: Mathematics: projective variety

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projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space ${\\displaystyle \\mathbb {P} ^{n}}$ 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 ${\\displaystyle \\mathbb {P} ^{n}}$ . 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
${\\displaystyle k[x_{0},\\ldots ,x_{n}]/I}$
is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Projective_variety)

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