8.3 Blowing up and resolution of singularities

Blowing up and resolving singularities are crucial techniques in algebraic geometry. They allow us to transform varieties by replacing subvarieties with new exceptional divisors, creating smoother versions of our original objects. These methods help us study complex geometric structures more easily.

In the context of rational maps and birational geometry, blowing up and resolving singularities are powerful tools. They enable us to find birational equivalences between varieties, simplify their structure, and uncover hidden properties. This process is fundamental for understanding and classifying algebraic varieties.

Blowing up varieties

Geometric interpretation and definition

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• Blowing up is a fundamental operation in algebraic geometry that transforms a variety by replacing a subvariety with a new exceptional divisor
• The blowup of a variety $X$ along a subvariety $Y$ is denoted by $Bl_Y(X)$ and is obtained by introducing new projective coordinates along $Y$
• Geometrically, the blowup separates the points of $Y$ and introduces new projective directions, resulting in a new variety that is birational to the original one
• The exceptional divisor $E$ introduced by the blowup is a projective bundle over the blown-up subvariety $Y$, with fibers being projective spaces of dimension equal to the codimension of $Y$ in $X$ minus one
  • For example, blowing up a surface ($2$-dimensional variety) at a point ($0$-dimensional subvariety) introduces an exceptional divisor that is a projective line ($1$-dimensional projective space)

Properties of the blowup map

• The blowup map $π: Bl_Y(X) → X$ is a birational morphism that is an isomorphism away from the exceptional divisor and contracts $E$ to the subvariety $Y$
• The blowup map is proper, which means that the preimage of a compact set is compact, and it is surjective
• The blowup map induces a bijection between the points of $Bl_Y(X) - E$ and the points of $X - Y$
• The fiber of the blowup map over a point $p \in Y$ is the projective space of dimension equal to the codimension of $Y$ in $X$ minus one

Computing blowups

Blowing up at a point

• To compute the blowup of a variety $X$ at a point $p$, choose local coordinates $(x_1, ..., x_n)$ around $p$ such that $p$ is the origin $(0, ..., 0)$
• The blowup of $X$ at $p$ is given by the equations $x_i * y_j = x_j * y_i$ for all $i, j$ in the product of $X$ with projective space $\mathbb{P}^{n-1}$ with homogeneous coordinates $[y_1 : ... : y_n]$
  • For example, blowing up the affine plane $\mathbb{A}^2$ at the origin $(0, 0)$ is given by the equation $x * v = y * u$ in $\mathbb{A}^2 \times \mathbb{P}^1$, where $[u : v]$ are homogeneous coordinates on $\mathbb{P}^1$
• The exceptional divisor $E$ is locally defined by the vanishing of the $x_i$ coordinates in the blowup equations, and it is isomorphic to the projective space $\mathbb{P}^{n-1}$

Blowing up along a subvariety

• For blowing up along a subvariety $Y$, choose local coordinates $(x_1, ..., x_k, y_1, ..., y_l)$ such that $Y$ is defined by the vanishing of $(x_1, ..., x_k)$
• The blowup of $X$ along $Y$ is given by the equations $x_i * z_j = x_j * z_i$ for all $i, j$ in the product of $X$ with a projective bundle over $Y$ with fibers being projective spaces $\mathbb{P}^{k-1}$ with homogeneous coordinates $[z_1 : ... : z_k]$
  • For instance, blowing up a threefold ($3$-dimensional variety) along a curve ($1$-dimensional subvariety) introduces an exceptional divisor that is a $\mathbb{P}^1$-bundle over the curve
• The exceptional divisor $E$ is locally defined by the vanishing of the $x_i$ coordinates in the blowup equations, and it is a projective bundle over the blown-up subvariety $Y$

Resolution of Singularities

Definition and existence

• Resolution of singularities is the process of finding a smooth variety that is birational to a given singular variety
• A resolution of singularities of a variety $X$ is a proper birational morphism $π: X' → X$ from a smooth variety $X'$ to $X$
• The existence of resolution of singularities is a fundamental result in algebraic geometry, proven by Hironaka for varieties over fields of characteristic zero
  • The proof is highly non-constructive and does not provide an explicit algorithm for finding a resolution
• Resolution of singularities is not known to exist in positive characteristic, and it is an active area of research

Significance and applications

• Resolution of singularities has numerous applications in algebraic geometry, including the study of birational geometry, the minimal model program, and the classification of algebraic varieties
• It allows the study of singular varieties by relating them to smooth ones, which are easier to understand and work with
• Resolution of singularities is used in the computation of invariants associated with singularities, such as the canonical divisor, discrepancies, and log canonical thresholds
• It plays a crucial role in the minimal model program, which aims to find the "simplest" birational model of a given variety
  • The minimal model program involves a sequence of contractions and flips, which are birational transformations that improve the singularities of the variety

Resolving Singularities with Blowups

Curves

• For algebraic curves ($1$-dimensional varieties), singularities can be resolved by a finite sequence of blowups at singular points
• The multiplicity of a singular point determines the number of blowups required to resolve the singularity
  • Ordinary double points (nodes) require a single blowup, while higher multiplicity points may require multiple blowups
• The genus of the smooth curve obtained by resolving the singularities is related to the genus of the original singular curve and the number and types of singularities
  • The genus formula for singular curves involves the delta invariants of the singularities, which measure the difference between the genus of the singular curve and the genus of its normalization

Surfaces

• For algebraic surfaces ($2$-dimensional varieties), the resolution of singularities is more involved and may require blowups along curves or singular points
• The resolution of surface singularities can be achieved by successively blowing up singular points or curves until the resulting surface becomes smooth
• The exceptional divisors introduced by the blowups in the resolution process encode important information about the singularities, such as their types and invariants (e.g., multiplicity, Milnor number)
  • For example, the resolution of a rational double point (ADE singularity) on a surface introduces a configuration of exceptional curves whose intersection graph is a Dynkin diagram of type A, D, or E
• The intersection graph of the exceptional divisors, known as the dual graph or resolution graph, provides a combinatorial description of the resolution process and the singularity structure
  • The dual graph encodes the self-intersections of the exceptional curves and their intersections with each other
• The canonical divisor of the resolved surface can be computed from the canonical divisor of the original surface and the data of the exceptional divisors using the adjunction formula