What are algebraic varieties?
Q: What are algebraic varieties?
A: Algebraic varieties are one of the central objects of study in algebraic geometry. They are defined as the set of solutions of a system of polynomial equations, over the real or complex numbers.
Q: How do modern definitions differ from the original definition?
A: Modern definitions try to preserve the geometric intuition behind the original definition while generalizing it. Some authors require that an "algebraic variety" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not.
Q: What is one difference between a variety and a manifold?
A: A variety may have singular points, while a manifold will not.
Q: What does the fundamental theorem of algebra establish?
A: The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex coefficients (an algebraic object) is determined by the set of its roots (a geometric object).
Q: What does Hilbert's Nullstellensatz provide?
A: Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets.
Q: How has this correspondence been used by mathematicians?
A: Mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory using this correspondence.
Q: What makes this particular area unique among other subareas within geometry? A: This strong correspondence between questions on algebraic sets and questions of ring theory makes this particular area unique among other subareas within geometry.
