🕴🏼Elementary Algebraic Geometry Unit 3 – Projective Varieties

What's This All About?

• Projective varieties are a central object of study in algebraic geometry, a branch of mathematics that combines techniques from algebra and geometry
• Projective varieties generalize the notion of algebraic varieties, which are geometric objects defined by polynomial equations, to include points "at infinity"
• Studying projective varieties allows us to understand the geometric properties of solutions to systems of polynomial equations in a more complete and unified way
• Projective varieties have rich geometric and algebraic structures that make them particularly interesting and useful in many areas of mathematics and its applications
• Understanding projective varieties requires familiarity with concepts from linear algebra, abstract algebra, and topology, making it a challenging but rewarding area of study

Key Concepts and Definitions

• Projective space $\mathbb{P}^n$ is an extension of affine $n$-space $\mathbb{A}^n$ that includes points at infinity, constructed by considering lines through the origin in $\mathbb{A}^{n+1}$
• Homogeneous coordinates are a way to represent points in projective space using ratios of coordinates, with the property that $[x_0 : \cdots : x_n] = [\lambda x_0 : \cdots : \lambda x_n]$ for any non-zero scalar $\lambda$
• A homogeneous polynomial is a polynomial $f(x_0, \ldots, x_n)$ where all terms have the same total degree, satisfying $f(\lambda x_0, \ldots, \lambda x_n) = \lambda^d f(x_0, \ldots, x_n)$ for some integer $d$
• A projective variety $V(f_1, \ldots, f_s)$ is the set of points $[x_0 : \cdots : x_n]$ in projective space that satisfy a system of homogeneous polynomial equations $f_1 = \cdots = f_s = 0$
  • The ideal $I(V)$ of a projective variety $V$ is the set of all homogeneous polynomials that vanish on $V$
  • The homogeneous coordinate ring $S(V)$ of $V$ is the quotient ring $k[x_0, \ldots, x_n] / I(V)$, graded by degree
• A morphism of projective varieties is a map $\varphi: V \to W$ induced by a homogeneous polynomial map on the ambient projective spaces satisfying $\varphi(V) \subseteq W$

Projective Space Basics

• Projective $n$-space $\mathbb{P}^n$ over a field $k$ is the set of equivalence classes of $(n+1)$-tuples $(x_0, \ldots, x_n) \in k^{n+1} \setminus \{0\}$ under the relation $(x_0, \ldots, x_n) \sim (\lambda x_0, \ldots, \lambda x_n)$ for all $\lambda \in k^\times$
• Points in projective space are denoted using homogeneous coordinates $[x_0 : \cdots : x_n]$, with the understanding that $[x_0 : \cdots : x_n] = [\lambda x_0 : \cdots : \lambda x_n]$ for any non-zero scalar $\lambda$
• Projective space has a natural topology, the Zariski topology, where closed sets are defined by homogeneous polynomial equations
• Projective space is compact, meaning that every sequence has a convergent subsequence, unlike affine space
• There is a natural map from affine space to projective space, sending $(x_1, \ldots, x_n)$ to $[1 : x_1 : \cdots : x_n]$, allowing us to view affine space as a subset of projective space
  • The complement of the image of this map in $\mathbb{P}^n$ is called the hyperplane at infinity, defined by the equation $x_0 = 0$

Homogeneous Polynomials and Projective Varieties

• A polynomial $f(x_0, \ldots, x_n)$ is homogeneous of degree $d$ if every term has total degree $d$, meaning that $f(\lambda x_0, \ldots, \lambda x_n) = \lambda^d f(x_0, \ldots, x_n)$ for any scalar $\lambda$
• A projective variety $V(f_1, \ldots, f_s)$ is the set of points $[x_0 : \cdots : x_n]$ in projective space that satisfy a system of homogeneous polynomial equations $f_1 = \cdots = f_s = 0$
  • Projective varieties are the closed sets in the Zariski topology on projective space
  • Every projective variety can be defined by a finite set of homogeneous polynomial equations
• The ideal $I(V)$ of a projective variety $V$ is the set of all homogeneous polynomials that vanish on $V$, forming a homogeneous ideal in the polynomial ring $k[x_0, \ldots, x_n]$
• The homogeneous coordinate ring $S(V)$ of a projective variety $V$ is the quotient ring $k[x_0, \ldots, x_n] / I(V)$, which inherits a grading by degree from the polynomial ring
  • The homogeneous coordinate ring encodes important algebraic and geometric information about the projective variety
• Projective varieties can be irreducible or reducible, depending on whether they can be written as the union of two proper subvarieties
  • The irreducible components of a projective variety correspond to the minimal prime ideals containing its defining ideal

Properties of Projective Varieties

• Projective varieties are invariant under projective transformations, which are invertible linear maps on the homogeneous coordinates
• The dimension of a projective variety is the transcendence degree of its homogeneous coordinate ring over the base field, which agrees with the intuitive notion of dimension
• Smooth projective varieties are those that locally look like projective space, without any singularities or self-intersections
  • Smoothness can be characterized by the Jacobian criterion, which states that a point is smooth if and only if the Jacobian matrix of the defining equations has full rank at that point
• Projective varieties can be complete intersections if their defining ideal is generated by exactly codim $V$ polynomials, where codim $V$ is the codimension of $V$ in the ambient projective space
• The degree of a projective variety is the number of intersection points with a generic linear subspace of complementary dimension, providing a measure of the "size" or "complexity" of the variety
• Projective varieties can have additional structures, such as group actions, vector bundles, or embeddings into other varieties, which can be used to study their properties and classify them

Important Examples and Special Cases

• Projective space $\mathbb{P}^n$ itself is the simplest example of a projective variety, defined by the trivial ideal $(0)$
• Projective subspaces, such as points, lines, planes, and hyperplanes, are projective varieties defined by linear equations
• Quadric hypersurfaces are projective varieties defined by a single homogeneous quadratic equation, such as the projective conic curves in $\mathbb{P}^2$ or the projective quadric surfaces in $\mathbb{P}^3$
• The Veronese variety $v_d(\mathbb{P}^n)$ is the image of the map $\mathbb{P}^n \to \mathbb{P}^{\binom{n+d}{d}-1}$ given by all monomials of degree $d$, providing an embedding of $\mathbb{P}^n$ into a higher-dimensional projective space
• The Segre variety $\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_k} \to \mathbb{P}^{(n_1+1)\cdots(n_k+1)-1}$ is the image of the product of projective spaces under the map given by all products of homogeneous coordinates, used to study tensor products and multilinear algebra
• Grassmannians $\operatorname{Gr}(k,n)$ are projective varieties parameterizing $k$-dimensional linear subspaces of an $n$-dimensional vector space, generalizing the projective space of lines in $\mathbb{P}^n$

Techniques and Calculations

• Elimination theory provides algorithms for computing the defining equations of projective varieties, such as the Buchberger algorithm for computing Gröbner bases
• Intersection theory allows us to compute the intersection multiplicities of projective varieties, using tools such as Bézout's theorem and the Chow ring
• Cohomology theories, such as sheaf cohomology and Hodge theory, provide powerful tools for studying the topology and geometry of projective varieties
  • The cohomology ring of a smooth projective variety often has a rich structure, such as Poincaré duality and the hard Lefschetz theorem
• The Riemann-Roch theorem relates the dimensions of the cohomology groups of a line bundle on a projective variety to its degree and the variety's intrinsic geometry, providing a key tool for studying divisors and linear systems
• Computational algebraic geometry software, such as Macaulay2, Singular, and Sage, can be used to perform calculations and experiments with projective varieties and their properties

Connections to Other Math Topics

• Projective varieties are closely related to affine varieties, with projective space being a natural compactification of affine space and projective varieties being the closures of affine varieties in this compactification
• The study of projective varieties is deeply connected to commutative algebra, with the homogeneous coordinate ring providing a bridge between the geometric and algebraic perspectives
• Projective varieties play a key role in complex geometry and Kähler geometry, with the Fubini-Study metric on projective space providing a canonical Kähler structure and many important examples of Kähler manifolds arising as projective varieties
• Projective varieties over the complex numbers are closely related to complex analytic spaces and Riemann surfaces, with the GAGA (Géométrie Algébrique et Géométrie Analytique) principle providing a correspondence between the algebraic and analytic theories
• The theory of moduli spaces, which parameterize geometric objects such as curves, surfaces, or vector bundles, often involves studying projective varieties that represent these objects, such as the Hilbert scheme or the moduli space of stable curves
• Projective varieties and their invariants, such as the Hodge structure or the Chow motive, are central objects of study in algebraic geometry and have deep connections to number theory, representation theory, and mathematical physics