Question

\(\mathrm{A}\) set \(V \subset \mathbb{P}^{n}(k)\) is called a linear subvariety of \(\mathbb{P}^{n}(k)\) if \(V=V\left(H_{1}, \ldots, H_{r}\right)\), where each \(H_{i}\) is a form of degree 1. (a) Show that if \(T\) is a projective change of coordinates, then \(V^{T}=T^{-1}(V)\) is also a linear subvariety. (b) Show that there is a projective change of coordinates \(T\) of \(\mathbb{P}^{n}\) such that \(V^{T}=V\left(X_{m+2}, \ldots, X_{n+1}\right)\), so \(V\) is a variety. (c) Show that the \(m\) that appears in part (b) is independent of the choice of \(T\). It is called the dimension of \(V(m=-1\) if \(V=\varnothing\) ).

Short answer

Based on the provided step-by-step solution, the short answer is: (a) V^T is a linear subvariety because after applying the invertible matrix T to each form H_i, we obtain new forms of degree 1, H'_i. Therefore, V^T is also defined by a set of forms of degree 1. (b) We can find a projective change of coordinates T that transforms V to a standard form by forming an invertible matrix T whose rows are linearly independent vectors formed by the coefficients of the H_i forms. The transformed variety V^T will be in the standard form V(x_m, x_{m+1}, ..., x_n). (c) The integer m representing the dimension of V is independent of the choice of T because any transformation T would lead to the same dimension. When V^T and V' represent the same variety V under different transformations T and T', the dimensions m and k must be equal, showing that m is independent of the choice of T.

Step-by-step solution

(a) Prove V^T is a linear subvariety

To show that V^T is a linear subvariety, we need to prove that it can be defined by a set of form of degree 1. Recall that projective change of coordinates T is given by an invertible matrix. We are given \(V = V(H_1, H_2, \dots, H_r)\), where each \(H_i\) is a form of degree 1. Let's consider the projective change of coordinates T. We have \(V^T = T^{-1}(V) = T^{-1}(V(H_1, H_2, \dots, H_r))\). Since T is invertible, applying \(T^{-1}\) to each \(H_i\) would result in forms of degree 1, say \(H'_i = [H'_i(x_0,\dots,x_n)]\). Hence, \(V^T = V(H'_1, H'_2, \dots, H'_r)\). This shows that \(V^T\) is also a linear subvariety.

(b) Find projective change of coordinates to standard form

To prove there exists projective change of coordinates T, we need to show that, after applying T, V becomes a variety \(V(X_{m+2}, \ldots, X_{n+1})\). Let \(V = V(H_1, H_2, \dots, H_r)\), where each \(H_i\) is a form of degree 1 of the form \(a_{i1}X_1+ a_{i2}X_2 + \dots + a_{in}X_n\). We can form an invertible matrix T whose rows are linearly independent vectors formed by the coefficients of the \(H_i\)s. Applying the projective change of coordinates T, we will have the transformed variety V^T as \(V(x_m, x_{m+1}, \dots, x_n)\) in standard form.

(c) Prove m is independent of the choice of T

To show that the integer m is independent of the choice of T, which is the dimension of V, we want to show that any transformation T would lead to the same dimension. Suppose we had another matrix T' that we could apply to V, so V transform by T' would become \(V'(x_k, x_{k+1}, \dots, x_n)\). Now consider the projective change of coordinates S from V^T to V' given by S = T'T^{-1}. Under change of coordinate, S also changes V^T as \(S^{-1}\left(V'\right) = V\) so that applying T' gives \(V'(x_k, x_{k+1}, \dots, x_n)\). However, as V^T is \(V(x_m, x_{m+1},\ldots,x_n)\) in standard form, and both V^T and V' are representing the same variety V, it follows that m=k and thus the dimension m is independent of the choice of T.

Key concepts

Projective Change of Coordinates

Understanding how a projective change of coordinates works is central to grasping problems in algebraic geometry. Imagine you have a set of equations defining a shape, and you twist, rotate or shift your viewpoint – this is what a projective transformation does to your set, geometrically speaking. Mathematically, a projective change of coordinates is represented by an invertible matrix. The beauty of this transformation lies in its ability to preserve the degree of the forms that define a variety.

When applied to linear subvarieties, a projective change keeps the defining equations linear, a property which we exploit to simplify and understand the structure of the variety in question better. For instance, if we have a variety defined by forms of degree 1, applying an invertible projective transformation will yield new forms of the same degree, maintaining the linearity of the subvariety. Thus, the transformed variety, denoted as V^T, remains a linear subvariety under the transformed coordinates.

Algebraic Geometry

At its heart, algebraic geometry is the study of solutions to algebraic equations and their geometric properties. It bridges abstract algebra and geometry, providing a rich framework for solving a wide range of problems.

In the context of the exercise, we deal with linear subvarieties within projective space \( \mathbb{P}^{n}(k) \), where the goal is to find a projective change of coordinates that simplifies our view of the variety. This simplification often involves finding a representation where the variety is defined by zeroing out certain coordinates, hence making the complex relationship between algebraic equations and their geometric interpretations more tractable.

The discipline often involves moving between different representations or coordinate systems to gain further insight or simplify problems. By employing tactics such as the projective change of coordinates, algebraic geometers can unravel the inherent properties of varieties independent of the perspective from which you view them.

Dimension of a Variety

The dimension of a variety is a fundamental concept that measures how 'big' the variety is in a certain sense. It is comparable to dimensions we are familiar with in everyday life – a line is one-dimensional, a plane is two-dimensional, and so on. However, in algebraic geometry, we extend this notion to deal with complex shapes defined by equations.

One of the exercises's key takeaways is understanding that the dimension of a variety remains constant under a projective change of coordinates. This invariance is an essential property since it implies that dimension is an intrinsic characteristic of the variety, not dependent on how we choose to describe it mathematically. In simpler terms, no matter how you rotate or shift the variety, you will not change its fundamental 'size'.

In our exercise, we denote the dimension by m, and we show that after any projective change of coordinates, the dimension stays the same. The variety, regardless of the transformation applied, will have the same number of 'free' coordinates \(X_{m+2}, \ldots, X_{n+1}\) – which speaks to the underlying structure and 'space-filling' nature of the variety.