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The Veronese surface arises naturally in the study of conics.A conic is a degree 2 plane curve, thus defined by an equation: + + + + + = The pairing between coefficients (,,,,,) and variables (,,) is linear in coefficients and quadratic in the variables; the Veronese map makes it linear in the coefficients and linear in the monomials.
The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor).
Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces; Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers , an algebraic surface has complex dimension two (as a complex manifold , when it is non-singular ) and so of dimension four as a smooth manifold .
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
An algebraic surface is an algebraic variety of dimension two. The Enriques-Kodaira classification gives an overview of the possibilities. Over the complex numbers, a non-singular algebraic surface is an example of a 4-manifold
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
Equivalent conditions are that the line bundle is big, or that the d-canonical map is generically injective (that is, a birational map to its image) for d sufficiently large. For example, a variety with ample canonical bundle is of general type. In some sense, most algebraic varieties are of general type.