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I am getting quite confused with the following material and I would greatly appreciate if someone could provide me an explanation for this. Suppose I have $F_1, ..., F_r \in \mathbb{Q}[x_0, .., x_n]$ homogeneous polynomials. Then the zero set $F_1(\mathbf{x}) = ... = F_r(\mathbf{x}) = 0$ defines a projective variety (here I don't necessarily mean irreducible).

We say that a point $P$ on $X$ is singular if the Jacobian of the polynomials have rank less than $n - \dim X$ at $P$.

I have two questions. (1) Suppose $X$ can be realized as a zero set of other polynomials $G_1(\mathbf{x}) = ... = G_s(\mathbf{x}) = 0$ as well. How do we know that $P$ in $X$ being singular is irrelevant to the choice of polynomials defining $X$? i.e. is the rank of the Jacobian with respect to $F_j$'s and $G_i$'s the same at every point on $X$?

(2) Suppose I have a homogeneous polynomial $F(\mathbf{x}) \in \mathbb{Q}[x_1, ..., x_n]$, and $H (\mathbf{x}) = a_0 x_0 + ... + a_n x_n$, say $a_n \not = 0$. Then the variety defined by $F(\mathbf{x}) = H(\mathbf{x}) = 0$ can be viewed as a zero set of one polynomial in $F_0(x_0, ..., x_{n-1})$ obtained by substitution. So $X$ can be viewed as a projective variety of $\mathbb{P}^{n-1}$ as well. How does $X \subseteq \mathbb{P}^n$ defined in first way relate to $X \subseteq \mathbb{P}^{n-1}$ defined in the second way. In particular, how do singularities relate each other?

Thank you very much!!

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The fact that a point is singular can be expressed in terms of its local ring: a point is non singular if its local ring is regular. A point which is not singular is singular. The local ring does not depend of the generators of the ideal which defines the variety.

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