Blowing up is a key technique for resolving singularities in algebraic geometry. It involves replacing a singular point with a new subvariety, separating tangent directions and creating a smoother space. This process is crucial for understanding and simplifying complex geometric objects.
is a fundamental problem in algebraic geometry. For curves and surfaces, it can be achieved through a series of blow-ups. This process transforms singular varieties into smooth ones, making them easier to study and analyze.
Blowing Up Varieties
Concept and Properties of Blowing Up
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Blowing up a variety X at a point or subvariety Y is a geometric transformation that creates a new variety X′ with a projection map back to X
The preimage of the blown-up point or subvariety Y under the projection map is called the E, which is a subvariety of X′
The exceptional divisor E has codimension one in X′
E is isomorphic to the projectivized normal bundle of Y in X
The separates the tangent directions at the point or subvariety Y, creating a new variety X′ that is less singular than the original variety X
Intuitively, the blow-up "pulls apart" the tangent directions at Y, replacing Y with a divisor E that encodes these directions
The blow-up resolves certain types of singularities, such as ordinary double points on curves (nodes)
Algebraic and Geometric Interpretations
The blow-up process can be understood algebraically by considering the projective closure of the graph of a rational map from X to a
Given a rational map f:X⇢Pn, the blow-up of X along the base locus of f is the closure of the graph of f in X×Pn
The blow-up of a variety at a point can be constructed as the projective spectrum of a certain graded algebra, called the Rees algebra
For an ideal I in a ring R, the Rees algebra is defined as ⨁n≥0In, where In is the n-th power of the ideal I
The projective spectrum of the Rees algebra ⨁n≥0In is isomorphic to the blow-up of Spec(R) along the subscheme defined by I
Constructing Blow-Ups
Affine and Projective Cases
To construct the blow-up of an affine variety X at a point p, consider the ideal I(p) of functions vanishing at p and form the graded algebra ⨁n≥0I(p)n. The projective spectrum of this algebra is the blow-up of X at p
Example: For the affine plane A2 and the origin p=(0,0), the ideal I(p) is generated by x and y. The blow-up of A2 at p is the projective spectrum of ⨁n≥0(x,y)n
For a subvariety Y of X, the blow-up of X along Y is constructed similarly using the ideal of Y
The blow-up of a projective variety X at a point p can be constructed by taking the closure of the graph of the rational map from X to the projective space of lines through p
Example: The blow-up of P2 at a point p is isomorphic to the Hirzebruch surface F1, which is a P1-bundle over P1
Explicit Equations and Exceptional Divisors
Explicitly, the blow-up of the affine plane A2 at the origin (0,0) is the subvariety of A2×P1 defined by the equation xv=yu, where (x,y) are coordinates on A2 and (u:v) are homogeneous coordinates on P1
The projection map from the blow-up to A2 is given by (x,y,(u:v))↦(x,y)
The exceptional divisor E in the blow-up of A2 at the origin is the preimage of (0,0) under the projection map, which is isomorphic to P1
In the above equations, E is defined by the equations x=y=0, which gives the projective line with coordinates (u:v)
Similar explicit constructions can be given for blow-ups of other varieties at points or subvarieties using local equations and coordinates
Resolving Singularities
Iterative Blow-Up Process
Resolving the singularities of a variety X means finding a smooth variety X′ and a proper birational morphism f:X′→X that is an isomorphism over the smooth locus of X
The variety X′ is called a resolution of singularities of X
The blow-up process can be used iteratively to resolve singularities by blowing up the singular points or subvarieties of X until a smooth variety is obtained
At each step, the blow-up separates the tangent directions at the singular point or subvariety, creating a less singular variety
The process terminates when all singular points have been resolved and the resulting variety is smooth
Curves and Surfaces
For curves, blowing up a singular point replaces it with a copy of P1, effectively separating the branches of the curve at that point. Repeated blow-ups will eventually resolve all singularities
Example: Consider the curve y2=x3 (a nodal cubic). Blowing up the origin once resolves the singularity, resulting in a smooth curve isomorphic to P1
For surfaces, the resolution of singularities may require a sequence of blow-ups. The intersection graph of the exceptional divisors created in the process is a useful tool for understanding the resolution
Example: The resolution of the singularity of the surface z2=xy (a cone) requires a sequence of two blow-ups. The first blow-up creates an exceptional divisor isomorphic to P1, and the second blow-up resolves the remaining singularity
The minimal resolution of a surface singularity is the resolution that introduces the fewest exceptional divisors. It can be obtained by blowing up only the singular points and not any smooth points
The minimal resolution is unique up to isomorphism and has important geometric and algebraic properties
Resolution of Singularities for Curves and Surfaces
Curves: Normalization and Blow-Ups
For curves, the existence of a resolution of singularities follows from the normalization theorem, which states that every reduced curve has a unique normalization (a smooth curve birational to the original curve)
The normalization of a curve is a resolution of singularities that minimizes the genus of the resulting smooth curve
The normalization of a curve can be constructed explicitly by blowing up the singular points repeatedly until a smooth curve is obtained
Each blow-up reduces the delta-invariant (a measure of singularity) of the singular point, and the process terminates when all points have delta-invariant zero (i.e., are smooth)
Example: The normalization of the cuspidal cubic y2=x3 is isomorphic to A1, obtained by a sequence of three blow-ups at the origin
Surfaces: Existence and Induction on Multiplicity
For surfaces, the existence of a resolution of singularities was proved by Walker (1935) and Zariski (1939) using the blow-up process
The proof involves showing that the singularities of a surface can be improved (i.e., made simpler) by a sequence of blow-ups, and that this process must terminate after a finite number of steps
The complexity of a singularity is measured by its multiplicity, which is the degree of the lowest degree term in the local equation of the surface at the singular point
A key step in the proof is the "induction on the multiplicity" argument, which shows that the multiplicity of a singular point decreases after a blow-up, and hence the process must eventually stop
More precisely, if the multiplicity of a singular point is m, then after a blow-up, the multiplicity of any singular point in the preimage is strictly less than m
The resolution of singularities for surfaces can also be proved using the concept of the "infinitely near points" and the "tree of infinitely near points" associated with a singular point
Infinitely near points are points on the exceptional divisors created by successive blow-ups, and the tree encodes the configuration of these points
The resolution process can be understood as a sequence of blow-ups that "untangles" the tree of infinitely near points, eventually resulting in a tree with only smooth points
Key Terms to Review (18)
Blow-up: A blow-up is a geometric transformation that replaces a point, often a singular point, in a variety with a projective space, creating a new variety that resolves the singularity. This technique is essential in studying the structure of varieties and understanding their local properties, particularly how they behave near singular points. It allows for the examination of tangent cones and provides insights into the resolution of singularities by enabling more manageable geometric analysis.
David Mumford: David Mumford is a prominent mathematician known for his significant contributions to algebraic geometry and his work on moduli spaces. His research has greatly influenced various areas of mathematics, including the study of curves, surfaces, and the classification of algebraic varieties, making him a pivotal figure in modern geometry.
Dominant Morphism: A dominant morphism is a type of morphism between two varieties where the image of the morphism is dense in the target variety. This concept is crucial because it indicates that the source variety can be effectively 'mapped' onto a significant part of the target variety, allowing for important interactions between them. Dominant morphisms help us understand relationships between geometric objects, especially when analyzing rational maps and the effects of processes like blowing up and resolution of singularities.
Exceptional divisor: An exceptional divisor is a type of divisor that arises during the process of blowing up a variety at a point, which resolves singularities. In essence, it represents the preimage of the point where the blow-up occurs and provides a way to understand the structure of the modified variety. This concept is crucial for analyzing how singularities can be 'resolved' or altered into a smoother geometric form.
Fiber product: The fiber product is a construction in category theory and algebraic geometry that combines two schemes over a common base scheme into a new scheme, reflecting the relationships between them. It allows for a geometric understanding of how different schemes relate when restricted to a specific base. This construction is especially useful when dealing with morphisms and studying properties of varieties under different projections.
Finite morphism: A finite morphism is a type of morphism between schemes that is affine and satisfies the condition of being finitely presented, meaning it corresponds to a finite type of ring homomorphism. In simpler terms, this means that the preimage of any affine open set is a scheme that is covered by finitely many affine opens. Finite morphisms are crucial in algebraic geometry as they allow for the analysis of algebraic structures while maintaining a level of compactness and control.
Heisuke Hironaka: Heisuke Hironaka is a prominent mathematician known for his groundbreaking work in the field of algebraic geometry, particularly for proving the resolution of singularities in characteristic zero. His contributions have significantly impacted the understanding of singularities and their resolutions, allowing mathematicians to analyze more complex algebraic structures. Hironaka's methods and theorems are foundational in blowing up singularities and are essential for advancing various theories in algebraic geometry.
Hironaka's Theorem: Hironaka's Theorem states that every algebraic variety over a field of characteristic zero has a resolution of singularities. This means that for any given algebraic variety, it is possible to replace it with another variety that is smooth and behaves nicely, thus making the study of its geometric properties more manageable. This theorem is pivotal in understanding the structure of algebraic varieties and their singular points, connecting deeply with methods of blowing up and classifying algebraic surfaces.
Irreducible Variety: An irreducible variety is a type of algebraic variety that cannot be expressed as a union of two or more proper subvarieties. This means that an irreducible variety is 'whole' in the sense that it cannot be decomposed into simpler pieces, reflecting the idea that it is defined by a single polynomial equation or a set of equations with no common factors. Understanding irreducible varieties is essential because they serve as the building blocks for more complex varieties and play a crucial role in various concepts like blowing up and the resolution of singularities, as well as in the study of affine varieties and polynomial rings.
Isolated Singularity: An isolated singularity is a point at which a function, curve, or surface fails to be well-defined or is not differentiable, but is surrounded by points where it behaves normally. This concept is crucial in understanding how singularities can be 'resolved' or 'blown up' to study the underlying geometry. In many cases, these singularities are pivotal in determining the overall structure and properties of algebraic varieties.
Non-Isolated Singularity: A non-isolated singularity is a point on an algebraic variety where the singularity is not a standalone feature; instead, it exists in a broader context of other singular points. This means that there are infinitely many nearby points that also exhibit singular behavior. Non-isolated singularities often arise in situations where multiple curves or surfaces intersect or overlap, leading to more complex geometric and algebraic structures.
Projective Space: Projective space is a fundamental concept in algebraic geometry that extends the idea of Euclidean space by adding 'points at infinity' to account for parallel lines meeting. This transformation allows for a more comprehensive understanding of geometric properties and relationships among various geometric objects, such as varieties, curves, and surfaces.
Proper Transform: A proper transform is a concept in algebraic geometry that refers to a specific way of modifying a geometric object when performing a blow-up. When you blow up a space along a subvariety, the proper transform of another subvariety represents how that subvariety changes under this blow-up operation. This idea is crucial for understanding how singularities are resolved, as the proper transform helps track how the original objects relate to the new geometries created in the process.
Reduced Scheme: A reduced scheme is a type of scheme where the underlying ring has no non-zero nilpotent elements. This means that for any element in the structure sheaf, if it is nilpotent, then it must be zero. Reduced schemes are crucial in understanding various geometric properties, especially when dealing with singularities and morphisms between schemes, as they maintain a level of 'non-degenerate' behavior.
Resolution of singularities: Resolution of singularities is a process in algebraic geometry that aims to replace a singular variety with a new variety that has no singularities. This is crucial for understanding the geometry and topology of spaces, as well as for simplifying calculations. By resolving singularities, we can gain insights into the behavior of functions near these problematic points and study the structure of varieties more effectively.
Ring of regular functions: The ring of regular functions on a variety is a collection of functions that are defined and behave well (like polynomials) on the variety, meaning they can be expressed as quotients of polynomials where the denominator does not vanish. This ring captures the algebraic structure of the variety and plays a critical role in understanding both its geometry and singularities, which is essential for working with concepts like blowing up and determining normality or Cohen-Macaulay properties.
Sheaf: A sheaf is a mathematical tool used to systematically track local data attached to the open sets of a topological space, allowing us to study global properties through local behavior. Sheaves enable the construction of cohomology theories and facilitate the resolution of singularities in algebraic varieties, providing a bridge between local and global geometric properties.
Zariski's Main Theorem: Zariski's Main Theorem is a fundamental result in algebraic geometry that establishes a relationship between the birational properties of algebraic varieties and their function fields. It essentially states that if two varieties are birationally equivalent, then their function fields are isomorphic, which implies that rational maps between these varieties can be defined. This theorem connects deeply with concepts like morphisms, resolutions of singularities, minimal models, and schemes, playing a pivotal role in understanding the structure and classification of algebraic varieties.