Singular Points On Affine Varieties: 2 Definitions

Hey guys! Ever found yourself scratching your head over singular points on affine varieties? If you're diving into algebraic geometry, you've probably stumbled upon this concept. It can seem a bit daunting at first, especially when you see different definitions floating around. But don't worry, we're going to break it down in a way that’s super easy to grasp. We'll explore two common definitions of singular points on possibly reducible affine varieties and see how they connect. So, let's jump right in!

Two Definitions of Singular Points

In the fascinating world of algebraic geometry, understanding singular points is crucial for characterizing the geometric properties of varieties. When we talk about varieties, we're essentially referring to the set of solutions to a system of polynomial equations. Now, these varieties can be smooth and well-behaved, or they can have these funky points called singularities, where things get a bit wild. Let's consider an affine variety ${ X }$, which might even be reducible, meaning it can be broken down into simpler components. Singular points on ${ X }$ are points where the local geometry is not as “nice” as we'd expect. There are two primary ways to define these singularities, and we’re going to dive deep into both.

Definition 1: The Tangent Space Approach

The first definition, often seen in Shafarevich's work, uses the concept of the tangent space. Imagine you're standing on a surface – the tangent space at a point is like the flat plane that best approximates the surface near that point. For a smooth, well-behaved point, the tangent space has a dimension that we expect based on the dimension of the variety itself. However, at a singular point, the tangent space becomes unexpectedly large, indicating something special (or not-so-special, depending on your perspective!) is happening there.

Specifically, a point ${ x }$ on ${ X }$ is considered singular if the dimension of its tangent space, denoted as ${ \Theta_{X,x} }$, is greater than the dimension of the variety ${ X }$ at that point. Mathematically, we express this as:

${ \dim(\Theta_{X,x}) > \dim_x(X) }$

Here, ${ \dim(\Theta_{X,x}) }$ represents the dimension of the tangent space at ${ x }$, and ${ \dim_x(X) }$ is the local dimension of ${ X }$ at ${ x }$. This definition intuitively captures the idea that at a singular point, the variety doesn't look like a smooth manifold; it might have self-intersections, cusps, or other irregularities that cause the tangent space to