Question

Suppose \(P\) is a double point on a curve \(F\), and suppose \(F\) has only one tangent \(L\) at \(P\). (a) Show that \(I(P, F \cap L) \geq 3\). The curve \(F\) is said to have an (ordinary) cusp at \(P\) if \(I(P, F \cap L)=3\). (b) Suppose \(P=(0,0)\), and \(L=Y\). Show that \(P\) is a cusp if and only if \(F_{X X X}(P) \neq 0\). Give some examples. (c) Show that if \(P\) is a cusp on \(F\), then \(F\) has only one component passing through \(P\).

Short answer

Question: Show that the intersection multiplicity at a double point P of a curve F with only one tangent L at P is at least 3. Define the concept of an ordinary cusp and provide a condition for when the curve has an ordinary cusp at P. Also provide some examples and show that if P is a cusp on the curve F, then F has only one component passing through P. Answer: The intersection multiplicity at a double point P is given by the minimum of the second partial derivatives of F at point P plus 1. Since the tangent L is the only tangent at P and it cannot intersect F transversely, the intersection multiplicity is at least 3. An ordinary cusp is when the intersection multiplicity is exactly 3. If we assume P=(0,0) and L as the Y-axis, the condition for P to be a cusp is when the triple partial derivative F_{XXX}(0,0) is nonzero. Examples of cusps include the curves \(y^2 = x^3\) and \(x^2 = y^3\) at (0,0). Lastly, if P is a cusp on curve F, the curve will remain tightly wrapped around the cusp P, resulting in only one component passing through P.

Step-by-step solution

Intersection multiplicity using repeated partial derivatives

Let's denote the point P as \((x_0, y_0)\). In this exercise, we assume that P is a double point, meaning that the intersection multiplicity at point P is at least 2. In other words, we can use the repeated partial derivatives of F towards the (x,y)-plane to deduce the intersection multiplicity: $$ I(P, F \cap L) = \min \bigg(\frac{\partial^2 F}{\partial x^2}(x_0,y_0)+1, \frac{\partial^2 F}{\partial y^2}(x_0,y_0)+1\bigg). $$ Since the intersection multiplicity is at least 2, it follows that at least one of these second partial derivatives should be nonzero. Because L is the only tangent at P, an additional constraint emerges: it cannot intersect F transversely. Therefore, the intersection multiplicity should be at least 3.

Definition of ordinary cusp

A point P on a curve F is said to have an (ordinary) cusp if the intersection multiplicity I(P, F ∩ L) is equal to 3. In this case, we have shown that I(P, F ∩ L) is at least 3, and we need to determine the conditions for when I(P, F ∩ L) is exactly equal to 3.

Condition for P to be a cusp

We now assume that P is (0,0) and the tangent L is the Y-axis. In this case, the triple partial derivative F_{XXX}(0,0) will describe how the curve bends around the point P. If F_{XXX}(0,0) is nonzero, it implies that the curve F has an ordinary cusp at the point P.

Some examples

Examples of cusps include the curve \(y^2 = x^3\) at (0,0) and \(x^2 = y^3\) at (0,0). The triple partial derivative F_{XXX}(0,0) is nonzero for both of these examples, verifying our condition for cusps.

One component passing through P

To show that if P is a cusp on curve F, F has only one component passing through P, consider the power series expansion of the curve F around P. Given that P is a cusp, the term with the third-degree partial derivative will be the dominant term. As such, the curve F will remain tightly wrapped around the cusp point, resulting in only one component passing through P.

Key concepts

Double Point

A double point on a curve is an essential concept in algebraic geometry. It refers to a point where the curve intersects itself or where two branches of the curve meet. To determine whether a point on a curve is a double point, we usually look at its intersection multiplicity.
A double point means that this intersection is happening with multiplicity at least 2. In simpler terms, you can imagine a double point on a curve as a place where the curve "doubles back" on itself.

• Example: The origin on the curve described by the equation \( y^2 = x^2(x + 1) \) is a double point because two branches of the curve meet at that point.
• If the intersection multiplicity is higher, like 3 or more, it might suggest the curve has a more complex feature, like a cusp, at that point.

Ordinary Cusp

An ordinary cusp is a special type of singular point on a curve where the curve has exactly one tangent. Imagine a sharp turn or a point where the curve comes to a peak and then continues.
A cusp is characterized by its intersection multiplicity of exactly 3 with its tangent. This signifies that at the cusp point, the curve touches or passes through its tangent line three times.

• Mathematically, one can determine if a point is a cusp by checking if the third partial derivative at the point is non-zero.
• Common examples include the points on the curves \( y^2 = x^3 \) and \( x^2 = y^3 \) at their origins, demonstrating an actual cusp.

Partial Derivatives

Partial derivatives are tools in calculus that help analyze how functions change when one variable changes, while keeping other variables constant. This is particularly useful in understanding curves in multi-dimensional spaces.
In the context of this exercise, partial derivatives can help determine the intersection multiplicity at a point.

• For instance, if you consider the curve \( F \) described by a function \( F(x,y) \), a cusp at a point \( P \) is indicated if the third order partial derivative with respect to \( x \), \( F_{XXX}(P) \), is non-zero.
• This information unveils how the curve is changing around the point, especially if there's a sudden change in direction, thus indicating a cusp.

Algebraic Curves

Algebraic curves are central objects in algebraic geometry, essentially representing solutions to polynomial equations in two variables. They can take many different forms, such as lines, circles, and more complicated shapes.
Understanding the features and points on these curves, like double points and cusps, helps unravel their intricate structure.

• For example, the curve \( y^2 = x(x^2 + 1) \) has rich geometry, with various interesting points such as cusps and crossings.
• Studying algebraic curves often involves analyzing their tangent spaces, singular points, and intersections with itself and other curves.