1.1 Historical overview and fundamental concepts

Algebraic geometry blends algebra and geometry, tracing its roots to ancient Greek studies of conic sections. It evolved through contributions from Descartes, Fermat, and others, culminating in modern advancements by Grothendieck and Serre.

This field explores varieties, which are solution sets of polynomial equations. It uses concepts like ideals, rings, and Zariski topology to study geometric objects algebraically, connecting to other math areas and finding applications in physics, computer science, and biology.

History of algebraic geometry

Early developments in algebraic geometry

• Algebraic geometry has its roots in ancient Greek mathematics, with the study of conic sections (ellipses, parabolas, hyperbolas) and the work of Apollonius of Perga on the classification of curves
• In the 17th century, René Descartes introduced the concept of coordinate geometry, laying the foundation for the merger of algebra and geometry by representing geometric objects using algebraic equations
• Pierre de Fermat and Blaise Pascal further developed analytic geometry, which allowed for the study of geometric problems using algebraic methods
• Joseph-Louis Lagrange and Leonhard Euler made significant contributions to the study of algebraic curves and surfaces in the 18th century

Advancements in the 19th and 20th centuries

• The 19th century saw significant advancements in algebraic geometry, with contributions from mathematicians such as Bernhard Riemann, who introduced the concept of Riemann surfaces, and David Hilbert, who developed the theory of algebraic invariants
• Julius Plücker introduced the concept of homogeneous coordinates, which allowed for the study of projective geometry and its connection to algebraic geometry
• In the 20th century, Oscar Zariski and André Weil made crucial contributions to the field, developing the modern approach to algebraic geometry using abstract algebraic techniques such as commutative algebra and algebraic topology
• Alexander Grothendieck revolutionized algebraic geometry in the 1960s with his development of scheme theory, which provided a unified framework for studying geometric objects using commutative algebra and category theory
• Jean-Pierre Serre, Pierre Deligne, and others further developed and expanded upon Grothendieck's ideas, leading to major advances in the field and its applications to other areas of mathematics

Concepts in algebraic geometry

Varieties and their properties

• An affine variety is the set of solutions to a system of polynomial equations in a finite-dimensional affine space over an algebraically closed field (complex numbers)
• A projective variety is the set of solutions to a system of homogeneous polynomial equations in a projective space, which allows for the study of points at infinity and the intersection of varieties
• Varieties can be classified by their dimension, which is the maximum number of independent parameters needed to describe points on the variety (curves have dimension 1, surfaces have dimension 2)
• The singular points of a variety are those where the tangent space has a higher dimension than the variety itself, while non-singular points are called smooth
• Birational equivalence is a key concept in algebraic geometry, where two varieties are considered equivalent if there exists a rational map with a rational inverse between them

Algebraic objects and the Zariski topology

• An ideal is a subset of a ring that is closed under addition and multiplication by elements of the ring. In algebraic geometry, ideals are used to define varieties as the zero sets of the polynomials in the ideal
• A ring is an algebraic structure with addition, subtraction, and multiplication operations satisfying certain axioms (associativity, distributivity, identity elements). Polynomial rings are commonly used in algebraic geometry to define varieties and study their properties
• The Zariski topology is a topology on algebraic varieties defined by taking the closed sets to be the algebraic subsets, which are the solutions to systems of polynomial equations
• Open sets in the Zariski topology are complements of algebraic subsets, and they form a basis for the topology
• The Zariski topology is much coarser than the Euclidean topology, meaning that there are fewer open sets, which allows for the study of global properties of varieties

Significance of algebraic geometry

Connections to other fields of mathematics

• Algebraic geometry provides a powerful framework for studying geometric objects using algebraic techniques, allowing for the solution of complex problems in geometry and the discovery of new connections between different areas of mathematics
• The tools and techniques of algebraic geometry have found applications in various branches of mathematics, such as number theory (elliptic curves, Diophantine equations), complex analysis (Riemann surfaces, Hodge theory), and topology (cohomology theories, intersection theory)
• Algebraic geometry has also been used to prove major results in other fields, such as the Weil conjectures in number theory and the Riemann-Roch theorem in complex analysis

Applications to science and technology

• Algebraic geometry has played a crucial role in the development of modern theoretical physics, particularly in the study of string theory and quantum field theory, where geometric concepts are used to describe physical phenomena (Calabi-Yau manifolds, gauge theory)
• In computer science, algebraic geometry has applications in coding theory (algebraic-geometric codes), cryptography (elliptic curve cryptography), and computer vision (3D reconstruction, camera calibration), where geometric objects and their properties are used to develop efficient algorithms and secure communication protocols
• Algebraic geometry has also found applications in biology and chemistry, particularly in the study of phylogenetics (evolutionary trees, statistical models) and the geometry of molecules (conformational spaces, molecular docking), where geometric techniques are used to analyze and classify complex biological and chemical structures
• Other areas where algebraic geometry has been applied include robotics (motion planning, control theory), economics (game theory, optimization), and engineering (computer-aided design, geometric modeling)