Algebraic geometry is the study of geometric objects defined by polynomial equations, and its central idea is that algebra is the language in which such objects become computable. A zero set of one equation in three variables is a surface, one equation in two variables is a curve, and a collection of equations cuts out a more general object called a variety, which need not be smooth and need not be connected. The passage between the picture and the equation is the whole content of the subject, and the fundamental theorem of algebra guarantees that over the complex numbers the geometry is generous: a single equation of degree a has exactly a solutions counted with multiplicity. Over the reals the situation is poorer, and the subject of real algebraic geometry, opened by Tarski with his decision procedure for real closed fields, develops tools for deciding whether a system of polynomial inequalities has a solution, which is the foundation of the quantifier elimination methods used in computer aided geometric design and in the analysis of geometric constraint systems. The central theorems count, and they connect geometry to algebra in a way that is easy to remember and hard to forget. Bezout's theorem asserts that two plane curves of degrees a and b, with no common component and counted with multiplicity, meet in exactly a times b points in the complex projective plane, and it is the reason that a conic and a cubic have at most six common points and a pair of quartics at most sixteen. The argument uses elimination, and the elimination itself is an algebraic process of the same family as the Euclidean algorithm. The dimension theorem, which in one form says that a system of polynomials in n variables whose variety has positive dimension admits a polynomial that vanishes on it, is the algebraic expression of the fact that equations can become redundant. Noether normalisation, which says that for a positive dimensional variety there is a finite map from the space onto affine space that is finite and preserves dimension, is the workhorse of reduction theory and of the theory of birational geometry. It also explains why the classification of curves of a given genus is a difficult problem even when the genus is small. Curves are the most accessible family. A nonsingular plane curve of degree a has a genus of the order of the quantity obtained from a factorial expression in a, namely one half of the product of a and the numbers a minus 1 and a minus 2, adjusted by the Plucker formula for singularities, so a conic has genus zero, a smooth cubic has genus one, and a quartic of maximal genus has genus three. Genus zero curves are rational, meaning that points on them can be written down by rational functions of a parameter, and this is the classical answer to the question of when a conic can be solved by square roots alone. The arithmetic of a curve is encoded in a sequence of numbers attached to rational functions on it, their degrees over a chosen point, and these numbers obey a pattern. Genus one curves are the elliptic curves, they carry a group law on their rational points, and that law turns Diophantine problems into group computations, which is the practical reason for the subject's importance in cryptography. Elliptic curves have the form of a plane cubic with a marked point of infinite order, and the theory is remarkably explicit. In the Weierstrass form of the equation of an elliptic curve, the parameters a and b are real or rational or over a finite field, and the discriminant is a combination of powers of a and b, and the curve has three distinct roots of the associated cubic exactly when the discriminant is not a set 4 equals 0. Two such curves are isomorphic if and only if a certain invariant, the so called modular function, takes the same value, and the value has a simple special case at 1728 and at 0, corresponding to the equianharmonic and lemniscatic cases, in which the extra automorphisms of the curve account for the coincidences. Over the rationals, Mordell proved that the group of rational points is finitely generated, so the Diophantine equation can be attacked by computing a descent, which reduces the question to a finite search, and the theorem that Fermat used to prove the impossibility of a solution to the equation whose terms are cubes of integers is an instance of exactly this descent. Singularities form a second pillar. A point at which the gradient of the defining equations vanishes is singular, and the local behaviour is classified into types, of which the ordinary node and the ordinary cusp are the simplest. Each singularity contributes a positive amount to the arithmetic genus relative to the genus of a smooth curve of the same degree, and the contribution is the delta invariant, which is a measure of how far the singularity is from being smooth. A curve with a single node, for instance, has arithmetic genus one more than the corresponding smooth curve has geometric genus. Milnor and Mumford showed that delta is finite, that it is the number of double points needed to smooth the curve, and that a finite bound in terms of the degree exists. From these facts, and from the classification of the topological types of isolated plane curve singularities into seven families, the subject gains a powerful language for describing the degenerations that appear in the solution of enumerative problems, and it is the language in which the braid monodromy of a family of curves is recorded. Algebraic geometry also supports an expanding collection of algorithms, implemented in systems such as Singular, Macaulay2, Magma and Sage, which compute Groebner bases, factor polynomials, classify singularities and settle questions about varieties, and these tools are now standard in the analysis of dynamical systems and in the statistics of discrete data.
