6.2 Blowing up and resolution of singularities

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.

Resolution of singularities 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 exceptional divisor $E$, 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 blow-up 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 projective space
  • Given a rational map $f: X \dashrightarrow \mathbb{P}^n$, the blow-up of $X$ along the base locus of $f$ is the closure of the graph of $f$ in $X \times \mathbb{P}^n$
• 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 $\bigoplus_{n \geq 0} I^n$, where $I^n$ is the $n$-th power of the ideal $I$
  • The projective spectrum of the Rees algebra $\bigoplus_{n \geq 0} I^n$ is isomorphic to the blow-up of $\text{Spec}(R)$ 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 $\bigoplus_{n \geq 0} I(p)^n$. The projective spectrum of this algebra is the blow-up of $X$ at $p$
  • Example: For the affine plane $\mathbb{A}^2$ and the origin $p = (0, 0)$, the ideal $I(p)$ is generated by $x$ and $y$. The blow-up of $\mathbb{A}^2$ at $p$ is the projective spectrum of $\bigoplus_{n \geq 0} (x, y)^n$
• For a subvariety $Y$ of $X$, the blow-up of $X$ along $Y$ is constructed similarly using the ideal sheaf 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 $\mathbb{P}^2$ at a point $p$ is isomorphic to the Hirzebruch surface $\mathbb{F}_1$, which is a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$

Explicit Equations and Exceptional Divisors

• Explicitly, the blow-up of the affine plane $\mathbb{A}^2$ at the origin $(0, 0)$ is the subvariety of $\mathbb{A}^2 \times \mathbb{P}^1$ defined by the equation $xv = yu$, where $(x, y)$ are coordinates on $\mathbb{A}^2$ and $(u : v)$ are homogeneous coordinates on $\mathbb{P}^1$
  • The projection map from the blow-up to $\mathbb{A}^2$ is given by $(x, y, (u : v)) \mapsto (x, y)$
• The exceptional divisor $E$ in the blow-up of $\mathbb{A}^2$ at the origin is the preimage of $(0, 0)$ under the projection map, which is isomorphic to $\mathbb{P}^1$
  • 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 $f: X' \to X$ 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 $\mathbb{P}^1$, effectively separating the branches of the curve at that point. Repeated blow-ups will eventually resolve all singularities
  • Example: Consider the curve $y^2 = x^3$ (a nodal cubic). Blowing up the origin once resolves the singularity, resulting in a smooth curve isomorphic to $\mathbb{P}^1$
• 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 $z^2 = xy$ (a cone) requires a sequence of two blow-ups. The first blow-up creates an exceptional divisor isomorphic to $\mathbb{P}^1$, 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 $y^2 = x^3$ is isomorphic to $\mathbb{A}^1$, 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