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Algebraic variety: Quiz

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Question 1: A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a ________.
Commutative ringSpectrum of a ringScheme (mathematics)Algebraic geometry

Question 2: When k is the ________, R, algebraic manifolds are called Nash manifolds.
Complex numberTranscendental numberReal numberIrrational number

Question 3: Let k be an ________ and let An be an affine n-space over k.
Algebraically closed fieldField (mathematics)Algebraic numberComplex number

Question 4: An affine algebraic set V is a variety ________ I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
If and only ifPropositional calculusFirst-order logicLogical connective

Question 5: Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of ________.
Affine spaceVector spaceMathematicsAffine combination

Question 6: The word "variety" is employed in the sense of a mathematical manifold, for which, in ________, cognates of the word "variety" are used.
Spanish languageFrench languageRomance languagesPortuguese language

Question 7: In ________, an algebraic variety is the set of solutions of a system of polynomial equations.
GeometrySet theoryMathematicsMathematical logic

Question 8: Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ________.
Ring (mathematics)Commutative ringRing theoryAlgebraic structure

Question 9: Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) ________.
CalculusNumber theoryAlgebraic geometryMathematics

Question 10: It is not well-defined to evaluate ƒ on points in Pn in ________.
Homogeneous coordinatesAffine transformationLinear mapBarycentric coordinates (mathematics)







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