Question

Let \(F\) be an irreducible cubic, \(P=[0: 0: 1]\) a cusp on \(F, Y=0\) the tangent line to \(F\) at \(P\). Show that \(F=a Y^{2} Z-b X^{3}-c X^{2} Y-d X Y^{2}-e Y^{3}\). Find projective changes of coordinates (i) to make \(a=b=1\); (ii) to make \(c=0\) (change \(X\) to \(\left.X-\frac{c}{3} Y\right)\); (iii) to make \(d=e=0(Z\) to \(Z+d X+e Y)\). Up to projective equivalence, there is only one irreducible cubic with a cusp: \(Y^{2} Z=X^{3}\). It has no other singularities.

Short answer

Question: Find the equation of an irreducible cubic with a cusp at point P, a tangent at point P given by the line Y=0, and satisfies conditions (i) a=b=1, (ii) c=0, and (iii) d=e=0. Answer: After a series of projective changes of coordinates, we have found that the equation of the irreducible cubic is \(Y^2Z = X^3\). This equation meets all the conditions and represents the cubic with a cusp at point P and no other singularities.

Step-by-step solution

Analyzing the problem

First, let's examine the given information. We have an irreducible cubic \(F\), a cusp point \(P\) with homogeneous coordinates \([0:0:1]\), and tangent line \(Y=0\) at \(P\). We want to find the equation of \(F\) and perform some projective changes of coordinates to satisfy the given conditions.

Finding the equation of F

Since \(F\) is an irreducible cubic and has a cusp at point \(P\), its equation can be written in the form: \[F = a Y^2 Z - b X^3 - c X^2Y - d XY^2 - e Y^3\] Now, the tangent line to \(F\) at \(P\) is given by \(Y = 0\); this means that \(F\) has two tangents at \(P\) with the same direction. To find the equation of \(F\), we'll differentiate it with respect to \(X\) and \(Y\) and evaluate the results at point \(P\). Let's calculate the partial derivatives with respect to \(X\) and \(Y\): \[\frac{\partial F}{\partial X} = -3bX^2 - 2cXY - dY^2\] \[\frac{\partial F}{\partial Y} = 2aYZ - cX^2 - 2dXY - 3eY^2\] Now, evaluate these partial derivatives at point \(P\): \[\frac{\partial F}{\partial X}(0,0,1) = 0\] \[\frac{\partial F}{\partial Y}(0,0,1) = 2a\] Since the tangent line at \(P\) has the same direction, the partial derivatives must be proportional: \[2a = 0\] This implies that \(a=0\).

Performing projective changes of coordinates (i)

We want to make a projective change of coordinates to achieve \(a=b=1\). Since we found that \(a=0\), we can set \(b=1\) without any further changes: \[F = Y^2Z - X^3 - cX^2Y - dXY^2 - eY^3\]

Performing projective changes of coordinates (ii)

To make \(c=0\), change \(X\) to \(X-\frac{c}{3}Y\): \[F = Y^2Z - (X-\frac{c}{3}Y)^3 - c(X-\frac{c}{3}Y)^2Y - d(X-\frac{c}{3}Y)Y^2 - eY^3\] Simplify the equation to obtain: \[F = Y^2Z- X^3 + 3cXY^2 - dXY^2 - eY^3\] Now, \(c\) is eliminated, and \(F\) has the form: \[F = Y^2Z - X^3 - (d-3c)XY^2 - eY^3\]

Performing projective changes of coordinates (iii)

To make \(d=e=0\), change \(Z\) to \(Z+dX+eY\): \[F = Y^2(Z+dX+eY) - X^3 - (d-3c)XY^2 - eY^3\] Simplify the equation to obtain: \[F = Y^2Z - X^3\]

Conclusion

We have shown that up to projective equivalence, there is only one irreducible cubic with a cusp and no other singularities, given by the equation \(Y^2Z = X^3\). This is the canonical form of the initial given problem after performing the projective changes of coordinates.

Key concepts

Projective Changes of Coordinates

Understanding projective changes of coordinates is essential in the study of algebraic curves and surfaces. These changes are akin to moving and rotating a camera to get a better view of a sculpture. In the context of our problem, projective changes allowed us to modify the equation of our irreducible cubic until it met certain desirable conditions, such as normalization and simplification of coefficients.

For example, setting a variable to a simpler form or eliminating specific terms greatly simplifies the study of the object. These changes are formalized as linear transformations on the homogeneous coordinates of points in the projective space. In algebraic terms, we alter the variables in the equation of our cubic in a way that preserves its geometric properties. For instance, when we change X to \(X - \frac{c}{3}Y\), we are essentially shifting the curve in the projective plane so that undesirable terms in the equation are removed, making the equation easier to analyze and understand.

Algebraic Geometry

Algebraic geometry lies at the heart of this exercise, dealing with the shapes and properties of solutions to polynomial equations. It's a vast field that marries algebra, especially polynomial algebra, with geometric intuitions. In simpler terms, just as painters use colors and shapes, algebraic geometers use equations to create and understand mathematical landscapes.

In our task, we examined an irreducible cubic, a type of algebraic curve that cannot be written as a product of simpler polynomials. Our objective was to understand its shape, specifically because it has a cusp, a point where the curve comes to a sharp point. The techniques we used are standard tools in algebraic geometry: examining tangent lines, using derivatives to understand behavior at a point, and applying projective transformations to simplify the form of the curve's equation.

Homogeneous Coordinates

To grasp the full picture of projective geometry, one must become acquainted with homogeneous coordinates. These are like the usual coordinates, but with an added twist that allows for the representation of points at infinity in a natural way, thus unifying the study of algebraic curves within the finite and the infinite realms.

In our cusp problem, the point P is given as \([0: 0: 1]\) in homogeneous coordinates. These coordinates represent points in the projective plane, and the colons indicate a proportion rather than specific values. This means that \([0: 0: 1]\) could represent an entire family of points in Cartesian coordinates, all located at the crucial spot where the curve has a cusp. Threading through algebraic geometry with homogeneous coordinates offers the flexibility to move around within the projective space, simplifying equations, and understanding the deep properties of algebraic objects.

Tangent Line to a Curve

The concept of the tangent line to a curve in algebraic geometry parallels that in calculus, but with added sophistication due to the diverse possibilities in curves we can encounter. For curves described by polynomials, as in our case, the tangent line at a point is essentially the line that just grazes the curve at that point without cutting through it.

In our example involving an irreducible cubic with a cusp, the tangent line at the cusp was a vital clue to uncovering the right equation for F. A cusp, unlike a smooth point, gives rise to a scenario where a single tangent catches the curve from both sides at the point of, well, the cusp. This property, encoded in the derivatives of the equation of the curve, helped us assert that certain coefficients must vanish, which in turn guided us towards the correct transformations to simplify our equation. The role of the tangent line here transcends the simplicity of touching the curve; it becomes a tool for shaping the very essence of the equation representing the curve.