Question

Let \(C=V\left(X^{2}+Y^{2}-Z^{2}\right) \subset \mathbb{P}^{2} .\) For each \(t \in k\), let \(L_{t}\) be the line between \(P_{0}=\) \([-1: 0: 1]\) and \(P_{t}=[0: t: 1]\). (Sketch this.) (a) If \(t \neq \pm 1\), show that \(L_{t} \cdot C=P_{0}+Q_{t}\), where \(Q_{t}=\left[1-t^{2}: 2 t: 1+t^{2}\right] .\) (b) Show that the map \(\varphi: \mathbb{A}^{1} \backslash\\{\pm 1\\} \rightarrow C\) taking \(t\) to \(Q_{t}\) extends to an isomorphism of \(\mathbb{P}^{1}\) with \(C\). (c) Any irreducible conic in \(\mathbb{P}^{2}\) is rational; in fact, a conic is isomorphic to \(\mathbb{P}^{1}\). (d) Give a prescription for finding all integer solutions \((x, y, z)\) to the Pythagorean equation \(X^{2}+Y^{2}=Z^{2}\).

Short answer

In summary, we have shown that any irreducible conic in \(\mathbb{P}^2\) is rational and isomorphic to \(\mathbb{P}^1\). A prescription for finding all integer solutions \((x, y, z)\) to the Pythagorean equation \(x^2 + y^2 = z^2\) is to substitute any odd integer \(y\) into the formulas \(x = \frac{4 - y^2}{4}\) and \(z = \frac{4 + y^2}{4}\) and perform the calculations to find the corresponding \(x\) and \(z\).

Step-by-step solution

Define the line Lt

We're given that for each \(t \in k\), \(L_{t}\) is the line between the points \(P_{0} = [-1: 0: 1]\) and \(P_{t} = [0: t: 1]\). To find the equation of \(L_{t}\), we can take the difference of the homogeneous coordinates of \(P_t\) and \(P_0\), then normalize by \(t\) to obtain: $$ L_t : \begin{cases} X = -tY \\ Z = Y + tX \end{cases} $$

Intersection of Lt and C

Next, we need to find the intersection points of \(L_t\) and \(C\). Substitute the equations of \(L_t\) into the equation of \(C\) to obtain: $$ (-tY)^2 + Y^2 - (Y + tX)^2 = 0 $$ Simplify the equation and solve for \(Y\): $$ Y = \frac{2t}{1 - t^2}X $$ Now substitute this equation for \(Y\) back into the equation for \(Z\) to obtain: $$ Z = X + \frac{2t^2}{1 - t^2}X $$ The homogeneous coordinates of the intersection points are then given by: $$ Q_t = \left[1 - t^2 : 2t : 1 + t^2\right] $$

Show Intersection between Lt and C

To show that \(L_t \cdot C = P_0 + Q_t\), note that \(Q_t\) is on the line segment \(L_t\) since the coordinates of \(Q_t\) are proportional to the difference of \(P_0\) and \(P_t\). So, the line \(L_t\) intersects \(C\) at \(Q_t\). Since \(P_0\) is also on the conic and \(L_t\), we have \(L_t \cdot C = P_0 + Q_t\).

Extend the map φ to an isomorphism

The map \(\varphi: \mathbb{A}^1 \backslash {-1, 1} \rightarrow C\) taking \(t\) to \(Q_t = [1 - t^2 : 2t : 1 + t^2]\) can be extended to an isomorphism by identifying the projective coordinates \((t : 1)\) and \((1 : t)\) with the homogeneous coordinates \((1 - t^2 : 2t : 1 + t^2)\). This map is an isomorphism since it is bijective and maps between projective varieties.

Conclude that conics are rational

We have shown that there exists an isomorphism between the conic \(C\) and \(\mathbb{P}^1\), which is a rational curve. Thus, any irreducible conic in \(\mathbb{P}^2\) is rational and isomorphic to \(\mathbb{P}^1\).

Prescribe integer solutions to Pythagorean equation

We want to find all integer solutions \((x, y, z)\) to the Pythagorean equation \(X^2 + Y^2 = Z^2\). To do this, we first observe that the points on the conic formed by this equation have homogeneous coordinates \((1 - t^2 : 2t : 1 + t^2)\), where \(t = \frac{y}{x}\). Now, we will use the relation between the homogeneous coordinates and the integers \(x\), \(y\), and \(z\): $$ \begin{cases} x = 1 - t^2 \\ y = 2t \\ z = 1 + t^2 \end{cases} $$ Substituting \(t = \frac{y}{2}\) into the equation for \(x\) and \(z\), we obtain: $$ \begin{cases} x = \frac{4 - y^2}{4} \\ z = \frac{4 + y^2}{4} \end{cases} $$ Thus, for any odd integer \(y\), we can find an integer solution \((x, y, z)\) by substituting \(y\) into these formulas for \(x\) and \(z\).

Key concepts

Rational Conic

A rational conic is a special type of algebraic curve that can be parameterized by rational functions. An algebraic curve is a graphical representation of solutions to a polynomial equation of two variables. When it comes to conic sections, such graphs typically depict ellipses, hyperbolas, or parabolas.

Now, why do we call certain conics 'rational'? It's because these conics have the property that you can describe every point on the conic with equations that involve only rational functions of a parameter, usually denoted as 't'. An example of such a parametric equation could look like this:
\( x(t) = \frac{1-t^2}{1+t^2},\, y(t) = \frac{2t}{1+t^2} \).

Each point on a rational conic corresponds to exactly one value of 't' (except perhaps a finite number of points), and each 't' corresponds to exactly one point on the curve. This bidirectional correspondence, or bijection, is a key part of what makes these structures isomorphic to projective lines, which are essentially the simplest form of projective varieties -- thus showing that they are indeed rational. This notion was demonstrated in the problem when the conic \(C=V\left(X^{2}+Y^{2}-Z^{2}\right)\) is shown to be isomorphic to \(\mathbb{P}^{1}\), meaning that its points can be represented by a single parameter t in projective space.

Projective Geometry

Projective geometry is a branch of mathematics that extends the concepts of geometry to a projective space. Unlike Euclidean geometry, projective geometry is not concerned with measurements and distances; instead, it focuses on the properties of figures that are invariant under projective transformations.

One of the fundamental objects in projective geometry is the projective plane, denoted as \(\mathbb{P}^{2}\). This plane can be visualized as the set of all lines passing through the origin in three-dimensional space. Points in projective space are represented by homogeneous coordinates, which are not unique: any multiple of a set of coordinates represents the same point.

In the context of algebraic curves like conics, projective geometry allows us to include 'points at infinity', which solves certain problems of classical Euclidean geometry. For example, in projective geometry, two distinct lines always intersect at exactly one point, which might be a point at infinity if the lines are parallel in the Euclidean plane.

The exercise given extends the Euclidean concept of the conic section to the projective plane, showing that we can establish a continuous mapping that accounts for every point on the curve, even at infinity. The line \(L_t\) that connects points \(P_{0}\) and \(P_{t}\) in projective space provides a clear illustration of projective geometry's principles; as the parameter 't' ranges over all possible values, every point on the conic is accounted for, effectively proving part (b) of the exercise.

Pythagorean Triples

Pythagorean triples are sets of three positive integers \(a\), \(b\), and \(c\) that fit the formula \(a^2 + b^2 = c^2\). This equation represents the Pythagorean theorem, which states that in a right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

These triples are fundamentally connected to rational points on the unit circle in the coordinate plane. If we scale such a circle to have a radius equal to the hypotenuse 'c', the rational points on this circle, expressed in terms of \(x\), \(y\), and \(z\), correspond to Pythagorean triples. This is the conceptual link used in part (d) of the given exercise.

The homogenization

Identification of Homogeneous Coordinates

of the Pythagorean Theorem's equation, \(X^2 + Y^2 = Z^2\), is given in the form of the conic \(C\) with homogeneous coordinates. By rationalizing these coordinates, we translate the geometric properties of the conic into a set of algebraic tools for finding all Pythagorean triples. The exercise solves for integer solutions, finding that each odd integer \(y\) gives us a new Pythagorean triple when we plug it into the derived formulas, demonstrating the intricate link between algebraic geometry and number theory.

Homogeneous Coordinates

Homogeneous coordinates are used in projective geometry to represent points in a projective space. They differ from Cartesian coordinates in that they use an extra dimension to account for scaling. This allows for the representation of points at infinity and the unification of various geometric calculations.

For instance, in the two-dimensional projective plane, a point is represented by three numbers, not necessarily unique, since any non-zero scalar multiple of the triplet represents the same point. These coordinates are written as \([X: Y: Z]\), where 'Z' can be zero, allowing us to represent points at infinity.

Dealing with Ratios

The beauty of homogeneous coordinates lies in their handling of ratios instead of absolute values. In the given exercise, homogeneous coordinates were used to define the intersection of a line \(L_t\) and a conic in projective space. The coordinates \([1-t^2 : 2t : 1+t^2]\) of the point \(Q_t\) are not dependent on the particular 't' values chosen; rather, it is the ratio between the numbers that preserves the geometric properties across the projective transformation. This representation was central to proving the isomorphism between the conic \(C\) and \(\mathbb{P}^1\), and also to finding integer solutions to the Pythagorean equation. Homogeneous coordinates make the connection between algebraic curves and projective geometry seamless and elegant.