Normal and are key concepts in algebraic geometry. They help us understand how well-behaved a variety is, especially when it comes to singularities and function extensions.
These properties are closely tied to the study of singularities. have no "holes" in codimension one, while Cohen-Macaulay varieties have nice cohomological properties. Understanding these concepts is crucial for analyzing and resolving singularities.
Normal varieties and their properties
Definition and integral closure
Top images from around the web for Definition and integral closure
Algebraic variety - Simple English Wikipedia, the free encyclopedia View original
Is this image relevant?
Algebraic variety - Simple English Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 1
Top images from around the web for Definition and integral closure
Algebraic variety - Simple English Wikipedia, the free encyclopedia View original
Is this image relevant?
Algebraic variety - Simple English Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 1
A variety X is normal if every OX,x is an integrally closed domain in its field of fractions
Integrally closed means that if an element z in the field of fractions satisfies a monic polynomial equation with coefficients in OX,x, then z must be in OX,x
Geometric interpretation: normal varieties have no "holes" or "self-intersections" in codimension one
Nonsingularity and extension of rational functions
Normal varieties are nonsingular in codimension one, meaning the singular locus has codimension at least two
Example: the cone over a smooth quadric surface in P3 is normal but singular at the vertex
A variety X is normal if and only if every rational function defined on an open dense subset of X can be extended to a regular function on the whole of X
Intuition: normal varieties allow for a well-behaved extension of functions defined almost everywhere
Local and global properties
Normality is a local property, i.e., X is normal if and only if it is normal at every point
Normal varieties are geometrically integral, meaning they remain integral under any base field extension
Example: a normal variety over C remains integral when base changed to C(t)
The normalization of a variety X is a finite surjective morphism from a normal variety X′ to X that is an isomorphism over the normal locus of X
Universal property: any finite morphism from a normal variety to X factors uniquely through the normalization
Cohen-Macaulay varieties and local rings
Definition and depth
A local noetherian ring R is Cohen-Macaulay if its equals its Krull , i.e., depth(R)=dim(R)
Depth measures the length of the longest regular sequence in the maximal ideal of R
A variety X is Cohen-Macaulay if the local ring OX,x is Cohen-Macaulay for every point x in X
Regular sequences and properties
In a Cohen-Macaulay local ring, every system of parameters is a regular sequence
A system of parameters is a sequence of elements generating an ideal of definition
The is preserved under localization and completion
Geometric interpretation: the Cohen-Macaulay property is stable under taking open subsets and completing at a point
A Cohen-Macaulay variety is equidimensional, meaning all its irreducible components have the same dimension
Characterizations and vanishing of cohomology
The Cohen-Macaulay property can be characterized by the vanishing of certain local cohomology modules
Hmi(R)=0 for all i<dim(R), where m is the maximal ideal
For a projective variety X, the Cohen-Macaulay property is equivalent to the vanishing of higher cohomology of twisted ideal sheaves
Hi(X,IX(n))=0 for all i>0 and n≫0, where IX is the ideal sheaf of X
Normality vs Cohen-Macaulay properties
Implication and proof
Normal varieties are Cohen-Macaulay
Key step: show that in a normal local ring, every system of parameters is a regular sequence
Induct on the dimension d, using the fact that quotients of normal domains by non-zerodivisors are still normal domains
Conclude that normal local rings are Cohen-Macaulay, and thus normal varieties are Cohen-Macaulay
Counterexamples and singularities
The converse is false: there exist Cohen-Macaulay varieties that are not normal
Example: the pinch point singularity (y2=x2z) is Cohen-Macaulay but not normal
Cohen-Macaulay singularities are a special class of singularities that allow the variety to maintain the Cohen-Macaulay property
Example: the cone over a smooth plane curve is Cohen-Macaulay but singular at the vertex
Not all Cohen-Macaulay singularities are normal singularities, and vice versa
Rational singularities (singularities with a resolution whose higher direct images of the structure sheaf vanish) are Cohen-Macaulay but not necessarily normal
Singularities and algebraic properties
Singular points and obstructions
Singular points can prevent a variety from being normal or Cohen-Macaulay
A variety with only isolated singularities (i.e., the singular locus has codimension at least 2) can still be normal
Example: the cone over a smooth quadric surface in P3 is normal but singular at the vertex
Non-isolated singularities (e.g., singular curves on a surface) often obstruct normality and the Cohen-Macaulay property
Resolution of singularities
Resolving the singularities of a variety can help restore good algebraic properties
Hironaka's theorem: in characteristic zero, every variety admits a resolution of singularities
The resolution is a proper birational morphism from a smooth variety, obtained by a sequence of blowups
The normalization of a variety can be viewed as a partial resolution of singularities
It resolves the singularities in codimension one but may leave higher-codimensional singularities intact
Key Terms to Review (18)
Affine Space: An affine space is a geometric structure that generalizes the properties of Euclidean space, allowing for the representation of points, vectors, and linear combinations without an inherent origin. In affine spaces, points are treated as distinct entities, and the focus is on the relationships between points, making it crucial for understanding various algebraic structures and their applications.
Cohen-Macaulay Property: The Cohen-Macaulay property refers to a condition in algebraic geometry where a ring or a variety has well-behaved homological properties, specifically that the depth of the ring equals its Krull dimension. This property indicates that the variety has a certain level of geometric regularity, which is significant in understanding singularities and their resolutions.
Cohen-Macaulay Varieties: Cohen-Macaulay varieties are a class of algebraic varieties that have desirable geometric properties, particularly in relation to their ring of regular functions. These varieties satisfy a crucial property: the depth of their coordinate ring matches the dimension of the variety, which implies that they have 'nice' singularities and behave well under various mathematical operations. This concept is fundamental in understanding the structure and classification of algebraic varieties, especially when investigating their normality and the interplay between geometry and commutative algebra.
Depth: Depth is a fundamental concept in commutative algebra and algebraic geometry that refers to the length of the longest regular sequence of elements in an ideal or a module. It provides insight into the structure of varieties, especially in distinguishing between different types of singularities and regularities, making it crucial for understanding properties such as normality and Cohen-Macaulayness.
Dimension: Dimension is a fundamental concept in mathematics that refers to the number of independent directions in which one can move within a given space. It plays a critical role in understanding the structure and behavior of various geometric objects, helping to classify them and analyze their properties across different contexts, such as varieties, singularities, and algebraic groups.
Dimension formula: The dimension formula is a crucial concept in algebraic geometry that expresses the relationship between the dimensions of various geometric objects, particularly in relation to varieties. This formula helps to quantify how the dimensions of subvarieties, irreducible components, and their intersections contribute to the overall structure of a variety, allowing for a deeper understanding of its geometric and algebraic properties.
Free resolution: A free resolution is an exact sequence of free modules that helps to study properties of modules over a ring. It essentially provides a way to break down modules into simpler components, which can reveal more about their structure, especially in the context of algebraic varieties. This concept becomes particularly important when examining normal and Cohen-Macaulay varieties, as free resolutions can highlight their regularity and depth.
Intersection Multiplicity: Intersection multiplicity is a numerical invariant that measures how many times two algebraic varieties intersect at a given point, taking into account both the dimensions and local behavior of the varieties near that point. It helps to understand the local structure of the varieties and their intersections, reflecting both algebraic and geometric properties. This concept is essential in studying projective varieties and offers insights into the properties of normal and Cohen-Macaulay varieties as well.
Local Ring: A local ring is a commutative ring with a unique maximal ideal, making it an essential structure in algebraic geometry and commutative algebra. This property allows us to focus on the behavior of functions or elements near a specific point or prime ideal, which is crucial for studying local properties of varieties and schemes. The local ring captures the essence of the geometry at that point, facilitating the analysis of singularities, dimensions, and other local features.
Module over a ring: A module over a ring is a mathematical structure that generalizes the concept of vector spaces by allowing scalars to come from a ring instead of a field. In this context, modules retain many properties of vector spaces but also exhibit unique features due to the properties of rings. Modules play a crucial role in understanding algebraic structures and are particularly important in the study of normal and Cohen-Macaulay varieties, as well as in primary decomposition and associated primes.
Normal Varieties: Normal varieties are algebraic varieties that satisfy a particular property of local rings: they are integrally closed in their function fields. This means that every element that is integral over the coordinate ring of the variety is already contained in that ring. A key aspect of normal varieties is their nice geometric properties, particularly that they do not have certain types of singularities, which relates closely to Cohen-Macaulay varieties and their role in algebraic geometry.
Projective Space: Projective space is a fundamental concept in algebraic geometry that extends the idea of Euclidean space by adding 'points at infinity' to account for parallel lines meeting. This transformation allows for a more comprehensive understanding of geometric properties and relationships among various geometric objects, such as varieties, curves, and surfaces.
Rees Algebra: Rees algebra is a construction used in algebraic geometry that helps study ideals in a polynomial ring by associating them with graded rings. It provides a framework for understanding the properties of varieties through their homogeneous coordinates, making it particularly useful in the study of normal and Cohen-Macaulay varieties, where the behavior of ideals can reveal deeper insights about the structure and singularities of these varieties.
Regularity: Regularity is a concept that captures the idea of smoothness and well-behaved structures in algebraic geometry, particularly focusing on the properties of varieties and sheaves. It often describes how closely a geometric object resembles a 'nice' or 'regular' one, which can greatly impact the application of various theorems like Serre duality and the Riemann-Roch theorem, as well as the classification of varieties as normal or Cohen-Macaulay. Understanding regularity helps in assessing both geometric and algebraic properties that are essential for deeper studies in this field.
Ring of regular functions: The ring of regular functions on a variety is a collection of functions that are defined and behave well (like polynomials) on the variety, meaning they can be expressed as quotients of polynomials where the denominator does not vanish. This ring captures the algebraic structure of the variety and plays a critical role in understanding both its geometry and singularities, which is essential for working with concepts like blowing up and determining normality or Cohen-Macaulay properties.
Serre's Criterion: Serre's Criterion is a set of conditions that provides a way to determine whether a variety is normal or Cohen-Macaulay. This criterion connects geometric properties of varieties with algebraic properties of their coordinate rings, playing a crucial role in understanding the structure of varieties. The criterion specifically uses the behavior of local rings and their dimensions to classify varieties, leading to significant implications for their geometric properties.
Singular Point: A singular point on a variety is a point where the variety fails to be 'smooth' or 'regular,' meaning that it does not have a well-defined tangent space. This concept is crucial in understanding the behavior of varieties at these points, as they can exhibit unusual geometrical properties and affect the overall structure of the variety. Singular points are often identified using tangent cones and play a significant role in classifying varieties as normal or Cohen-Macaulay.
Smooth point: A smooth point on a variety is a point where the local structure of the variety behaves nicely, specifically where the tangent space has the expected dimension. At a smooth point, the variety does not exhibit any singular behavior, which means that it locally resembles an affine space and allows for well-defined tangent vectors. Understanding smooth points is crucial as they are essential in defining properties like normality and Cohen-Macaulayness of varieties.