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
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An between two varieties is a 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 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→Y and g:Y→X such that g∘f=idX and f∘g=idY
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 Pn to a higher-dimensional projective space PN, defined by monomials of a fixed degree d
The d-uple Veronese embedding of Pn is given by [x0:…:xn]↦[…:x0a0…xnan:…], where a0+…+an=d
Allows studying higher degree hypersurfaces as linear subspaces of a higher-dimensional space
Example: The Veronese surface is the image of P2 under the 2-uple Veronese embedding, given by [x:y:z]↦[x2:xy:y2:yz:z2:xz]
Segre Embedding
The Segre embedding is a map from the product of two projective spaces Pn×Pm to a higher-dimensional projective space P(n+1)(m+1)−1
The Segre embedding of Pn×Pm is given by ([x0:…:xn],[y0:…:ym])↦[…:xiyj:…], where 0≤i≤n and 0≤j≤m
Allows studying the product of projective spaces as a subvariety of a single projective space
Example: The Segre embedding of P1×P1 into P3 is given by ([x0:x1],[y0:y1])↦[x0y0:x0y1:x1y0:x1y1]
Other Embeddings
Rational normal curve: Embedding of P1 into Pn given by [x:y]↦[xn:xn−1y:…:xyn−1:yn]
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