Concept

Projective variety

Summary
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space \mathbb{P}^n over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb{P}^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring :k[x_0, \ldots, x_n]/I is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and t
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