6.3 Normal and Cohen-Macaulay varieties

Normal and Cohen-Macaulay varieties 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. Normal varieties 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 local ring $O_{X,x}$ 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 $O_{X,x}$, then $z$ must be in $O_{X,x}$
• 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 $\mathbb{P}^3$ 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 $\mathbb{C}$ remains integral when base changed to $\mathbb{C}(t)$
• 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 depth equals its Krull dimension, i.e., $\operatorname{depth}(R) = \dim(R)$
• 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 $O_{X,x}$ 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 Cohen-Macaulay property 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
  • $H_\mathfrak{m}^i(R) = 0$ for all $i < \dim(R)$, where $\mathfrak{m}$ 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
  • $H^i(X, \mathcal{I}_X(n)) = 0$ for all $i > 0$ and $n \gg 0$, where $\mathcal{I}_X$ 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 $(y^2 = x^2z)$ 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 $\mathbb{P}^3$ 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