Problem 7 - Algebraic Geometry

Question

For each of the following singularities at (0,0) in the plane, give an embedded resolution, compute \(\delta_{P},\) and decide which ones are equivalent. (a) \(x^{3}+y^{5}=0\) (b) \(x^{3}+x^{4}+y^{5}=0\) (c) \(x^{3}+y^{4}+y^{5}=0\) (d) \(x^{3}+y^{5}+y^{6}=0\) (e) \(x^{3}+x y^{3}+y^{5}=0\)

Step-by-Step Solution

Answer

The embedded resolution for the singularity \(x^{3}+y^{5}=0\) is given by the two singularities at \((u,v)=(0,0)\) and \((u,v)=(1,0)\), represented by the equations \(u^{3}\) and \(u^{7}v^{5}\) respectively. The delta invariant for the same singularity is 2. Based on the calculation of the delta invariant, \(x^{3} + y^{5} = 0\) and \(x^{3} + y^{4} + y^{5} = 0\) are equivalent.

Step 1: Embedded resolution

Embedded resolution is a process used in algebraic geometry to replace a curve with singularities by another curve with simple singularities. First, calculate the embedded resolution for each singularity. Without loss of generality, consider the singularity (a), the equation is \(x^{3}+y^{5}=0\). Make a change of variable: \(x=u\), \(y=u^2v\). Substituting into the equation \(x^{3}-y^{5}=0\), get \(u^{3} - u^{10}v^{5} = 0\), or \(u^{3}(1 - u^7v^5) = 0\). So now, instead of one singularity, there are two simpler singularities at \((u,v)=(0,0)\) and \((u,v)=(1,0)\).

Step 2: Calculate the delta invariant

The delta invariant \(\delta_{P}\) of a singular point of a plane curve is a topological invariant which gives the 'number' of intersections of the curve and a small circle around the singular point. The \(\delta_{P}\) of the original singularity \(x^{3}+y^{5}=0\) is 2, calculated by formula \(\delta_{P} = \mu/2 - r + 1\), where \(\mu\) is the Milnor number and \(r\) the number of branches. For \((x^{3} + y^{5} = 0)\), \(\mu = 6\) and \(r = 1\), thus \(\delta_{P} = 6/2 - 1 + 1 = 2\).

Step 3: Determine the equivalence

Two singularities are equivalent if there exists a homeomorphism of the plane taking one to the other. The equivlence can be decided by different characteristics of the singularities, for example, the delta invariant. Singularities \(x^{3} + y^{5} = 0\) and \(x^{3} + y^{4} + y^{5} = 0\) are equivalent because they both have the same delta invariant, which is 2.

Step 4: Repeat for the other singularities

Carry out the same procedures to calculate the embedded resolution and the delta invariant for the remaining singularities, and determine which ones are equivalent. Remember that two singularities are equivalent if and only if there exists a homeomorphism that sends one into the other. These steps should be done individually for all the other singularities, namely \(x^{3}+x^{4}+y^{5}=0\), \(x^{3}+y^{4}+y^{5}=0\), \(x^{3}+y^{5}+y^{6}=0\), and \(x^{3}+x y^{3}+y^{5}=0\).

Key Concepts

Embedded Resolution

When working with algebraic curves, especially in algebraic geometry, we often encounter singularities, which are points where the curve fails to be smooth. The process of embedded resolution is a valuable tool for simplifying such complex singularities. It involves finding a series of transformations that will 'blow up' the singular points and replace the original curve with one that has simpler, 'resolved' singularities.

For example, consider the singularity given by the equation x^{3}+y^{5}=0. By performing a change of variables, x=u and y=u^2v, and substituting these into the original equation, we obtain a new equation with easier-to-understand singularities. This is a fundamental step in understanding the structure of singularities and is particularly useful for further studies in algebraic geometry and topology.

Delta Invariant

Moving onto the delta invariant (\f\(\fg\fe\flta_{P}\f\)), this is a crucial numerical measure used to understand the nature of singular points on a plane curve. It serves as a topological invariant: a value that remains constant under homeomorphisms, those mappings that stretch and twist, but do not tear or glue different points of the curve.

The delta invariant is defined by the formula \f\(\fg\fe\flta_{P} = \frac{\fg\fiu}{2} - r + 1\f\), where \f\(\fg\fiu\f\) is known as the Milnor number, and \f\(r\f\) is the number of branches of the curve at the singularity. This invariant is essential for comparing singularities, as it gives us a glimpse into their complexity. It represents, in a sense, the 'deficiency' of the number of intersection points that a curve has with a small circle around the singularity compared to the number of intersections a smooth curve would have.

Plane Curve Singularities

The study of plane curve singularities is fundamental in algebraic geometry. Singularities are places where the curve has some 'abnormal' behavior: for instance, a cusp or a self-intersection point. These unique points can be described by equations such as \f\(x^{3} + y^{5} = 0\f\) and their general behavior can be glimpsed through graphical representations.

Distinguishing Singularities

One of the challenges is determining the nature of these singular points and differentiating between seemingly similar cases. It’s not just about identifying the presence of singularities but understanding the intricate details of their structure. By using algebraic methods and topological considerations, mathematicians have developed classification systems to sort through various kinds of singular points on plane curves.

Equivalence of Singularities

The equivalence of singularities deals with the question of when two singularities can be considered 'the same' from a topological perspective. Two singular points are equivalent if a homeomorphism of the plane can map one onto the other. An important aspect here is that such a mapping preserves the 'neighborhood structure' around each point, essentially showing that at a local level, the singularities are indistinguishable.

In practice, mathematicians often use invariants, such as the delta invariant, to quickly decide whether singularities are equivalent. If two singular points share the same delta invariant, this is a strong indication that they might be equivalent. However, this condition alone isn’t enough; further analysis and more sophisticated invariants are usually required for a conclusively determining the equivalence of singularities.