🕴🏼Elementary Algebraic Geometry Unit 1 – Intro to Algebraic Geometry

Key Concepts and Definitions

• Algebraic geometry studies geometric objects defined by polynomial equations and investigates their properties using abstract algebra
• Affine space $\mathbb{A}^n$ is the set of all $n$-tuples of elements from a field $k$ ($k^n$) without any notion of distance or angle
• Affine varieties are subsets of affine space defined by polynomial equations
• Projective space $\mathbb{P}^n$ extends affine space by adding points at infinity, allowing the study of solutions to homogeneous polynomial equations
• Coordinate rings are rings of polynomial functions on an affine variety, capturing its algebraic structure
• Function fields are fields of rational functions on a variety, generalizing the concept of coordinate rings
• Morphisms are structure-preserving maps between varieties that respect the underlying algebraic structure
• Rational maps are morphisms defined on dense open subsets of varieties, allowing more flexibility than regular morphisms

Foundations of Algebraic Geometry

• Algebraic geometry combines techniques from abstract algebra, particularly commutative algebra, with classical geometric concepts
• The fundamental objects of study are algebraic varieties, which are geometric spaces defined by polynomial equations
• Affine varieties are the building blocks of algebraic geometry and are studied in affine space
• Projective varieties are studied in projective space and provide a more general setting for investigating geometric properties
• Algebraic geometry uses the correspondence between geometric objects and their algebraic counterparts (rings, ideals, modules) to gain insights
• The Zariski topology, defined by taking algebraic sets as closed sets, provides a topology on varieties that captures their algebraic structure
• Sheaves are important tools in algebraic geometry that allow the study of local properties of varieties and the transition between local and global information

Affine Varieties and Ideals

• An affine variety $V(I)$ is the set of points in affine space that satisfy all polynomial equations in an ideal $I$
• Ideals are subsets of polynomial rings that are closed under addition and multiplication by ring elements
• The ideal-variety correspondence establishes a bijection between affine varieties and radical ideals in polynomial rings
  • Every affine variety determines a unique radical ideal, and every radical ideal determines a unique affine variety
• The Zariski topology on affine space has affine varieties as its closed sets
• Irreducible varieties are varieties that cannot be written as the union of two proper subvarieties
  • Irreducibility corresponds to the notion of prime ideals in the associated coordinate ring
• The dimension of an affine variety is the transcendence degree of its function field over the base field
• Singular points on a variety are points where the tangent space has higher dimension than expected, while non-singular points are called smooth

Coordinate Rings and Function Fields

• The coordinate ring $A(V)$ of an affine variety $V$ is the ring of polynomial functions on $V$
  • It is obtained by quotienting the polynomial ring by the ideal defining the variety
• The coordinate ring captures the algebraic structure of the variety and allows the study of its properties using ring-theoretic techniques
• Regular functions on an affine variety are elements of its coordinate ring, i.e., polynomial functions restricted to the variety
• The function field $K(V)$ of a variety $V$ is the field of rational functions on $V$, obtained by localizing the coordinate ring at all non-zero elements
• The function field of an irreducible variety is an extension field of the base field, and its transcendence degree equals the dimension of the variety
• Birational equivalence is an equivalence relation between varieties, where two varieties are birationally equivalent if they have isomorphic function fields
  • Birational equivalence captures the idea of varieties being "almost" isomorphic, differing only on lower-dimensional subsets

Projective Spaces and Projective Varieties

• Projective space $\mathbb{P}^n$ is the set of equivalence classes of $(n+1)$-tuples of elements from a field $k$, where two tuples are equivalent if they differ by a non-zero scalar multiple
  • Points in projective space are represented by homogeneous coordinates $[x_0 : \cdots : x_n]$
• Projective space extends affine space by adding points at infinity, allowing the study of solutions to homogeneous polynomial equations
• Projective varieties are subsets of projective space defined by homogeneous polynomial equations
• The homogeneous coordinate ring of a projective variety is a graded ring, with elements of each degree corresponding to homogeneous polynomials of that degree
• Projective varieties can be studied using techniques similar to those used for affine varieties, with some modifications to account for the graded structure of the coordinate ring
• The projective closure of an affine variety is the smallest projective variety containing it, obtained by homogenizing the defining equations and adding points at infinity
• Projective space and projective varieties provide a natural setting for studying geometric properties that are invariant under projective transformations, such as incidence relations and cross-ratios

Morphisms and Rational Maps

• A morphism between affine varieties is a map that can be represented by polynomial functions in each coordinate
  • Morphisms are the algebraic geometry analogue of continuous maps in topology
• Morphisms between projective varieties are defined by homogeneous polynomials of the same degree in each coordinate
• Isomorphisms are morphisms with an inverse that is also a morphism; they capture the notion of two varieties being "the same" algebraically and geometrically
• Rational maps are maps between varieties that are defined by rational functions (quotients of polynomials) rather than just polynomials
  • Rational maps are not necessarily defined everywhere, but only on a dense open subset of the domain
• Birational maps are rational maps with a rational inverse; they establish birational equivalence between varieties
• The graph of a rational map is a subvariety of the product of the domain and codomain, which can be used to study properties of the map
• Blow-ups are a type of birational morphism that can be used to resolve singularities or indeterminacy of rational maps

Applications and Examples

• Algebraic geometry has applications in various fields, including number theory, cryptography, coding theory, and robotics
• Elliptic curves, which are projective varieties of dimension 1, play a crucial role in cryptography and the proof of Fermat's Last Theorem
• Algebraic geometry is used in the study of Diophantine equations, which are polynomial equations with integer coefficients
  • The Mordell-Weil theorem, which states that the set of rational points on an elliptic curve forms a finitely generated abelian group, is a key result in this area
• Algebraic geometry techniques are employed in the study of error-correcting codes, such as algebraic-geometric codes (AG codes)
• Toric varieties, which are varieties defined by combinatorial data (fans), have applications in geometric modeling and robotics
• The resolution of singularities, a fundamental result in algebraic geometry, has implications for the classification of algebraic varieties and the study of their properties
• The Riemann-Roch theorem, which relates the dimension of the space of global sections of a line bundle to its degree and the genus of the variety, is a powerful tool in the study of curves and surfaces

Common Pitfalls and Study Tips

• Algebraic geometry involves abstract concepts and heavy use of algebraic techniques, so a solid foundation in abstract algebra (particularly commutative algebra) is essential
• It is important to understand the interplay between the geometric and algebraic aspects of varieties, as many results and proofs rely on this correspondence
• When working with projective varieties, be mindful of the differences between affine and projective spaces, particularly regarding the use of homogeneous coordinates and the behavior at infinity
• Pay attention to the role of irreducibility and dimension in the study of varieties, as these concepts are crucial for understanding their structure and properties
• Be comfortable with the notion of sheaves and their role in capturing local-to-global properties of varieties
• Practice computing examples of varieties, coordinate rings, and morphisms to develop intuition and familiarity with the objects of study
• Engage with the geometric intuition behind the algebraic concepts, as this can provide valuable insights and guide problem-solving approaches
• Consult multiple resources (textbooks, lecture notes, papers) to gain different perspectives on the material and reinforce understanding