplane sections of 3-folds". Geometric and Functional Analysis. 9 (2): 370–392. doi:10
Michael McQuillan (mathematician)
Michael_McQuillan_(mathematician)
Geometric transformation that preserves lines but not angles nor the origin
that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane. A generalization of an affine transformation
Affine_transformation
Space formed by the ''n''-tuples of real numbers
vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra
Real_coordinate_space
Sum of terms, each multiplied with a scalar
non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes
Linear_combination
1922a, vol 1, p. 62) model 1. A variety whose points (or sometimes hyperplane sections) correspond to elements of some family. Similar to what is now called
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Equation that does not involve powers or products of variables
More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n
Linear_equation
Mathematical concept
space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space. Suppose that the eye of the artist is the origin in Rn+1
Complex_projective_space
Property of a mathematical space
that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety. An algebraic set being
Dimension
Mathematical theorem of complex manifolds
Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69 (3): 713–717, doi:10.2307/1970034
Andreotti–Frankel_theorem
Invariant measure of fractal dimension
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Hausdorff_dimension
Type of geometric algebra
euclideanly orthogonal to (−1,a,b)—i.e., a plane; or in n dimensions, a hyperplane through the origin. This would cut another plane not through the origin
Conformal_geometric_algebra
Geometric arrangements of points, foundational to Lie theory
the set Φ {\displaystyle \Phi } is closed under reflection through the hyperplane perpendicular to α {\displaystyle \alpha } . (Integrality) If α {\displaystyle
Root_system
vertex figure. It is self-dual, and its dual coincides with itself. Hyperplane sections of this honeycomb include 3-dimensional honeycombs . The honeycomb
Witting_polytope
calculus) Pierre Samuel, Sections hyperplanes des variétés normales, d'après A. Seidenberg (algebraic geometry, hyperplane sections, normal variety) Jacques
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
Fundamental space of geometry
space of dimension n is a set of n + 1 points that are not contained in a hyperplane. An affine basis defines barycentric coordinates for every point. Many
Euclidean_space
Economic Model
Define the price hyperplane ⟨p,q⟩=⟨p,∑jyj⟩. Since it's a supporting hyperplane of 𝑃𝑃𝑆, and 𝑃𝑃𝑆 is a convex cone, the price hyperplane passes the origin
Arrow–Debreu_model
Mathematical object
intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection
3-sphere
Concept in mathematics
generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant
Complex_reflection_group
Mathematical algorithm
then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search
Coordinate_descent
Four-dimensional number system
orientations of key-frames in computer graphics. For the remainder of this section, i, j, and k will denote both the three imaginary basis vectors of ℍ and
Quaternion
Multidimensional search tree for points in k dimensional space
generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the
K-d_tree
Function of a polynomial's coefficients that gives information about its roots
the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected indeterminate. For example
Discriminant
Surface in 3D space defined by an implicit function of three variables
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Implicit_surface
Abstraction of linear independence of vectors
r-1} is called a hyperplane, or co-atoms or copoints. These are the maximal proper flats; that is, the only superset of a hyperplane that is also a flat
Matroid
Concept in linear algebra
a⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both proj b a {\displaystyle \operatorname
Vector_projection
Solvability theorem for finite systems of linear inequalities
than 90°. The hyperplane normal to this vector has the vectors ai on one side and the vector b on the other side. Hence, this hyperplane separates the
Farkas'_lemma
Coordinate system using perpendicular axes
the signed distances from the point to n mutually perpendicular fixed hyperplanes. Cartesian coordinates are named for René Descartes, whose invention
Cartesian_coordinate_system
Overview of and topical guide to geometry
Convex Convex hull Coxeter group Euclidean distance Homothetic center Hyperplane Lattice Ehrhart polynomial Leech lattice Minkowski's theorem Packing Sphere
Outline_of_geometry
Locus of the zeros of a polynomial of degree two
trivial linear equation which defines a hyperplane. Hence P ⊥ {\displaystyle P^{\perp }} is either a hyperplane or P {\displaystyle {\mathcal {P}}} . For
Quadric
Smooth approximation of one-hot arg max
linear constraint that all output sum to 1, meaning that it lies on a hyperplane. Along the main diagonal (x,x,…,x), softmax is just the uniform distribution
Softmax_function
Completion of the usual space with "points at infinity"
any n + 1 of them are independent; that is, they are not contained in a hyperplane. If V is an (n + 1)-dimensional vector space, and p is the canonical projection
Projective_space
Concept in mathematical optimization
{\alpha } )} . Since the idea of this approach is to find a supporting hyperplane on the feasible set Γ = { x ∈ X : g i ( x ) ≤ 0 , i = 1 , … , m } {\displaystyle
Karush–Kuhn–Tucker_conditions
Statistical method
example, the hyperplane is just a 2-dimensional plane defined by the two factor vectors. The projection of the data vectors onto the hyperplane is given by
Factor_analysis
Non-orientable surface with one edge
strip, one that is fully four-dimensional and for which all cuts by hyperplanes separate it into two parts that are topologically equivalent to disks
Möbius_strip
Concept in projective geometry
pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double
Duality_(projective_geometry)
Mathematical description of spacetime used in relativity
Suppose x ∈ M is timelike. Then the simultaneous hyperplane for x is {y : η(x, y) = 0}. Since this hyperplane varies as x varies, there is a relativity of
Minkowski_spacetime
Mathematical concept
plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity
Infinity
In mathematics, invariant of square matrices
and "coplanar" and "tetrahedron" replaced with "belonging to the same hyperplane" and "simplex". Besides the change of matrix dimensions, the main change
Determinant
Australian and American mathematician (born 1975)
singular integral operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to Lp
Terence_Tao
and such that they are compatible. Let L be a line bundle on X and s a section of it such that s : O X ↪ L {\displaystyle s:{\mathcal {O}}_{X}\hookrightarrow
Relative effective Cartier divisor
Relative_effective_Cartier_divisor
Bowers) The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated
Stericated_5-simplexes
Motion of a certain space that preserves at least one point
other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of
Rotation_(mathematics)
Notation system for crystal lattice planes
than three dimensions, involving the intersection of more than one such hyperplane.) Crystal structure Crystal habit Geometrical crystallography before X-rays
Miller_index
Geometric space with five dimensions
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Five-dimensional_space
Invariance of operations under geometric translation
| n ∈ Z} = p + Z a. Fundamental domains are e.g. H + [0, 1] a for any hyperplane H for which a has an independent direction. This is in 1D a line segment
Translational_symmetry
Algorithm for generating pseudo-randomized numbers
points in an n-dimensional space, the points will lie on, at most, n√n!⋅m hyperplanes (Marsaglia's theorem, developed by George Marsaglia). This is due to
Linear_congruential_generator
Coordinate system that is defined by points instead of vectors
the complement of a hyperplane. The projective completion is unique up to an isomorphism. The hyperplane is called the hyperplane at infinity, and its
Barycentric_coordinate_system
Mathematical theorem
{1, 2, . . . , n}, the set {(s1,…,sn)∈Is,tn∣si=sj} is contained in a hyperplane, hence by an application of Fubini's theorem its measure with respect
Grönwall's_inequality
Unbounded quadric surface
w = r } , {\displaystyle H_{r}=\lbrace p\ :\ w=r\rbrace ,} which is a hyperplane. Then P ∩ H r {\displaystyle P\cap H_{r}} is the sphere with radius r
Hyperboloid
All points whose relative distances to two circles are same
hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two non-concentric hyperspheres. Michel Chasles, C. H. Schnuse: Die
Radical_axis
Family of polynomials
fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplane; these latter
Gaussian_binomial_coefficient
Geometric space with four dimensions
plane, two-dimensional beings in this plane would only observe a cross-section of the three-dimensional object within this plane. For example, if a sphere
Four-dimensional_space
− 1 ) {\displaystyle {\mathcal {O}}_{X}(-1)} . It is also called the hyperplane bundle. O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} 1. If D is an
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Group that admits a formal description in terms of reflections
given two hyperplanes meeting at an angle of π / k {\displaystyle \pi /k} , the composite of the two reflections about these hyperplanes is a rotation
Coxeter_group
Quantum search algorithm
The operator U ω {\displaystyle U_{\omega }} is a reflection at the hyperplane orthogonal to | ω ⟩ {\displaystyle |\omega \rangle } for vectors in the
Grover's_algorithm
parallel planes in three-dimensional Euclidean space or between two hyperplanes in higher dimensions. A slab can also be defined as a set of points:
Slab_(geometry)
Pseudorandom number generator
absolute values of all the coefficients of the hyperplanes in standard form. That is, if the hyperplanes are of the form Ax1 + Bx2 + Cx3 = some integer
RANDU
Four-dimensional geometrical object
4-space. The maximal cross-section of the runcinated 5-cell with a 3-dimensional hyperplane is a cuboctahedron. This cross-section divides the runcinated
Runcinated_5-cell
Russian mathematician (born 1966)
exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes.[P89] As such, his construction provided further obstruction to the extension
Grigori_Perelman
Geometric figure
on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit
Unit_hyperbola
Measure of relativistic velocity
orthogonality whatever the value of w. The appropriate x-axis is the hyperplane of simultaneity corresponding to rapidity w at the origin. The hyperbola
Rapidity
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