Checking if a specific variety is smooth

Question

Suppose we have the following variety $X = V(z_{0}^d + \ldots + z_{n + 1}^d) \subset \mathbb{P}^{n + 1}$. I want to check that this variety is smooth. One condition that the book that I am reading stated that we could use is that if the the cone $C(X) - \{0\}$ is smooth. How can we check that it is smooth ? I want to know also what is $C(X) - \{0\}$ ?

Answer 1

1) As you probably know $\mathbb{P}^{n}$ may be constructed as a quotient of $\mathbb{A}^{n+1}-\{0\}$ by the scaling action of $\mathbb{C}^\times$. Then cone $C(X)$ of $X$ is the vanishing locus of the same polynomials defining $X$ but in $\mathbb{A}^{n+1}$ instead of $\mathbb{P}^n$.

2) Using the Jacobian criterion for smoothness you can verify that the cone will almost always be singular at the origin (it certainly is when $1<d$, what may fail if $d=1$?). The key fact in here is that $X$ is defined by homogeneous polynomials.

3) Let me assume that $X$ is equidimensional. To see why checking that the cone is smooth implies the variety is smooth we argue as follows. We can certainly check smoothness locally, so consider the affine variety $X_i:=X\cap U_i$, for some $i\in\{0,\dots,n\}$. Take $i=0$ for simplicity. Then, if $I=(f_1,\dots,f_k)$ is the homogeneous radical ideal defining $X$, the dehomogenized ideal of $X_i$ will be $I_i=(g_1,\dots,g_k)$ where $g_j:=f_j(1,x_1,\dots,x_n)$.

To check smoothness we need to check that the rank of the Jacobian matrix (at each point) is maximal, this is, $\operatorname{rank} J_i=n-\dim X_i$. Thus, we need to show that the Jacobian matrix of the cone is full rank if and only if all the matrices $J_i$ are. Observe that the Jacobian matrix $J_i$ has the same rank as the submatrix obtained from $J$ by removing the $i$th column (or row, depending on your conventions). Therefore, all $X_i$ are smooth $\Leftrightarrow$ $C(X)$ is smooth. (Use that the rank of a matrix is the size of the largest square submatrix which is full rank.)

(PS. I believe my argument in 3) is correct, however, if anyone finds a mistake please let me know!)