Question

Let \(V=V\left(X^{2}-Y^{3}, Y^{2}-Z^{3}\right) \subset \mathbb{A}^{3}, P=(0,0,0), \mathfrak{m}=\mathfrak{m}_{P}(V) .\) Find \(\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)\).

Short answer

Answer: The dimension of \(\mathfrak{m}/\mathfrak{m}^2\) is 3.

Step-by-step solution

Finding coordinates ring and ideal of the variety V at P

The ring of polynomials in the three variables is given by \(k[X,Y,Z]\). We are given the variety \(V(X^2-Y^3, Y^2-Z^3)\), and the ideal of the variety is given by \(I(V) = \langle X^2-Y^3, Y^2-Z^3 \rangle\). The coordinate ring of \(V\) is the quotient ring $$\mathbb{A}_V = \frac{k[X,Y,Z]}{I(V)}.$$ Now let's focus on the point \(P\). In our case, \(P = (0,0,0)\), and \(\mathfrak{m} = \mathfrak{m}_P(V)\) is the maximal ideal of \(\mathbb{A}_V\) corresponding to \(P\).

Compute the Jacobian matrix of the regular map

We consider the regular map given by the polynomials \(f(X,Y,Z) = X^2 - Y^3\) and \(g(X,Y,Z) = Y^2 - Z^3\). The Jacobian matrix of that regular map is given by $$ J(f,g)= \begin{bmatrix} \frac{\partial f}{\partial X} & \frac{\partial f}{\partial Y} & \frac{\partial f}{\partial Z} \\ \frac{\partial g}{\partial X} & \frac{\partial g}{\partial Y} & \frac{\partial g}{\partial Z} \end{bmatrix} = \begin{bmatrix} 2X & -3Y^2 & 0 \\ 0 & 2Y & -3Z^2 \end{bmatrix}. $$

Evaluating the Jacobian matrix at point P

Now we need to compute the rank of the Jacobian matrix evaluated at the point \(P=(0,0,0)\). Substituting the coordinates of \(P\) into the Jacobian matrix, we get $$ J(f,g)|_{P} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}. $$ The rank of this matrix is 0.

Compute the dimension of the cotangent space

According to the theorem about the dimension of the cotangent space, we have $$ \operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right) = \operatorname{dim}(V) - \operatorname{rank}(J(f,g)). $$ In our case, \(V\subset\mathbb{A}^3\). Since the rank of the Jacobian matrix is 0, the equation above becomes $$ \operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right) = 3 - 0 = 3. $$ Therefore, we have found that \(\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right) = \boxed{3}\).

Key concepts

Jacobian Matrix

The Jacobian matrix is an essential tool in algebraic geometry, particularly when analyzing regular maps. In simple terms, it is a matrix of all first-order partial derivatives of a vector-valued function. Its main purpose is to capture how a small change in input variables will affect the output, linearizing the function near a point of interest.

Imagine you're working with the functions - \(f(X,Y,Z) = X^2 - Y^3\)- \(g(X,Y,Z) = Y^2 - Z^3\) The Jacobian matrix for these functions can be expressed as:\[J(f,g)= \begin{bmatrix}\frac{\partial f}{\partial X} & \frac{\partial f}{\partial Y} & \frac{\partial f}{\partial Z} \\frac{\partial g}{\partial X} & \frac{\partial g}{\partial Y} & \frac{\partial g}{\partial Z}\end{bmatrix}\]

This matrix essentially contains each derivative linking changes in each variable with changes in the functions. When you evaluate this Jacobian at the point \((0,0,0)\), you are checking how those functions behave at that specific point. Here, it becomes:\[J(f,g)|_{P} = \begin{bmatrix}0 & 0 & 0 \0 & 0 & 0\end{bmatrix}\]
This result shows that the rank, or the maximum number of linearly independent rows or columns, is 0, meaning there is no local linear independence of the derivatives at that point.

Coordinate Ring

The concept of the coordinate ring arises when studying varieties, which are sets of zeros of polynomial equations. For a variety like \(V = V(X^2 - Y^3, Y^2 - Z^3)\), the coordinate ring is a specific algebraic structure that represents it.

In simpler terms, it's a way to compress all polynomial relationships within the variety into a manageable form. The coordinate ring is constructed as a quotient of the polynomial ring.

• For the variety \(V\), the polynomial ring is \(k[X, Y, Z]\), which indicates a set of all polynomials in variables \(X, Y, Z\).
• The ideal \(I(V) = \langle X^2-Y^3, Y^2-Z^3 \rangle\) consists of all polynomials that vanish on \(V\).
• Then, the coordinate ring is \(\mathbb{A}_V = \frac{k[X,Y,Z]}{I(V)}\), meaning we look at polynomials under the condition they are equivalent modulo the ideal \(I(V)\).

This way, the coordinate ring essentially captures the essence of the variety, allowing us to manipulate and study it algebraically.

Cotangent Space

The cotangent space is a fundamental concept in algebraic geometry, offering a natural setting to study the properties and structure of a variety at a specific point. Consider it as a collection of linear functionals that describe how functions behave infinitesimally at the point of interest.

The cotangent space at a point \(P\) on a variety \(V\) is represented by \(\mathfrak{m} / \mathfrak{m}^2\), where \(\mathfrak{m}\) is the maximal ideal of the coordinate ring at \(P\).

• \(\mathfrak{m}\) consists of elements of the coordinate ring that vanish at \(P\).
• \(\mathfrak{m}^2\) represents elements formed by the product of any two elements in \(\mathfrak{m}\).

Now, calculating \(\mathrm{dim}_k(\mathfrak{m} / \mathfrak{m}^2)\) gives us the dimension of the cotangent space, an indicator of the variety's 'directional possibilities' at \(P\). In our example, the dimension was found using the relation:\[\operatorname{dim}_{k}\left(\mathfrak{m} / \mathfrak{m}^{2}\right) = \operatorname{dim}(V) - \operatorname{rank}(J(f,g))\]With a rank of 0 and \(V\) in \(\mathbb{A}^3\), the dimension, and thus insight into the infinitesimal structure at \((0,0,0)\), is 3.