🌿Computational Algebraic Geometry Unit 1 – Intro to Algebraic Geometry

Key Concepts and Definitions

• Algebraic geometry studies geometric objects defined by polynomial equations and the properties of these objects that are invariant under algebraic transformations
• Affine varieties are defined as the zero locus of a set of polynomials in affine space $\mathbb{A}^n$
• Projective varieties are defined as the zero locus of a set of homogeneous polynomials in projective space $\mathbb{P}^n$
• Coordinate rings are the rings of polynomial functions on an affine variety
  • Consists of all polynomial functions that are well-defined on the variety
  • Provides an algebraic description of the geometric properties of the variety
• Function fields are the fields of rational functions on a variety
• Morphisms are maps between varieties that are defined by polynomial functions
  • Allows for the study of relationships between different varieties
• Computational algebraic geometry uses algorithmic and computational methods to study and solve problems in algebraic geometry
  • Includes techniques such as Gröbner bases, resultants, and elimination theory

Foundations of Algebraic Geometry

• Algebraic geometry combines concepts from abstract algebra, particularly commutative algebra, with classical geometric notions
• Polynomial equations and algebraic varieties form the foundation of algebraic geometry
• Hilbert's Nullstellensatz establishes a correspondence between algebraic varieties and ideals in polynomial rings
  • States that every maximal ideal in a polynomial ring over an algebraically closed field is the ideal of a point
• Zariski topology is a topology on algebraic varieties defined by taking closed sets to be the zero loci of polynomial equations
  • Provides a framework for studying the geometric properties of varieties
• Sheaves are a fundamental tool in algebraic geometry that allow for the study of local properties of varieties
  • A sheaf is a collection of algebraic objects (rings, modules, etc.) attached to the open sets of a topological space, with certain compatibility conditions
• Schemes are a generalization of algebraic varieties that provide a unified framework for studying both affine and projective varieties
  • A scheme is a locally ringed space that is locally isomorphic to the spectrum of a commutative ring

Affine and Projective Varieties

• Affine varieties are algebraic varieties that can be described as the zero locus of a set of polynomials in affine space
  • Affine space $\mathbb{A}^n$ is the set of all $n$-tuples of elements from a field $k$
• Projective varieties are algebraic varieties that can be described as the zero locus of a set of homogeneous polynomials in projective space
  • 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
• Veronese embeddings are a class of embeddings of projective spaces into higher-dimensional projective spaces
  • Defined by mapping a point $[x_0: \cdots : x_n]$ to all monomials of a fixed degree $d$
• Segre embeddings are a class of embeddings of the product of two projective spaces into a higher-dimensional projective space
  • Defined by mapping a pair of points $([x_0: \cdots : x_n], [y_0: \cdots : y_m])$ to the tensor product of their homogeneous coordinates
• Grassmannians are projective varieties that parameterize linear subspaces of a fixed dimension in a vector space
  • The Grassmannian $\mathrm{Gr}(k, n)$ parameterizes $k$-dimensional subspaces of an $n$-dimensional vector space

Coordinate Rings and Function Fields

• The coordinate ring of an affine variety $V$ is the ring of polynomial functions on $V$
  • Denoted by $A(V)$ or $k[V]$, where $k$ is the base field
  • Consists of all polynomial functions that are well-defined on the variety
  • Isomorphic to the quotient ring $k[x_1, \ldots, x_n] / I(V)$, where $I(V)$ is the ideal of polynomials vanishing on $V$
• The function field of an irreducible variety $V$ is the field of rational functions on $V$
  • Denoted by $K(V)$ or $k(V)$, where $k$ is the base field
  • Consists of all rational functions (quotients of polynomial functions) that are well-defined on a dense open subset of $V$
  • Can be constructed as the field of fractions of the coordinate ring $A(V)$
• The local ring of a variety $V$ at a point $p$ is the ring of rational functions that are well-defined in a neighborhood of $p$
  • Denoted by $\mathcal{O}_{V,p}$
  • Consists of all rational functions that can be written as the quotient of two polynomial functions, where the denominator does not vanish at $p$
• The maximal ideal of the local ring $\mathcal{O}_{V,p}$ consists of all rational functions that vanish at $p$
  • Denoted by $\mathfrak{m}_p$
  • Allows for the study of the local properties of the variety at the point $p$

Morphisms and Maps

• A morphism between two varieties $V$ and $W$ is a map $\varphi: V \to W$ that is defined by polynomial functions
  • For affine varieties, a morphism is a map that pulls back polynomial functions on $W$ to polynomial functions on $V$
  • For projective varieties, a morphism is a map that pulls back homogeneous polynomial functions on $W$ to homogeneous polynomial functions on $V$
• Isomorphisms are morphisms that have an inverse morphism
  • Two varieties are isomorphic if there exists an isomorphism between them
  • Isomorphic varieties have the same geometric and algebraic properties
• Rational maps are maps between varieties that are defined by rational functions
  • A rational map $\varphi: V \dashrightarrow W$ is a morphism defined on a dense open subset of $V$
  • Rational maps allow for the study of maps between varieties that are not everywhere defined
• Birational maps are rational maps that have a rational inverse
  • Two varieties are birational if there exists a birational map between them
  • Birational varieties have the same function field and share many geometric properties
• Finite morphisms are morphisms that have finite fibers
  • A morphism $\varphi: V \to W$ is finite if the preimage of every point in $W$ is a finite set
  • Finite morphisms are important in the study of ramification and branching of maps between varieties

Computational Techniques and Tools

• Gröbner bases are a key computational tool in algebraic geometry
  • A Gröbner basis is a particular generating set of an ideal in a polynomial ring that has nice algorithmic properties
  • Gröbner bases allow for the effective computation of many algebraic and geometric properties of varieties
• Buchberger's algorithm is a method for computing Gröbner bases
  • Provides a way to transform any generating set of an ideal into a Gröbner basis
  • Implemented in many computer algebra systems (Mathematica, Maple, Sage)
• Elimination theory studies the problem of eliminating variables from a system of polynomial equations
  • Allows for the computation of projections and images of varieties under morphisms
  • Closely related to the computation of Gröbner bases and resultants
• Resultants are a tool for eliminating variables from systems of polynomial equations
  • The resultant of two polynomials is a polynomial in their coefficients that vanishes if and only if the polynomials have a common root
  • Resultants can be used to compute the intersection of varieties and to study the fibers of morphisms
• Toric varieties are a class of algebraic varieties that are described by combinatorial data
  • Defined by a fan, which is a collection of cones in a lattice
  • Toric varieties have a rich combinatorial structure and are important in computational algebraic geometry
• Homotopy continuation is a numerical method for solving systems of polynomial equations
  • Based on the idea of deforming a simple system of equations into a more complicated one while tracking the solutions
  • Allows for the computation of all isolated solutions of a system of polynomial equations

Applications and Examples

• Algebraic statistics uses algebraic geometry to study statistical models and inference problems
  • Algebraic varieties can be used to represent statistical models, such as Bayesian networks and phylogenetic trees
  • Gröbner bases and elimination theory can be used for model selection and parameter estimation
• Coding theory uses algebraic geometry to construct and study error-correcting codes
  • Algebraic geometric codes are a class of linear codes that are constructed from algebraic curves
  • The geometric properties of the curve determine the parameters and performance of the code
• Cryptography uses algebraic geometry to design and analyze cryptographic systems
  • Elliptic curve cryptography is based on the arithmetic of elliptic curves, which are a type of algebraic curve
  • The discrete logarithm problem on elliptic curves is used as a basis for many cryptographic protocols
• Robotics uses algebraic geometry to study the kinematics and motion planning of robots
  • The configuration space of a robot can be modeled as an algebraic variety
  • Gröbner bases and elimination theory can be used to compute the forward and inverse kinematics of a robot
• Computer vision uses algebraic geometry to study the geometry of images and 3D scenes
  • The set of all possible images of a 3D scene can be modeled as an algebraic variety (the image variety)
  • Techniques from algebraic geometry can be used for 3D reconstruction, camera calibration, and object recognition

Further Reading and Resources

• "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea is a classic textbook on computational algebraic geometry
  • Covers the basics of Gröbner bases, elimination theory, and the algebra-geometry dictionary
  • Includes many examples and exercises
• "Algebraic Geometry and Statistical Learning Theory" by Watanabe is a monograph on the applications of algebraic geometry to machine learning
  • Covers the use of algebraic varieties and singularities in statistical learning theory
  • Includes topics such as the algebraic geometry of neural networks and the resolution of singularities
• "Computational Algebraic Geometry" by Schenck is a textbook on the computational aspects of algebraic geometry
  • Covers Gröbner bases, resultants, and toric varieties
  • Includes chapters on the applications of algebraic geometry to robotics, computer vision, and geometric modeling
• "Algebraic Geometry and Commutative Algebra" by Eisenbud is a comprehensive textbook on the foundations of algebraic geometry
  • Covers the basics of commutative algebra, schemes, and sheaves
  • Includes many examples and exercises
• The Journal of Symbolic Computation is a leading journal in computational algebra and geometry
  • Publishes research papers on the development and application of symbolic algorithms in algebra and geometry
  • Includes topics such as Gröbner bases, resultants, and elimination theory
• The Journal of Algebra is a leading journal in algebra and algebraic geometry
  • Publishes research papers on the foundations and applications of algebra and algebraic geometry
  • Includes topics such as commutative algebra, representation theory, and algebraic geometry
• The arXiv (arxiv.org) is a preprint server that includes many research papers in algebraic geometry and related fields
  • Allows for quick access to the latest research developments
  • Includes preprints of papers that may not yet be published in peer-reviewed journals