Isomorphisms and embeddings are crucial concepts in algebraic geometry, helping us understand the relationships between varieties. Isomorphisms reveal when varieties are essentially the same, while embeddings allow us to view one variety as part of another.

These ideas are key to the broader study of morphisms and rational maps. They provide tools for classifying varieties, simplifying their analysis, and uncovering hidden connections between seemingly different geometric objects.

Isomorphisms vs Embeddings of Varieties

Defining Isomorphisms and Embeddings

• An isomorphism between two varieties is a bijective morphism with a morphism inverse
  • Meaning the two varieties are essentially the same and have identical structure
  • Isomorphic varieties have the same geometric properties (dimension, degree, singularities)
  • They also share algebraic properties (coordinate ring structure, Picard group, cohomology groups)
• An embedding of a variety $X$ into a variety $Y$ is an injective morphism from $X$ to $Y$
  • Allows viewing $X$ as a subvariety of $Y$, inheriting some properties from the ambient space $Y$
  • Embeddings are generally not invertible, unlike isomorphisms

Comparing Properties and Uses

• Isomorphisms preserve the intrinsic structure of varieties, while embeddings allow studying varieties as subvarieties of other spaces
  • Isomorphic varieties are essentially identical, having the same geometric and algebraic properties
  • Embedded varieties inherit some properties from the ambient space but may have additional structure
• Isomorphisms are used for classifying varieties up to equivalence (varieties with the same intrinsic properties)
  • Help identify when seemingly different varieties are actually the same
  • Simplify the study of algebraic geometry by grouping equivalent varieties
• Embeddings are used to study varieties as subvarieties of other spaces, often simplifying their analysis
  • Allow studying varieties using tools and techniques available in the ambient space (linear algebra in projective spaces)

Proving Isomorphisms with Morphisms

Constructing Isomorphisms

• Two varieties $X$ and $Y$ are isomorphic if there exist morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$ and $f \circ g = id_Y$
  • To prove isomorphism, construct explicit morphisms between the varieties
  • Show that the compositions of the morphisms are the identity morphisms on respective varieties
• Isomorphisms can be proved using the properties of the varieties
  • Defining equations of the varieties can be used to construct morphisms
  • Coordinate rings of the varieties can be used to study isomorphisms algebraically

Isomorphisms and Coordinate Rings

• If two affine varieties have isomorphic coordinate rings, then the varieties are isomorphic
  • Coordinate rings encode the algebraic structure of affine varieties
  • Isomorphisms between coordinate rings induce isomorphisms between the corresponding affine varieties
• Isomorphisms between projective varieties can be studied using the homogeneous coordinate rings
  • Homogeneous coordinate rings capture the structure of projective varieties
  • Isomorphisms between homogeneous coordinate rings induce isomorphisms between the corresponding projective varieties

Examples of Embeddings: Veronese and Segre

Veronese Embedding

• The Veronese embedding is a map from the projective space $\mathbb{P}^n$ to a higher-dimensional projective space $\mathbb{P}^N$, defined by monomials of a fixed degree $d$
  • The $d$-uple Veronese embedding of $\mathbb{P}^n$ is given by $[x_0 : \ldots : x_n] \mapsto [\ldots: x_0^{a_0}\ldots x_n^{a_n} : \ldots]$, where $a_0 + \ldots + a_n = d$
  • Allows studying higher degree hypersurfaces as linear subspaces of a higher-dimensional space
  • Example: The Veronese surface is the image of $\mathbb{P}^2$ under the 2-uple Veronese embedding, given by $[x:y:z] \mapsto [x^2:xy:y^2:yz:z^2:xz]$

Segre Embedding

• The Segre embedding is a map from the product of two projective spaces $\mathbb{P}^n \times \mathbb{P}^m$ to a higher-dimensional projective space $\mathbb{P}^{(n+1)(m+1)-1}$
  • The Segre embedding of $\mathbb{P}^n \times \mathbb{P}^m$ is given by $([x_0 : \ldots : x_n], [y_0 : \ldots : y_m]) \mapsto [\ldots: x_i y_j : \ldots]$, where $0 \leq i \leq n$ and $0 \leq j \leq m$
  • Allows studying the product of projective spaces as a subvariety of a single projective space
  • Example: The Segre embedding of $\mathbb{P}^1 \times \mathbb{P}^1$ into $\mathbb{P}^3$ is given by $([x_0:x_1],[y_0:y_1]) \mapsto [x_0y_0:x_0y_1:x_1y_0:x_1y_1]$

Other Embeddings

• Rational normal curve: Embedding of $\mathbb{P}^1$ into $\mathbb{P}^n$ given by $[x:y] \mapsto [x^n:x^{n-1}y:\ldots:xy^{n-1}:y^n]$
• Plücker embedding of the Grassmannian: Embedding of the Grassmannian $G(k,n)$ into a projective space using Plücker coordinates
• Canonical embedding of a variety using its canonical divisor: Embedding of a variety into a projective space using sections of the canonical line bundle

Significance of Isomorphisms in Classification

Classifying Varieties up to Isomorphism

• Isomorphisms provide a way to classify varieties up to equivalence
  • Isomorphic varieties share the same intrinsic properties and are essentially the same
  • Classification often involves finding invariants that distinguish non-isomorphic varieties
  • Varieties with the same invariants are then shown to be isomorphic
• Isomorphisms preserve geometric properties that can be used to distinguish varieties
  • Dimension, degree, and singularities are preserved under isomorphisms
  • These properties can be used as invariants to classify varieties

Algebraic Invariants and Isomorphisms

• Isomorphisms preserve algebraic properties of varieties
  • Structure of the coordinate ring, Picard group, and cohomology groups are preserved
  • These algebraic invariants can be used to distinguish non-isomorphic varieties
  • Varieties with isomorphic algebraic invariants are likely to be isomorphic
• Understanding isomorphisms helps simplify the study of algebraic geometry
  • Identifies when seemingly different varieties are actually the same
  • Allows focusing on representative varieties from each isomorphism class
  • Reduces the complexity of the subject by grouping equivalent varieties together