5.4 Blowing up and resolution of singularities

Blowing up is a key technique in algebraic geometry for transforming varieties. It replaces points or subvarieties with higher-dimensional objects, introducing exceptional divisors. This process can improve or resolve singularities, making it crucial for studying geometric properties.

Resolution of singularities uses blowups to find smooth varieties birational to singular ones. It's always possible for curves and surfaces, while Hironaka's theorem proves it for higher dimensions in characteristic zero. This powerful tool connects to topology, classification, and other math areas.

Blowing up varieties

Definition and notation

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• Blowing up is a fundamental operation in algebraic geometry that transforms a variety by replacing a point or subvariety with a higher-dimensional object, such as a projective space or an exceptional divisor
• The blowup of a variety $X$ at a point $p$ is denoted by $Bl_p(X)$ and is defined as the closure of the graph of the rational map from $X$ to the projective space of lines through $p$
• The blowup of a variety $X$ along a subvariety $Y$ is denoted by $Bl_Y(X)$ and is obtained by blowing up each point of $Y$ in $X$ and taking the closure of the resulting space

Properties of blowups

• The blowup is an isomorphism away from the center of the blowup (the point or subvariety being blown up) and introduces an exceptional divisor over the center
• The exceptional divisor is a projective space or a bundle over the center of the blowup, depending on whether the center is a point or a subvariety
  • For example, blowing up a point in a surface introduces an exceptional divisor isomorphic to $\mathbb{P}^1$
  • Blowing up a curve in a threefold introduces an exceptional divisor that is a $\mathbb{P}^1$-bundle over the curve

Local structure after blowing up

Effect on smooth and singular points

• Blowing up a smooth point on a variety results in a smooth variety, with the exceptional divisor being a projective space of codimension one
• Blowing up a singular point can improve the singularity, making it less severe or even resolving it completely
  • The multiplicity of the singularity decreases after blowing up
  • For example, blowing up an ordinary double point on a surface (a node) results in a smooth surface with an exceptional divisor of self-intersection $-2$

Intersection and self-intersection

• The exceptional divisor introduced by the blowup intersects the strict transform (the proper transform) of the original variety, and the intersection multiplicity provides information about the singularity
• The self-intersection number of the exceptional divisor is related to the type of singularity being blown up
  • For example, blowing up an ordinary double point on a surface results in an exceptional divisor with self-intersection $-2$
  • The self-intersection number decreases by $1$ each time a point on the exceptional divisor is blown up

Successive blowups and local study

• Successive blowups can be used to study the local structure of a variety near a singularity, as each blowup provides a new model of the variety with a modified singularity
• The sequence of blowups and the configuration of the exceptional divisors encode information about the type of singularity
  • For example, the resolution of a rational double point (ADE singularity) on a surface can be described by a sequence of blowups, with the final configuration of exceptional divisors forming a Dynkin diagram of the corresponding simple Lie algebra

Resolving singularities with blowups

Curves and surfaces

• Resolution of singularities is the process of finding a smooth variety birational to a given singular variety, achieved through a sequence of blowups
• For curves, resolution of singularities is always possible and can be achieved by a finite sequence of blowups at singular points
  • The final result is a smooth curve birational to the original one
  • For example, a curve with a node can be resolved by blowing up the node once, resulting in a smooth curve with an exceptional divisor of self-intersection $-2$
• For surfaces, resolution of singularities is also always possible, but the process may involve blowing up singular points and curves on the surface
  • The sequence of blowups is not unique, but the final result is a smooth surface birational to the original one
  • The resolution of rational double points (ADE singularities) on surfaces can be explicitly described using the Dynkin diagrams of the corresponding simple Lie algebras
  • The intersection matrix of the exceptional divisors in the resolution of a surface singularity encodes important information about the singularity type

Higher dimensions and Hironaka's theorem

• Resolution of singularities in higher dimensions is a deep and active area of research, with significant progress made by Hironaka's theorem
• Hironaka's theorem proves the existence of resolution of singularities for varieties over fields of characteristic zero
  • The theorem states that for any variety $X$ over a field of characteristic zero, there exists a sequence of blowups along smooth centers that results in a smooth variety $\tilde{X}$ birational to $X$
  • The proof of Hironaka's theorem is highly complex and involves intricate arguments from commutative algebra and algebraic geometry

Significance of resolution of singularities

Studying geometry and topology of varieties

• Resolution of singularities is a powerful tool for studying the geometry and topology of algebraic varieties, as it allows one to work with smooth models while preserving the birational equivalence class
• Many invariants and properties of varieties, such as the genus, Euler characteristic, and Hodge numbers, are preserved under birational equivalence and can be computed using a smooth model obtained by resolution of singularities
  • For example, the genus of a curve can be computed using a smooth model obtained by resolving its singularities
  • The Hodge numbers of a singular variety can be defined as the Hodge numbers of a smooth model obtained by resolution of singularities

Classification and minimal model program

• The existence of resolution of singularities has important consequences in the classification of algebraic varieties
  • For example, it implies that every complex algebraic surface is birational to a smooth projective surface
• Resolution of singularities plays a crucial role in the minimal model program, which aims to find a "simplest" representative in each birational equivalence class of varieties
  • The minimal models are obtained by performing a sequence of contractions and flips, starting from a smooth model obtained by resolution of singularities
  • The minimal model program has been successfully completed for surfaces and threefolds, and it is an active area of research in higher dimensions

Connections with other areas of mathematics

• The study of singularities and their resolutions has deep connections with other areas of mathematics, such as topology, differential geometry, and representation theory
• The McKay correspondence relates the resolution of certain singularities (quotient singularities) to the representation theory of finite subgroups of $SL(2,\mathbb{C})$
  • The exceptional divisors in the resolution of a quotient singularity correspond to the irreducible representations of the finite subgroup
  • The intersection matrix of the exceptional divisors is related to the McKay quiver of the finite subgroup
• The study of singularities and their resolutions has applications in string theory and mathematical physics, particularly in the context of compactifications and dualities