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...Hartshorne s algebraic geometry varieties anna antal dimacs reu at rutgers university june some initial denitions let k be a xed algebraically closed eld n an ane space over denoted is the set of all tuples elements p with called point and i are coordinates polynomial ring in variables we interpret element f as function given t zero z for sets zariski topology asubset y if there exists subset asuch that proposition union two intersection any family subsets empty whole dene on by taking open to complements this irreducibility nonempty topological x irreducible it cannot expressed proper example line every ideal principle can show using remainder theorem so zeros single since nonzero written c thus because its only nite yet innite now variety induced...