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In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1 , which is embedded in an ambient space of dimension n , generally a Euclidean space , an affine space or a projective space . [ 1 ]
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface =
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.
Similarly, if M is a hypersurface in a Riemannian manifold N, then the principal curvatures are the eigenvalues of its second-fundamental form. If k 1 , ..., k n are the n principal curvatures at a point p ∈ M and X 1 , ..., X n are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of M at p is given by
Geometrically, each F i defines a hypersurface; the intersection of these hypersurfaces should be V. The intersection of n − m hypersurfaces will always have dimension at least m, assuming that the field of scalars is an algebraically closed field such as the complex numbers.
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial = + + + + Computing the partial derivatives of gives the four polynomials = = = = = Since the only points where they vanish is given by the coordinate axes in , the vanishing locus is empty since [::::] is not a point in .
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number h n,0 (equal to h 0,n by Serre duality), that is, the dimension of the canonical linear system plus one.
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