4.1 Weil and Cartier divisors

Weil and Cartier divisors are fundamental concepts in algebraic geometry, representing subvarieties of codimension 1. Weil divisors are formal sums of subvarieties, while Cartier divisors are defined using local rational functions. Understanding their relationship is crucial for studying line bundles and intersection theory.

On smooth varieties, Weil and Cartier divisors coincide, but they differ on singular varieties. This distinction helps measure singularities and is essential for understanding the geometry of algebraic varieties. Exploring these concepts lays the groundwork for more advanced topics in algebraic geometry.

Weil vs Cartier Divisors

Definition and Properties of Weil Divisors

• A Weil divisor on an algebraic variety $X$ is a formal finite sum of codimension-1 subvarieties of $X$ with integer coefficients
  • The group of Weil divisors on $X$ is denoted by $Div(X)$
• A prime divisor is an irreducible subvariety of codimension 1
  • Every Weil divisor can be uniquely written as a finite sum of prime divisors with integer coefficients
  • Example prime divisors: a point on a curve, a line on a surface
• The Weil divisors form an abelian group under formal addition of subvarieties
  • The zero divisor is the empty sum
  • The negative of a divisor $\sum a_i V_i$ is the divisor $\sum (-a_i) V_i$

Definition and Properties of Cartier Divisors

• A Cartier divisor on an algebraic variety $X$ is a global section of the quotient sheaf $K^*/O^*$
  • $K^*$ is the sheaf of total quotient rings
  • $O^*$ is the sheaf of invertible regular functions
  • The group of Cartier divisors on $X$ is denoted by $CDiv(X)$
• A Cartier divisor can be represented by an open cover $\{U_i\}$ of $X$ and rational functions $f_i$ on each $U_i$
  • On the overlap $U_i \cap U_j$, the transition function $f_i/f_j$ must be a regular invertible function
  • Example: on $\mathbb{P}^n$, a Cartier divisor is given by a homogeneous polynomial
• The Cartier divisors form an abelian group under multiplication of local defining functions
  • The zero divisor is represented by the constant function 1 on each open set
  • The negative of a divisor represented by $\{(U_i, f_i)\}$ is represented by $\{(U_i, 1/f_i)\}$

Relationship of Weil and Cartier Divisors

Cartier to Weil Map

• Every Cartier divisor determines a Weil divisor by taking the divisor of zeros and poles of the local defining rational functions
  • This gives a homomorphism $CDiv(X) \to Div(X)$
  • Example: on $\mathbb{P}^n$, a homogeneous polynomial $f$ defines a Cartier divisor, and the corresponding Weil divisor is the hypersurface $\{f=0\}$
• The homomorphism $CDiv(X) \to Div(X)$ is always injective
  • Different Cartier divisors always give different Weil divisors
  • Locally principal Weil divisors correspond bijectively to Cartier divisors

Smooth vs Singular Varieties

• On a smooth variety, every Weil divisor is locally principal, and thus determines a Cartier divisor
  • In this case, the homomorphism $CDiv(X) \to Div(X)$ is an isomorphism
  • Examples of smooth varieties: $\mathbb{P}^n$, smooth projective curves, smooth hypersurfaces
• On a singular variety, a Weil divisor may not be locally principal, and thus may not determine a Cartier divisor
  • In this case, the homomorphism $CDiv(X) \to Div(X)$ is injective but not surjective
  • Example: on a curve with a node, the node is a Weil divisor but not a Cartier divisor
• The quotient group $Div(X)/CDiv(X)$ measures the failure of Weil divisors to be Cartier
  • This is a measure of the singularities of $X$
  • Example: for a curve with a node, $Div(X)/CDiv(X) \cong \mathbb{Z}/2\mathbb{Z}$, reflecting the presence of the node

Computing Divisor Groups

Projective Spaces and Smooth Curves

• For the projective space $\mathbb{P}^n$, $Div(\mathbb{P}^n) \cong \mathbb{Z}$, generated by a hyperplane
  • Every Weil divisor is Cartier, so $CDiv(\mathbb{P}^n) \cong Div(\mathbb{P}^n) \cong \mathbb{Z}$
  • A divisor $aH$, where $H$ is a hyperplane, has degree $a$
• For a smooth projective curve $C$, $Div(C)$ is the free abelian group on the points of $C$
  • Every Weil divisor is Cartier, so $CDiv(C) \cong Div(C)$
  • A divisor $\sum a_i P_i$ has degree $\sum a_i$
  • Example: on $\mathbb{P}^1$, a divisor is a formal sum of points $\sum a_i [x_i:y_i]$

Singular Curves

• For the projective plane curve with a node singularity given by $y^2 = x^2(x+1)$:
  • $Div(X)$ is freely generated by the prime divisors (the irreducible components and the node)
  • The node is not a Cartier divisor
  • $CDiv(X)$ is isomorphic to the subgroup of $Div(X)$ consisting of divisors that are even multiples of the node
• For the affine plane curve with a cusp singularity given by $y^2 = x^3$:
  • $Div(X)$ is freely generated by the prime divisors (the irreducible component and the cusp)
  • The cusp is not a Cartier divisor
  • $CDiv(X)$ is trivial, as no non-zero multiple of the cusp is Cartier

Properties of Divisors and Equivalence

Principal Divisors

• A principal divisor is a Weil divisor that is the divisor of a rational function
  • The principal divisors form a subgroup of $Div(X)$, denoted by $Prin(X)$
  • Example: on $\mathbb{P}^n$, the divisor of a rational function $f/g$, where $f$ and $g$ are homogeneous of the same degree, is principal
• A Cartier divisor is principal if it is represented by a global rational function
  • The principal Cartier divisors form a subgroup of $CDiv(X)$, also denoted by $Prin(X)$
  • Example: on $\mathbb{P}^n$, a principal Cartier divisor is given by a global rational function $f/g$, where $f$ and $g$ are homogeneous of the same degree

Linear Equivalence

• Two Weil divisors $D$ and $D'$ are linearly equivalent if their difference $D - D'$ is a principal divisor
  • Linear equivalence is an equivalence relation on $Div(X)$
  • The group of linear equivalence classes is the Picard group $Pic(X) := Div(X)/Prin(X)$
• Two Cartier divisors are linearly equivalent if their difference is a principal Cartier divisor
  • The group of linear equivalence classes of Cartier divisors is isomorphic to $Pic(X)$
  • Example: on $\mathbb{P}^n$, all hyperplanes are linearly equivalent, as their differences are principal
• The degree is a homomorphism $deg : Div(X) \to \mathbb{Z}$ or $deg : CDiv(X) \to \mathbb{Z}$ that is invariant under linear equivalence
  • Thus the degree descends to a homomorphism $deg : Pic(X) \to \mathbb{Z}$
  • Example: on $\mathbb{P}^n$, $Pic(\mathbb{P}^n) \cong \mathbb{Z}$, generated by the class of a hyperplane, and the degree is an isomorphism