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 ________.  

Question 2: When k is the ________, R, algebraic manifolds are called Nash manifolds.  

Question 3: Let k be an ________ and let A^{n} be an affine nspace over k.  

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.  

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

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

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

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 ________.  

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

Question 10: It is not welldefined to evaluate ƒ on points in P^{n} in ________.  

