Numerical Equivalence Of Exceptional Components In Birational Morphisms
Introduction
In the realm of algebraic geometry, particularly within the study of birational geometry, the behavior of exceptional components under birational morphisms is a fascinating and crucial area of investigation. This article delves into a specific question regarding the numerical equivalence of exceptional components arising from a birational morphism between smooth projective varieties. Specifically, we address the query: Can exceptional components in a birational morphism be numerically equivalent? To explore this, we consider a birational morphism π:X→Y between smooth projective varieties, where the exceptional locus E is non-empty and composed of irreducible components E₁, ..., Eᵣ. This article aims to provide a comprehensive exploration of this question, offering insights and a detailed analysis.
Birational Morphisms and Exceptional Loci: Foundations of the Discussion
To understand the intricacies of numerical equivalence among exceptional components, it's essential to first establish a strong foundation in birational geometry. At the heart of our discussion lies the concept of a birational morphism. A birational morphism π: X → Y between algebraic varieties X and Y is a morphism that induces an isomorphism between open subsets U ⊂ X and V ⊂ Y. In simpler terms, it's a map that's almost an isomorphism, except possibly on some lower-dimensional subsets. These subsets are where the interesting behavior often occurs, particularly within the exceptional locus.
The exceptional locus, denoted as E, is the set of points in X where the morphism π fails to be an isomorphism. Formally, it's the set of points x ∈ X such that π is not an isomorphism in any neighborhood of x. This locus plays a pivotal role in understanding how birational morphisms transform varieties. When the exceptional locus is non-empty, it indicates that the birational morphism is more than just a simple isomorphism; it involves some geometric transformation. The irreducible components of the exceptional locus, denoted as E₁, ..., Eᵣ, are the fundamental building blocks of this transformation. Each component represents a subvariety that is 'collapsed' or 'blown down' by the morphism π.
Understanding the nature of these components – their geometry, their intersections, and their numerical properties – is crucial for unraveling the global behavior of the birational morphism. The numerical equivalence of these components, which we will delve into shortly, provides a powerful lens through which to examine these transformations. The properties of divisors and their behavior under blow-ups, another key concept in birational geometry, further enrich this analysis, providing tools to dissect the structure of exceptional loci and their components. Exploring these foundational concepts sets the stage for a deeper investigation into the central question of this article: whether these exceptional components can be numerically equivalent.
Numerical Equivalence: A Key Concept in Algebraic Geometry
The concept of numerical equivalence is a cornerstone in algebraic geometry, especially when dealing with divisors and cycles on algebraic varieties. To understand whether exceptional components can be numerically equivalent, we must first define what numerical equivalence means in this context. Two divisors, D₁ and D₂, on a smooth projective variety X are said to be numerically equivalent if their intersection numbers with any curve C on X are the same. Mathematically, this is expressed as D₁ ⋅ C = D₂ ⋅ C for all curves C on X. In essence, numerical equivalence captures the idea that two divisors behave similarly in terms of intersections with curves, even if they are not linearly equivalent or even Cartier divisors.
This notion is weaker than other forms of equivalence, such as linear equivalence, which requires that the difference D₁ - D₂ is the divisor of a rational function. Numerical equivalence focuses solely on the intersection properties, ignoring other algebraic or geometric differences. This focus makes it a powerful tool for studying the coarse geometric properties of divisors and varieties. For instance, it plays a crucial role in the study of the Néron-Severi group, which is the group of divisors modulo numerical equivalence. The rank of this group, known as the Picard number, is a fundamental invariant of the variety, reflecting its complexity and the richness of its divisor structure.
Understanding numerical equivalence allows us to classify divisors based on their intersection behavior. Divisors that are numerically equivalent belong to the same numerical class, which provides a coarser classification than linear equivalence. This classification is particularly useful when studying birational transformations, as numerical equivalence is often preserved under such transformations, while linear equivalence may not be. The question of whether exceptional components can be numerically equivalent taps into this idea of classification and behavior under transformations. If two exceptional components are numerically equivalent, it implies a certain symmetry or balance in how they intersect curves on the variety, which can have significant implications for the geometry of the birational morphism. Delving into the conditions under which this can occur requires a careful examination of the geometry of blow-ups and the behavior of divisors under these transformations.
Blow-Ups and Exceptional Divisors: The Building Blocks of Birational Maps
Blow-ups are fundamental operations in birational geometry, and they are intricately linked to the creation of exceptional divisors. To address the question of whether exceptional components can be numerically equivalent, it is crucial to understand how blow-ups give rise to these components and how they behave. A blow-up is a birational morphism that replaces a subvariety of a given variety with the projectivization of its normal bundle. The most common example is blowing up a smooth point on a variety. In this case, the point is replaced by a projective space, which becomes the exceptional divisor.
Formally, let Y be a smooth variety, and let Z be a smooth subvariety of Y. The blow-up of Y along Z, denoted as X = Bl_Z(Y), is a morphism π: X → Y such that π restricts to an isomorphism outside of Z, and π⁻¹(Z) = E is the exceptional divisor. The exceptional divisor E is isomorphic to the projectivization of the normal bundle of Z in Y. When we blow up a smooth point, the exceptional divisor is a projective space of one dimension less than the dimension of Y. For instance, blowing up a point on a surface creates an exceptional divisor that is isomorphic to the projective line P¹. The blow-up process introduces a new variety X and a morphism π that modifies Y in a controlled way, replacing the subvariety Z with the exceptional divisor E.
The exceptional divisor E is crucial because it encodes information about the singularity that was resolved by the blow-up. In the case of blowing up a point, the exceptional divisor records the directions approaching the point. More generally, the exceptional divisor provides a way to separate tangent directions and resolve singularities. The intersection properties of the exceptional divisor and its components with other curves and divisors on X are of paramount importance. These intersection numbers dictate the numerical behavior of the exceptional components and, consequently, influence whether they can be numerically equivalent. The self-intersection of the exceptional divisor, for example, is a key invariant that reflects the nature of the blow-up. Understanding the geometry of blow-ups and the properties of exceptional divisors is essential for tackling the central question of this article: the numerical equivalence of exceptional components in birational morphisms.
Numerical Equivalence of Exceptional Components: Possibilities and Constraints
The central question we address is whether exceptional components in a birational morphism can be numerically equivalent. Considering a birational morphism π: X → Y between smooth projective varieties with a non-empty exceptional locus E and irreducible components E₁, ..., Eᵣ, we delve into the conditions under which these components can be numerically equivalent.
To reiterate, two divisors Eᵢ and Eⱼ are numerically equivalent if their intersection numbers with any curve C on X are the same, i.e., Eᵢ ⋅ C = Eⱼ ⋅ C for all curves C. This condition implies a certain symmetry in how these components intersect curves on the variety. However, the geometry of blow-ups and birational morphisms imposes significant constraints on when this can occur. One primary constraint arises from the fact that exceptional components are often contracted by the morphism π. This contraction has implications for their intersection numbers and, consequently, their numerical equivalence.
Consider the case where π is a single blow-up of a smooth subvariety Z in Y. The exceptional divisor E is the projectivization of the normal bundle of Z in Y. In this scenario, the components of E are closely related to the geometry of Z and its embedding in Y. If E has multiple irreducible components, their intersection properties are determined by the singularities of Z or the way different components of Z intersect. In simpler blow-up scenarios, such as blowing up a smooth point on a surface, the exceptional divisor is irreducible (e.g., a projective line), so the question of numerical equivalence between components does not arise. However, in more complex situations, such as blowing up a singular curve or a subvariety with multiple components, the exceptional locus can have multiple irreducible components.
For the exceptional components to be numerically equivalent, their intersection numbers with all curves must be identical. This requires a high degree of symmetry in the geometry of the blow-up. For instance, if two components intersect a curve C in different ways (e.g., one intersects transversally, and the other does not intersect at all), they cannot be numerically equivalent. Similarly, if the components have different self-intersection numbers, they cannot be numerically equivalent. Self-intersection numbers are particularly important because they reflect the intrinsic geometry of the components and their embedding in the variety. Numerical equivalence would imply that these components are, in some sense, geometrically interchangeable, which is a strong condition.
Thus, while it is possible for exceptional components to be numerically equivalent, it requires very specific geometric configurations and symmetries. The constraints imposed by the nature of blow-ups and birational morphisms significantly limit the scenarios in which this can occur. Exploring specific examples and cases can further illuminate these possibilities and constraints.
Examples and Cases: Illustrating the Possibilities
To better understand the conditions under which exceptional components can be numerically equivalent, let's explore some illustrative examples and cases. These examples will highlight both the possibilities and the constraints discussed earlier.
Case 1: Blow-up of a Smooth Point on a Surface
Consider the simplest scenario: blowing up a smooth point on a smooth surface. Let Y be a smooth surface, and let X = Blₚ(Y) be the blow-up of Y at a smooth point p. The exceptional divisor E in this case is isomorphic to the projective line P¹. Since there is only one irreducible component (E itself), the question of numerical equivalence between components does not arise. This case serves as a basic example where the exceptional locus is simple, and there are no multiple components to compare.
Case 2: Blow-up of a Node on a Surface
Now, consider a more complex scenario: blowing up a node (an ordinary double point) on a surface. Let Y be a surface with a node at point p, and let X = Blₚ(Y) be the blow-up of Y at p. In this case, the exceptional locus E consists of two irreducible components, E₁ and E₂, which are both isomorphic to the projective line P¹. These components intersect at a single point. While they are geometrically similar, they are not numerically equivalent. To see this, consider their intersection matrix. The self-intersection numbers are E₁² = E₂² = -1, and the intersection number E₁ ⋅ E₂ = 1. Since the components intersect each other, but each has a negative self-intersection, they cannot be numerically equivalent. For example, a curve that intersects E₁ but not E₂ will have different intersection numbers with the two components.
Case 3: Simultaneous Blow-up of Multiple Points
Consider blowing up multiple smooth points simultaneously on a surface. Let Y be a smooth surface, and let X be the blow-up of Y at n distinct smooth points p₁, ..., pₙ. The exceptional locus E consists of n irreducible components E₁, ..., Eₙ, each isomorphic to P¹, corresponding to the blow-up of each point. If the points are chosen generically, the exceptional components Eᵢ are disjoint and each has self-intersection -1. In this case, the components are numerically equivalent because for any curve C on X, if C is not one of the exceptional curves, then Eᵢ ⋅ C = 0. If C = Eᵢ, then Eᵢ ⋅ Eᵢ = -1, and if C = Eⱼ for i ≠ j, then Eᵢ ⋅ Eⱼ = 0. Thus, all the Eᵢ are numerically equivalent.
Case 4: Blow-up of a Singular Curve
Finally, consider blowing up a singular curve on a surface. This case can lead to more intricate exceptional loci with multiple components. The numerical equivalence of these components would depend on the specific singularities of the curve and how they are resolved by the blow-up. The intersection properties of the components would need to be carefully analyzed to determine if numerical equivalence is possible. This is a scenario where symmetry in the resolution process could potentially lead to numerically equivalent components.
These examples illustrate that the numerical equivalence of exceptional components is not a given. It requires specific geometric conditions and symmetries, and it is not a common occurrence in general birational morphisms. The geometry of the blow-up, the nature of the blown-up subvariety, and the intersection properties of the resulting components all play a crucial role in determining whether numerical equivalence is possible.
Conclusion
In conclusion, the question of whether exceptional components in a birational morphism can be numerically equivalent is a nuanced one. While the possibility exists, it is constrained by the inherent geometry of blow-ups and birational transformations. The specific configurations and symmetries required for numerical equivalence to hold are not universally present, making it a somewhat rare phenomenon in the broader landscape of birational geometry.
The key takeaway is that numerical equivalence among exceptional components implies a significant degree of symmetry in how these components interact with curves on the variety. This symmetry is not automatic; it arises from specific geometric circumstances, such as simultaneous blow-ups of points or symmetric resolutions of singularities. The examination of various cases, from blowing up smooth points to singular curves, reveals the spectrum of possibilities and constraints.
Further exploration of this topic could delve into more complex birational transformations and the role of numerical equivalence in minimal model programs. Understanding the numerical properties of exceptional components is crucial for classifying varieties and understanding their birational behavior. The interplay between geometry and arithmetic, as captured by numerical equivalence, continues to be a vibrant area of research in algebraic geometry.