In the realm of algebraic geometry, divisors play a crucial role in understanding the geometric properties of algebraic varieties. A divisor is a formal linear combination of subvarieties of codimension one. Two divisors are said to be linearly equivalent if their difference is the divisor of a rational function. This notion is closely related to the concept of numerical equivalence, which is a weaker equivalence relation that focuses on the intersection behavior of divisors. While linearly equivalent divisors are always numerically equivalent, the converse does not hold in general. This article explores the relationship between these two equivalence relations and highlights the differences between them.
Linear Equivalence
Linear equivalence is a fundamental notion in the theory of divisors. Two divisors $D_1$ and $D_2$ on an algebraic variety $X$ are said to be linearly equivalent, denoted by $D_1 \sim D_2$, if there exists a rational function $f \in K(X)$ such that
$ D_1 - D_2 = \text{div}(f), $
where $\text{div}(f)$ is the divisor of $f$. This means that the difference between the two divisors is given by the poles and zeros of the rational function $f$.
Example: Consider the projective space $\mathbb{P}^2$ with homogeneous coordinates $[x:y:z]$. The divisors $D_1 = 2[x=0] - [y=0]$ and $D_2 = [x=0] - [z=0]$ are linearly equivalent because their difference is given by the divisor of the rational function $f = \frac{x^2}{yz}$, i.e.,
$ D_1 - D_2 = (2[x=0] - [y=0]) - ([x=0] - [z=0]) = [x=0] + [z=0] - [y=0] = \text{div}(f). $
Properties of Linear Equivalence:
- Reflexivity: $D \sim D$.
- Symmetry: If $D_1 \sim D_2$, then $D_2 \sim D_1$.
- Transitivity: If $D_1 \sim D_2$ and $D_2 \sim D_3$, then $D_1 \sim D_3$.
Linear equivalence captures the notion of divisors being "algebraically equivalent" in the sense that their difference is represented by a rational function. This equivalence relation plays a crucial role in the study of line bundles and the Picard group of an algebraic variety.
Numerical Equivalence
Numerical equivalence is a weaker equivalence relation than linear equivalence. Two divisors $D_1$ and $D_2$ on an algebraic variety $X$ are said to be numerically equivalent, denoted by $D_1 \equiv D_2$, if
$ D_1 \cdot C = D_2 \cdot C $
for all curves $C$ on $X$. Here, the dot product denotes the intersection number of divisors.
Example: Consider the projective space $\mathbb{P}^2$ with homogeneous coordinates $[x:y:z]$. The divisors $D_1 = [x=0] + [y=0]$ and $D_2 = [x=0] + [z=0]$ are numerically equivalent because they intersect any line in $\mathbb{P}^2$ at the same number of points. However, they are not linearly equivalent because there is no rational function whose divisor is their difference.
Properties of Numerical Equivalence:
- Reflexivity: $D \equiv D$.
- Symmetry: If $D_1 \equiv D_2$, then $D_2 \equiv D_1$.
- Transitivity: If $D_1 \equiv D_2$ and $D_2 \equiv D_3$, then $D_1 \equiv D_3$.
Numerical equivalence captures the notion of divisors being "geometrically equivalent" in the sense that they intersect curves in the same way. This equivalence relation is often easier to work with than linear equivalence, especially when dealing with complex intersections of divisors.
Relation between Linear and Numerical Equivalence
As mentioned earlier, linearly equivalent divisors are always numerically equivalent. This is because if $D_1 \sim D_2$, then their difference is the divisor of a rational function $f$. The intersection number of $\text{div}(f)$ with any curve $C$ is zero because the poles and zeros of $f$ cancel each other out along $C$. Therefore, $D_1 \cdot C = D_2 \cdot C$ for all curves $C$, implying $D_1 \equiv D_2$.
However, numerically equivalent divisors are not necessarily linearly equivalent. This is because numerical equivalence only considers intersection behavior, while linear equivalence takes into account the global structure of divisors.
Example: Consider an elliptic curve $E$ with a point $P$ on it. The divisors $D_1 = P$ and $D_2 = P + (P - P)$ are numerically equivalent because they both intersect any line on $E$ at one point. However, they are not linearly equivalent because their difference is the divisor of a non-constant function, which cannot be represented by a rational function on $E$.
Significance of Numerical Equivalence
Numerical equivalence plays a significant role in the study of algebraic varieties. It allows us to define the Néron-Severi group of an algebraic variety, which is the group of divisors modulo numerical equivalence. This group captures the geometric properties of divisors and is a powerful tool for studying the geometry of algebraic varieties.
Moreover, numerical equivalence is closely related to the concept of ample divisors. An ample divisor is a divisor that has a positive intersection number with any curve. Ample divisors play a crucial role in the study of projective varieties and are essential for understanding their geometric properties.
In conclusion, linearly equivalent divisors are numerically equivalent, but numerically equivalent divisors are not necessarily linearly equivalent. The concept of numerical equivalence provides a weaker, but often more useful, equivalence relation for divisors. It allows us to study the geometry of algebraic varieties in terms of intersection behavior, leading to a deeper understanding of their geometric properties. The notion of numerical equivalence is crucial for understanding the Néron-Severi group and ample divisors, which are fundamental concepts in algebraic geometry.
