Algebraic geometry is a branch of mathematics that studies geometric objects defined by polynomial equations. It connects algebra with geometry by interpreting solutions as points of a space and studying that space through its algebraic functions. Classical objects include curves, surfaces, and higher-dimensional varieties; modern foundations extend the subject to schemes, which also encode infinitesimal structure and arithmetic information. Its questions concern the structure, classification, intersections, and transformations of these objects, rather than merely finding individual solutions. (jmilne.org)
Polynomial equations and varieties
Let (k) be a field, and let (f_1,\ldots,f_r) be polynomials in (n) variables. Their common zero set in affine space is [ V(f_1,\ldots,f_r)= {a\in k^n:f_i(a)=0\text{ for every }i}. ] For example, (y-x^2=0) defines a parabola. More generally, polynomial zero sets are called algebraic sets. Under a common classical convention, an algebraic variety is an irreducible algebraic set: it cannot be expressed as the union of two proper closed algebraic subsets. Terminology varies between treatments. (jmilne.org)
Equations generate an ideal in the polynomial ring (k[x_1,\ldots,x_n]). Conversely, the polynomials vanishing on an algebraic set form an ideal. Over an algebraically closed field, Hilbert’s Nullstellensatz establishes an inclusion-reversing correspondence between algebraic sets and radical ideals. Thus geometric containment becomes an algebraic relation between ideals. The coordinate ring [ k[V]=k[x_1,\ldots,x_n]/I(V) ] records polynomial functions on (V), making algebraic calculations a means of investigating geometry. (jmilne.org)
Projective geometry and maps
Projective geometry supplies spaces in which points have homogeneous coordinates ([x_0:\cdots:x_n]), with nonzero coordinate tuples identified under multiplication by a common nonzero scalar. Homogeneous polynomials define projective algebraic sets. Affine space appears as a coordinate chart, while its projective completion includes points outside that chart, often described as points at infinity. This framework treats many intersection problems more uniformly than affine geometry does. (jmilne.org)
Maps between algebraic spaces must respect their algebraic structures. Regular maps are defined everywhere on their domains; rational maps are represented by regular maps on dense open subsets, with representatives identified when they agree appropriately. A rational map therefore need not extend across every point. This distinction is central when comparing spaces through their functions rather than through a particular embedding. (stacks.math.columbia.edu)
Two integral varieties are birationally equivalent when they contain isomorphic dense open subsets. This is weaker than isomorphism: birational models may differ along exceptional subsets. For curves, normalization and extension theorems give especially strong links between function fields and nonsingular projective models. Higher-dimensional birational geometry involves additional phenomena because modifications can change entire divisors or more complicated subspaces. (math.stanford.edu)
Local structure and global invariants
Dimension, tangent spaces, and singularities describe the local geometry of an algebraic space. For an algebraic set over an algebraically closed field, dimension can be expressed through chains of irreducible closed subsets. Tangent spaces are obtained by linearizing equations at a point. A Jacobian matrix provides a computational description of these linear conditions, linking local geometry with linear algebra. Singular points are places where the expected nonsingular local structure fails. (math.stanford.edu)
Global questions require tools that relate information from different open subsets. Sheaves organize locally defined functions and other algebraic data, while sheaf cohomology measures aspects of their global compatibility. For projective curves, the Riemann–Roch theorem relates spaces of sections to degree and genus. Genus distinguishes important classes of curves, but does not alone determine a curve’s isomorphism class. (math.stanford.edu)
Schemes and modern foundations
A scheme is a space locally modeled on affine schemes (\operatorname{Spec}(A)). For a commutative ring (A), the points of its spectrum are prime ideals. These points carry the Zariski topology and a structure sheaf of rings. A scheme is therefore not simply a point set: its locally defined algebraic functions are part of the object. Scheme morphisms preserve this combined topological and algebraic structure. (stacks.math.columbia.edu)
Schemes retain information that ordinary zero sets discard. For example, the rings (k[x]/(x)) and (k[x]/(x^2)) define different schemes despite having the same underlying point set; the second retains a nonzero nilpotent element. Alexander Grothendieck’s foundational work incorporated such structures and emphasized families and functorial constructions. This approach allows geometry over arbitrary fields and more general rings, including arithmetic spaces built from the integers. (jmilne.org)
Computation and research problems
Computational algebraic geometry studies effective procedures for manipulating equations and extracting geometric information. Gröbner bases transform generators of polynomial ideals into forms suited to ideal-membership tests, elimination, and related calculations. They extend some of the computational roles played by row reduction in linear algebra, although polynomial computations can require much greater resources. Complexity depends strongly on the equations and the chosen monomial ordering. (arxiv.org)
Symbolic methods manipulate exact algebraic expressions. Numerical algebraic geometry instead uses approximate solutions and techniques such as homotopy continuation and monodromy to investigate polynomial systems. These approaches can complement one another: exact calculations establish algebraic relations, while numerical methods help explore solution sets that are difficult to handle symbolically. Research problems include determining components, dimensions, singularities, and intersections of spaces presented by equations. (pi.math.cornell.edu)