Question

Show that \(\left[x_{1}: \ldots: x_{n}\right] \mapsto\left[x_{1}: \ldots: x_{n}: 0\right]\) gives an isomorphism of \(\mathbb{P}^{n-1}\) with \(H_{\infty} \subset \mathbb{P}^{n} .\) If a variety \(V\) in \(\mathbb{P}^{n}\) is contained in \(H_{\infty}, V\) is isomorphic to a variety in \(\mathbb{P}^{n-1}\). Any projective variety is isomorphic to a closed subvariety \(V \subset \mathbb{P}^{n}\) (for some \(n\) ) such that \(V\) is not contained in any hyperplane in \(\mathbb{P}^{n}\).

Short answer

Show that the map \(\phi: \mathbb{P}^{n-1} \to H_{\infty} \subset \mathbb{P}^{n}\), defined as \(\phi(\left[x_{1}: \ldots: x_{n}\right]) = \left[x_{1}: \ldots: x_{n}: 0\right]\), is an isomorphism. To do this, we need to show that the map is both injective and surjective. We show injectivity by proving that if \(\phi(p) = \phi(q)\), then \(p = q\), which means that \(\left[x_{1}: \ldots: x_{n}: 0\right] = \left[y_{1}: \ldots: y_{n}: 0\right]\) and \(x_i = k \cdot y_i\) for all \(i = 1, \ldots, n\) and some \(k \neq 0\). Therefore, the points \(p\) and \(q\) are the same in \(\mathbb{P}^{n-1}\). We show surjectivity by considering any point \(p\) in \(H_{\infty}\), i.e., \(p = \left[x_{1}: \ldots: x_{n}: 0\right]\). Then, we have \(\phi(\left[x_1: \ldots: x_n\right]) = \left[x_{1}: \ldots: x_{n}: 0\right] = p\). As \(\phi\) is both injective and surjective, it gives a bijection between \(\mathbb{P}^{n-1}\) and \(H_{\infty} \subset \mathbb{P}^{n}\), making it an isomorphism.

Step-by-step solution

Define the map given

Let's consider the map \(\phi: \mathbb{P}^{n-1} \to \mathbb{P}^{n}\) defined as: $$ \phi(\left[x_{1}: \ldots: x_{n}\right]) = \left[x_{1}: \ldots: x_{n}: 0\right] $$

Prove that the map is injective

To show that the map is injective, we need to prove that if \(\phi(p) = \phi(q)\), then \(p = q\): $$ \phi(p) = \phi(q) \implies \left[x_{1}: \ldots: x_{n}: 0\right] = \left[y_{1}: \ldots: y_{n}: 0\right] $$ Since in the projective space two points are equal if and only if their coordinates are proportional, then \(x_i = k \cdot y_i\) for all \(i = 1, \ldots, n\) and some \(k \neq 0\). Therefore, the points \(p = \left[x_{1}: \ldots: x_{n}\right]\) and \(q = \left[y_{1}: \ldots: y_{n}\right]\) are the same in \(\mathbb{P}^{n-1}\), proving injectivity.

Prove that the map is surjective onto H_{infty}

Recall that \(H_{\infty} = \{[x_1:\ldots:x_n:0]\} \subset \mathbb{P}^n\). Let \(p \in H_{\infty}\), i.e., \(p = \left[x_{1}: \ldots: x_{n}: 0\right]\). Then we have: $$ \phi(\left[x_1: \ldots: x_n\right]) = \left[x_{1}: \ldots: x_{n}: 0\right] = p $$ This shows that for any point \(p\) in \(H_{\infty}\), there exists a point in \(\mathbb{P}^{n-1}\) that maps to it by \(\phi\). Thus, the map is surjective onto \(H_{\infty}\).

Conclude that the map is an isomorphism

As \(\phi\) is both injective and surjective, it gives a bijection between \(\mathbb{P}^{n-1}\) and \(H_{\infty} \subset \mathbb{P}^{n}\). Thus, \(\phi\) is an isomorphism between these two spaces.

Show that varieties contained in H_{infty} is isomorphic to varieties in P^{n-1}

Let \(V \subseteq H_{\infty}\) be a projective variety. We know that \(H_{\infty}\) is isomorphic to \(\mathbb{P}^{n-1}\). Since \(V\) is a subvariety of \(H_{\infty}\), we can restrict \(\phi\) to \(V\), which means that \(V\) is isomorphic to a variety in \(\mathbb{P}^{n-1}\).

Show that any projective variety is isomorphic to a closed subvariety not contained in any hyperplane

Let \(X\) be a projective variety. If \(X\) is not contained in any hyperplane, then we are done. If \(X\) is contained in a hyperplane, say \(H\), then \(X\) is a subvariety of \(H\). Since \(H\) is isomorphic to some \(\mathbb{P}^{n-1}\) as shown before, there exists a variety in \(\mathbb{P}^{n-1}\) that is isomorphic to \(X\). This procedure can be repeated until we obtain a variety that is not contained in any hyperplane. Thus, any projective variety is isomorphic to a closed subvariety \(V \subset \mathbb{P}^{n}\) such that \(V\) is not contained in any hyperplane in \(\mathbb{P}^{n}\).

Key concepts

Algebraic Curves

Algebraic curves represent a fundamental concept in algebraic geometry, akin to the strings that bind the fabric of this mathematical field. Imagine a gently arching bridge or the graceful outline of a mountain range. In a mathematical sense, these can be similarly described by algebraic curves, which are the solutions to polynomial equations in two variables.

When we focus on projective algebraic curves, we're talking about curves that exist in projective spaces. These can include familiar shapes like lines and conics, extending to more exotic varieties with complex curves. An essential property of projective curves is that they have no 'ends'; they loop back on themselves, creating a continuum. This endless nature is born from the projective space itself, where every line meets at a point at infinity.

So when dealing with exercises like mapping from \( \mathbb{P}^{n-1} \) to a hyperplane in \( \mathbb{P}^{n} \) via homogeneous coordinates, we are essentially interacting with the underpinnings of algebraic curves. Each point on the curve is represented by a set of coordinates that satisfies both the original polynomial equation and the transformation properties of projective spaces.

Algebraic Geometry

At its heart, algebraic geometry is the study of solutions to polynomial equations, and it beautifully marries algebra with geometry. Polynomial equations are like the genes of algebraic geometry; they encode the instructions for creating a diverse array of shapes called varieties, including points, curves, surfaces, and more complicated multidimensional figures.

Within this realm, understanding the relationship between varieties and the ambient space they live in, such as \( \mathbb{P}^{n} \) or \( \mathbb{P}^{n-1} \) is fundamental. Whether these varieties are lines, planes, or more intricate surfaces, they each tell a unique story about the connection between equations and geometric forms.

With the context of the exercise, the isomorphism between the projective spaces and the inclusion of varieties into these spaces exemplifies how algebraic geometry strives to unify the abstract algebraic definitions with geometric intuition. Solving such problems is more than just performing algebraic manipulations; it is about visualizing the geometric transformations that take place.

Projective Spaces

Imagine if you could stretch your view to infinity, wrapping around so that opposite directions somehow touch. This is essentially what projective spaces do. Projective spaces, denoted as \( \mathbb{P}^{n} \) where 'n' indicates the dimension, are geometric settings where every line extends to meet at points at infinity, no matter the direction it heads towards.

In our transformed view into projective spaces, parallel lines in the Euclidean sense are no longer parallel; they converge at points at infinity. This remarkable property is what allows projective spaces to be so instrumental in algebraic geometry, as it ensures that every pair of lines intersects exactly once, either in the conventional space or at infinity.

The exercise's task to show an isomorphism between \( \mathbb{P}^{n-1} \) and a hyperplane in \( \mathbb{P}^{n} \) ties into this concept. By defining an isomorphism, we create a bridge between these different projective spaces, maintaining the integrity of their geometric structure despite adding an extra dimension.

Homogeneous Coordinates

To navigate the vast landscapes of projective spaces, we need a reliable compass, and homogeneous coordinates are precisely that. They are a set of coordinates that provide more flexibility than Cartesian coordinates when representing points in projective space.

Homogeneous coordinates include an extra dimension that allows us to distinguish between points that would otherwise look identical in standard coordinates. The brilliant twist of these coordinates is that they treat multiple representations as the same point, as long as one set can be obtained from another by scaling all coordinates by the same non-zero factor.

In the given exercise, we're using homogeneous coordinates to transition from \( \mathbb{P}^{n-1} \) to a hyperplane in \( \mathbb{P}^{n} \) by appending a 0 as the last coordinate. This trick preserves the essence of the points and curves while seamlessly integrating them into a higher-dimensional projective space. It is a vivid demonstration of the power and elegance that homogeneous coordinates possess in the sphere of algebraic geometry.