🌿Algebraic Geometry Unit 6 – Singularities and Resolution

What's the Big Idea?

• Singularities are points on algebraic varieties where the variety fails to be smooth or regular
• Resolution of singularities involves finding a smooth variety that is birationally equivalent to the original singular variety
• The process of resolving singularities provides valuable insights into the structure and properties of algebraic varieties
• Singularities can be classified based on their type and complexity (isolated singularities, non-isolated singularities, rational singularities)
• Understanding singularities and their resolution is crucial for studying the geometry and topology of algebraic varieties
• The resolution of singularities has far-reaching implications in various areas of mathematics, including algebraic geometry, complex analysis, and number theory

Key Concepts to Know

• Algebraic varieties are geometric objects defined by polynomial equations in affine or projective space
• Smooth varieties have a well-defined tangent space at every point, while singular varieties have points where the tangent space is not well-defined
• Blowups are a fundamental tool in the resolution of singularities, involving replacing a point with a projective space of lines through that point
• The exceptional divisor is the subvariety introduced during the blowup process, which is mapped to the singular point in the original variety
• The multiplicity of a singularity measures its complexity and is related to the order of vanishing of the defining equations at the singular point
• Normalization is a process of resolving singularities in codimension one by considering the integral closure of the coordinate ring
• The Jacobian criterion is a method for detecting singularities by examining the rank of the Jacobian matrix of the defining equations

The Math Behind It

• Algebraic varieties are defined using polynomial equations in affine or projective space over a field $k$
  • Affine varieties are subsets of affine space $\mathbb{A}^n$ defined by polynomial equations $f_1(x_1, \ldots, x_n) = \cdots = f_m(x_1, \ldots, x_n) = 0$
  • Projective varieties are subsets of projective space $\mathbb{P}^n$ defined by homogeneous polynomial equations $F_1(x_0, \ldots, x_n) = \cdots = F_m(x_0, \ldots, x_n) = 0$
• The local ring of a point $p$ on a variety $X$, denoted by $\mathcal{O}_{X,p}$, captures the local behavior of functions near $p$
• The tangent space at a point $p$ on a variety $X$ is the dual of the cotangent space, which is the $k$-vector space $\mathfrak{m}_p/\mathfrak{m}_p^2$, where $\mathfrak{m}_p$ is the maximal ideal of $\mathcal{O}_{X,p}$
• Blowups are defined using the Proj construction, which generalizes the projective space construction to arbitrary graded rings
  • Given a variety $X$ and a subvariety $Y$, the blowup of $X$ along $Y$ is $Bl_Y(X) = Proj(\bigoplus_{d \geq 0} I_Y^d)$, where $I_Y$ is the ideal sheaf of $Y$ in $X$
• The multiplicity of a singularity can be computed using the Hilbert-Samuel multiplicity, which measures the growth of the local ring $\mathcal{O}_{X,p}$

Important Theorems and Proofs

• Hironaka's Theorem states that for any algebraic variety $X$ over a field of characteristic zero, there exists a resolution of singularities $f: \tilde{X} \to X$, where $\tilde{X}$ is smooth and $f$ is a birational morphism
  • The proof of Hironaka's Theorem is highly complex and involves a careful analysis of the structure of singularities and the use of blowups to systematically reduce their complexity
• The Jacobian Criterion states that a point $p$ on a variety $X$ defined by equations $f_1 = \cdots = f_m = 0$ is singular if and only if the Jacobian matrix $(\partial f_i/\partial x_j)$ has rank less than the codimension of $X$ at $p$
• The Theorem of Bierstone and Milman provides a constructive proof of resolution of singularities in characteristic zero using a sequence of blowups along smooth centers
• The Theorem of Jung and Hirzebruch gives a resolution of singularities for surfaces over any algebraically closed field using a finite sequence of normalization and blowups
• Zariski's Main Theorem relates the local properties of a variety to its birational geometry, stating that any birational morphism between quasi-projective varieties over a field can be factored into a sequence of blowups and blowdowns along smooth centers

Real-World Applications

• Singularity theory has applications in robotics and control theory, where understanding the singularities of the configuration space is crucial for motion planning and control
• In computer vision and image processing, techniques from singularity theory are used to analyze and classify features in images, such as edges, corners, and textures
• Singularities and their resolution play a role in string theory and theoretical physics, where they arise in the study of compactifications and the geometry of spacetime
• In data analysis and machine learning, singularity theory is used to study the structure and topology of high-dimensional data sets and to develop algorithms for dimensionality reduction and feature extraction
• Singularities and resolution techniques are applied in the study of caustics and wave fronts in optics and wave propagation, where they describe the behavior of light and sound in the presence of obstacles and boundaries

Common Pitfalls and How to Avoid Them

• One common pitfall is confusing the notion of smoothness with that of regularity. A variety can be regular (the local ring at every point is a regular local ring) without being smooth (the variety is locally isomorphic to affine space)
  • To avoid this, always check the definitions and properties of smoothness and regularity in the context of the problem at hand
• Another pitfall is forgetting to consider the field of definition when studying singularities and their resolution. The properties of singularities and the existence of resolutions can depend on the characteristic of the field
  • Always be mindful of the field of definition and any assumptions on its characteristic when applying theorems and techniques
• It is easy to overlook the global structure of a variety when focusing on local properties like singularities. However, understanding the global geometry is often crucial for resolving singularities and studying the variety as a whole
  • Regularly step back and consider the global picture, using tools like sheaves, cohomology, and intersection theory to relate local and global properties
• When computing blowups and resolutions, it is common to make algebraic errors or forget important steps in the process. This can lead to incorrect results or incomplete resolutions
  • Double-check your computations and make sure to follow the steps of the resolution process carefully. Use explicit examples to test your understanding and catch any errors

Solving Problems Step-by-Step

1. Identify the algebraic variety and its defining equations. Determine the field of definition and any relevant assumptions on its characteristic.
2. Locate the singular points of the variety using the Jacobian criterion or by examining the local rings at each point. Classify the singularities by their type and multiplicity.
3. Choose an appropriate resolution technique based on the characteristics of the variety and its singularities. Consider methods such as blowups, normalization, or algorithmic approaches like Bierstone-Milman or Jung-Hirzebruch.
4. If using blowups, determine the center of the blowup (the subvariety to be blown up) and compute the blowup using the Proj construction. Analyze the resulting variety and its singularities.
5. Repeat steps 3-4 as necessary, iteratively resolving singularities until a smooth variety is obtained. Keep track of the exceptional divisors introduced at each step.
6. Verify that the resolution is complete by checking the smoothness of the final variety and the properties of the resolution map (birational, proper, etc.).
7. Interpret the resolution in terms of the original variety, studying the preimages of singular points, the exceptional divisors, and any changes in the geometry or topology of the variety.

Further Reading and Resources

• "Resolution of Singularities" by Heisuke Hironaka, a seminal paper introducing the concept of resolution of singularities in characteristic zero
• "Lectures on Resolution of Singularities" by János Kollár, a comprehensive textbook covering the theory and techniques of resolution of singularities
• "Singularities in Algebraic Geometry" by Eduard Looijenga, a graduate-level textbook exploring the classification and properties of singularities in algebraic varieties
• "Algorithms in Real Algebraic Geometry" by Saugata Basu, Richard Pollack, and Marie-Françoise Roy, a book presenting algorithmic approaches to problems in real algebraic geometry, including resolution of singularities
• "The Geometry of Schemes" by David Eisenbud and Joe Harris, a textbook introducing schemes and their applications in algebraic geometry, with a chapter on resolution of singularities
• Online resources such as the Stacks Project (https://stacks.math.columbia.edu/), MathOverflow (https://mathoverflow.net/), and nLab (https://ncatlab.org/) provide a wealth of information and discussions on singularities, resolution, and related topics in algebraic geometry.