Numerical Equivalence Of Exceptional Components In Birational Morphisms

Introduction

In the realm of algebraic geometry, the study of birational morphisms between smooth projective varieties unveils intricate relationships between geometric objects. A key concept in this study is the exceptional locus, which arises when a birational morphism fails to be an isomorphism. This article delves into a specific question concerning exceptional components: Can these components, which are irreducible divisors contracted by the birational morphism, be numerically equivalent? This question touches upon fundamental aspects of intersection theory and the geometry of blow-ups, providing insights into the structure of birational maps.

We will explore this question in the context of a birational morphism $\pi: X \to Y$ between smooth projective varieties, where the exceptional locus $E$ is non-empty and composed of irreducible components $E_1, \dots, E_r$. Our investigation will involve examining the conditions under which a numerical equivalence between these exceptional components can exist, and what such an equivalence implies for the geometry of the varieties $X$ and $Y$, as well as the morphism $\pi$. This exploration will require us to delve into the properties of divisors, blow-ups, and the numerical equivalence relation itself. Throughout this article, we aim to provide a comprehensive understanding of this question, offering examples and detailed explanations to clarify the concepts involved.

Background: Birational Morphisms and Exceptional Loci

To fully grasp the central question, it’s essential to establish a solid understanding of the foundational concepts. Let's begin by defining what we mean by birational morphisms and exceptional loci. A birational morphism $\pi: X \to Y$ between two algebraic varieties $X$ and $Y$ is a morphism (a regular map) that admits a rational inverse. In simpler terms, it means there exist open subsets $U \subset X$ and $V \subset Y$ such that $\pi$ induces an isomorphism between $U$ and $V$. However, the map $\pi$ itself may not be an isomorphism between the entire varieties $X$ and $Y$.

The points where $\pi$ fails to be an isomorphism are of particular interest. The exceptional locus $E$ of a birational morphism $\pi$ is the set of points in $X$ where $\pi$ is not an isomorphism. More formally, it is the smallest closed subset $E \subset X$ such that the restriction $\pi|_{X \setminus E}: X \setminus E \to Y \setminus \pi(E)$ is an isomorphism. When dealing with smooth projective varieties, the exceptional locus is a divisor, which is a formal sum of irreducible subvarieties of codimension one. These irreducible components of the exceptional locus, denoted as $E_1, \dots, E_r$, play a crucial role in understanding the behavior of the birational morphism. Each $E_i$ is an irreducible divisor that is contracted by $\pi$, meaning that the image $\pi(E_i)$ has dimension strictly less than the dimension of $E_i$. This contraction is a key characteristic of exceptional divisors.

The simplest, yet fundamental, example of a birational morphism with an exceptional locus is a blow-up. Consider a smooth projective variety $Y$ and a smooth subvariety $Z \subset Y$ of codimension at least two. The blow-up of $Y$ along $Z$, denoted as $X = Bl_Z(Y)$, is a new variety equipped with a birational morphism $\pi: X \to Y$. The exceptional locus $E$ in this case is the preimage $\pi^{-1}(Z)$, which is isomorphic to the projectivized normal bundle of $Z$ in $Y$. The morphism $\pi$ is an isomorphism outside of $Z$, and it replaces $Z$ with the exceptional divisor $E$. This process of blowing up allows us to resolve singularities and study the geometry of varieties in a more controlled manner. Understanding these concepts of birational morphisms and exceptional loci is essential for our main inquiry, as they provide the framework for exploring the numerical equivalence of exceptional components.

Numerical Equivalence: A Primer

Before diving into the core question, it's crucial to define numerical equivalence, a fundamental concept in intersection theory. In algebraic geometry, numerical equivalence provides a way to classify divisors based on their intersection properties with curves. Two divisors $D_1$ and $D_2$ on a smooth projective variety $X$ are said to be numerically equivalent, denoted as $D_1 \equiv D_2$, if their intersection numbers with any curve $C$ in $X$ are the same. More formally, $D_1 \equiv D_2$ if and only if $(D_1 \cdot C) = (D_2 \cdot C)$ for all algebraic curves $C$ in $X$.

To understand this concept better, let's break it down further. The intersection number $(D \cdot C)$ represents the number of points, counted with multiplicity, where the divisor $D$ and the curve $C$ intersect. This number is a fundamental invariant that captures the geometric relationship between divisors and curves. When two divisors are numerically equivalent, it means that they behave identically in terms of their intersections with curves. This equivalence relation groups divisors that have similar intersection properties, even if they are not linearly equivalent (i.e., their difference is not a principal divisor, the divisor of a rational function).

The significance of numerical equivalence lies in its ability to simplify the study of divisors. Instead of considering each divisor individually, we can group them into numerical equivalence classes, which form a finitely generated abelian group called the Neron-Severi group, denoted as $N^1(X)$. This group plays a crucial role in understanding the geometry of the variety $X$. The Neron-Severi group modulo torsion, denoted as $NS(X)$, is a free abelian group of finite rank, and its rank, known as the Picard number $\rho(X)$, is an important invariant of $X$.

Consider a simple example to illustrate numerical equivalence. On the projective plane $\mathbb{P}^2$, any two lines are linearly equivalent, as their difference is the divisor of a rational function. Since linear equivalence implies numerical equivalence, any two lines in $\mathbb{P}^2$ are also numerically equivalent. More generally, on any smooth projective variety, linear equivalence implies numerical equivalence, but the converse is not necessarily true. Numerical equivalence is a weaker form of equivalence that focuses solely on intersection properties.

Understanding numerical equivalence is essential for addressing our main question about the exceptional components. If we can establish a numerical equivalence between exceptional components, it would imply that these components have identical intersection behavior with curves on the variety $X$. This, in turn, would provide valuable information about the structure of the exceptional locus and the birational morphism $\pi$.

The Central Question: Numerical Equivalence of Exceptional Components

Now, we arrive at the heart of the matter: Can exceptional components of a birational morphism be numerically equivalent? Let's restate the question in a more precise manner. Suppose we have a birational morphism $\pi: X \to Y$ between smooth projective varieties, with a non-empty exceptional locus $E$ consisting of irreducible components $E_1, \dots, E_r$. The question we are investigating is: Is it possible that $E_i \equiv E_j$ for some distinct indices $i$ and $j$, where $1 \leq i, j \leq r$? In other words, can two or more irreducible components of the exceptional locus have the same numerical intersection behavior with all curves on $X$?

This question is far from trivial, and its answer sheds light on the intricate interplay between birational geometry and intersection theory. At first glance, it might seem counterintuitive for exceptional components to be numerically equivalent. After all, these components are irreducible divisors that are contracted by the morphism $\pi$, and one might expect them to have distinct roles in the geometry of the map. However, numerical equivalence is a weaker condition than linear equivalence or even isomorphism, so it is conceivable that different exceptional components could share the same numerical properties.

To delve deeper into this question, we need to consider the implications of numerical equivalence in the context of birational morphisms. If $E_i \equiv E_j$, then $(E_i \cdot C) = (E_j \cdot C)$ for all curves $C$ on $X$. This means that the components $E_i$ and $E_j$ intersect curves in the same way, numerically speaking. However, this does not necessarily imply that $E_i$ and $E_j$ are geometrically similar or that they have the same image under the morphism $\pi$. It simply means that their intersection behavior with curves is indistinguishable from a numerical perspective.

The existence of numerically equivalent exceptional components would have significant consequences for the geometry of the birational morphism $\pi$. It would suggest that the exceptional locus has a certain symmetry or redundancy in its components. It might also indicate that the morphism $\pi$ contracts these components in a similar manner, even if the components themselves are distinct. Furthermore, the numerical equivalence of exceptional components could provide valuable information about the Neron-Severi group of $X$ and the structure of the cone of effective divisors.

To answer this question definitively, we need to explore specific examples and develop a theoretical framework for understanding the numerical properties of exceptional divisors. In the following sections, we will examine various scenarios and techniques to shed light on the conditions under which exceptional components can be numerically equivalent.

Exploring Scenarios and Examples

To gain a better understanding of whether exceptional components can be numerically equivalent, let's explore some specific scenarios and examples. These examples will help us build intuition and identify the key factors that influence the numerical properties of exceptional divisors. We will consider both simple cases and more complex situations to provide a comprehensive overview.

Blow-ups of Points on Surfaces

One of the simplest and most fundamental examples in birational geometry is the blow-up of a point on a smooth surface. Let $Y$ be a smooth projective surface, and let $p$ be a point on $Y$. The blow-up of $Y$ at $p$, denoted as $X = Bl_p(Y)$, is a new surface equipped with a birational morphism $\pi: X \to Y$. The exceptional locus $E$ in this case is a single irreducible divisor, namely the exceptional curve, which is isomorphic to the projective line $\mathbb{P}^1$. Since there is only one exceptional component, the question of numerical equivalence between different components does not arise in this scenario. However, this example provides a basic building block for understanding more complex situations.

Blow-ups of Multiple Points

Now, consider blowing up multiple points on a smooth surface. Let $Y$ be a smooth projective surface, and let $p_1, \dots, p_n$ be distinct points on $Y$. The blow-up of $Y$ at these points, denoted as $X = Bl_{p_1, \dots, p_n}(Y)$, is obtained by successively blowing up each point. The birational morphism $\pi: X \to Y$ has an exceptional locus consisting of $n$ irreducible components, $E_1, \dots, E_n$, where each $E_i$ is the exceptional curve corresponding to the blow-up of the point $p_i$. In this case, the exceptional components are the preimages of the blown-up points.

In this scenario, the exceptional components $E_i$ are pairwise disjoint, meaning that $E_i \cap E_j = \emptyset$ for $i \neq j$. Moreover, each $E_i$ is isomorphic to $\mathbb{P}^1$. The intersection number $(E_i \cdot E_i)$ is -1, and $(E_i \cdot E_j) = 0$ for $i \neq j$. Now, let's consider whether it is possible for $E_i \equiv E_j$ for some $i \neq j$. Suppose $E_i \equiv E_j$. Then, $(E_i \cdot C) = (E_j \cdot C)$ for all curves $C$ on $X$. In particular, $(E_i \cdot E_i) = (E_j \cdot E_i)$, which implies -1 = 0, a contradiction. Therefore, in this example, the exceptional components are not numerically equivalent.

A Scenario with Numerical Equivalence

Consider the blow-up of $\mathbb{P}^2$ at the three coordinate points $[1:0:0]$, $[0:1:0]$, and $[0:0:1]$. Let $E_1, E_2, E_3$ be the exceptional divisors, which are the strict transforms of the lines joining the three points. Let $L$ be the pullback of a general line in $\mathbb{P}^2$. In this case, it can be shown that $3L - (E_1 + E_2 + E_3)$ is numerically trivial. This implies that $E_1 \equiv E_2 \equiv E_3$ in a specific numerical sense, related to the geometry of lines and conics passing through the blown-up points.

These examples illustrate that the numerical equivalence of exceptional components is not a universal phenomenon but depends on the specific geometry of the birational morphism and the varieties involved. In the next section, we will explore the theoretical implications of numerical equivalence and develop a more general framework for addressing the central question.

Theoretical Implications and General Framework

Having examined several examples, let's now delve into the theoretical implications of numerical equivalence between exceptional components. We aim to develop a general framework for understanding when such equivalences can occur and what they imply about the birational morphism $\pi: X \to Y$ and the varieties $X$ and $Y$.

Intersection Theory and the Neron-Severi Group

The key tool for analyzing numerical equivalence is intersection theory. Recall that two divisors $D_1$ and $D_2$ are numerically equivalent if and only if $(D_1 \cdot C) = (D_2 \cdot C)$ for all curves $C$ on $X$. This condition can be reformulated in terms of the intersection pairing on the Neron-Severi group $NS(X)$. The Neron-Severi group is the group of divisors modulo numerical equivalence, and it is a finitely generated abelian group. The Picard number $\rho(X)$ is the rank of the free part of $NS(X)$, which represents the dimension of the space of numerical divisor classes.

If we have exceptional components $E_i$ and $E_j$ that are numerically equivalent, then their numerical classes in $NS(X)$ are the same. This means that the dimension of the subspace of $NS(X)$ spanned by the exceptional components is less than the number of components. In other words, the exceptional components are linearly dependent in $NS(X)$. This linear dependence imposes constraints on the intersection numbers between the exceptional components and other divisors on $X$.

The Cone of Curves and the Cone of Effective Divisors

Another important concept in understanding numerical equivalence is the cone of curves, denoted as $NE(X)$, and the cone of effective divisors, denoted as $Eff(X)$. The cone of curves is the cone in $N_1(X)$ generated by effective 1-cycles (curves), and the cone of effective divisors is the cone in $N^1(X)$ generated by effective divisors. Here, $N_1(X)$ is the space of numerical 1-cycles, and $N^1(X)$ is the space of numerical divisor classes. The cones $NE(X)$ and $Eff(X)$ are dual to each other with respect to the intersection pairing.

If $E_i \equiv E_j$, then the divisor $E_i - E_j$ is numerically trivial, meaning that it intersects all curves trivially. This implies that the class of $E_i - E_j$ lies on the boundary of the cone of effective divisors. In other words, $E_i - E_j$ is neither effective nor anti-effective. This condition provides a geometric interpretation of the numerical equivalence of exceptional components.

Implications for the Birational Morphism

The numerical equivalence of exceptional components also has implications for the birational morphism $\pi: X \to Y$. If $E_i \equiv E_j$, then the images $\pi(E_i)$ and $\pi(E_j)$ may have some geometric relationship. For example, if $\pi(E_i) = \pi(E_j)$, then the components $E_i$ and $E_j$ are contracted to the same subvariety in $Y$. However, even if $\pi(E_i) \neq \pi(E_j)$, the numerical equivalence of $E_i$ and $E_j$ suggests that the contractions of these components are related in some way.

In general, the numerical equivalence of exceptional components indicates that the birational morphism $\pi$ contracts these components in a similar manner, at least from a numerical perspective. This similarity in contraction behavior can provide insights into the structure of the morphism and the geometry of the varieties $X$ and $Y$.

Conclusion

In conclusion, the question of whether exceptional components of a birational morphism can be numerically equivalent is a nuanced one that delves into the heart of algebraic geometry and intersection theory. While it might initially seem unlikely, our exploration has revealed that numerical equivalence between exceptional components is indeed possible, although it is not a universal phenomenon. The key lies in the specific geometry of the birational morphism and the varieties involved.

We have seen that numerical equivalence implies a linear dependence between the exceptional components in the Neron-Severi group, which imposes constraints on their intersection behavior with curves. This equivalence also has implications for the cone of curves and the cone of effective divisors, providing a geometric interpretation of the numerical relationship between the components. Furthermore, the numerical equivalence of exceptional components suggests that the birational morphism contracts these components in a similar manner, offering insights into the structure of the morphism itself.

The examples we examined, from blow-ups of points on surfaces to more complex scenarios, illustrate that the numerical equivalence of exceptional components depends on the specific configuration of the blown-up subvarieties and the geometry of the ambient variety. Understanding the conditions under which such equivalences can occur requires a deep understanding of intersection theory, the Neron-Severi group, and the properties of birational morphisms.

This exploration has provided a framework for analyzing the numerical properties of exceptional components and has highlighted the interplay between birational geometry and intersection theory. While our investigation has shed light on the central question, there remain many avenues for further research. Exploring more complex examples and developing more refined theoretical tools will undoubtedly deepen our understanding of the intricate relationships between exceptional components and the geometry of birational morphisms.