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Intersection of irreducible hypersurface with tangent hyperplane in a non-singular point is singular

Mathematics Asked by lupidupi on December 23, 2020

Prove that the intersection of an irreducible hypersurface $V(F) subseteq mathbb{A}^{n}$ with the tangent hyperplane $T(V)_p$ in a non-singular point $p in V$ is singular at $p$. Note: Here we define the intersection by the ideal $(F, L)$ where $L$ is the linear equation for the tangent hyperplane.

Also show that this need not be true when one defines the intersection with reduced structure, that is, by the radical of the ideal $(F, L)$

I found a similar question and answer in this math.stack here. Since I don’t have enough reputation to comment, I cannot ask questions to the OP and the person who answered there.

The difference between my question and the one linked is that I am in affine space rather than in projective space. Is the solution the same for the affine case I am in? If yes, then there are certain things that I do not understand: What does it mean that the "problem is local at $P$" and how does this imply that the tangent hyperplane is given by $x_1 = 0$; or is it perhaps in my case $X_1 = a_1$? And how does one know that the polynomial defining the (affine) hypersurface is given by $x_1 + f_2(x_2, ldots, x_n) + ldots + f_r(x_2, ldots, x_n)$, where the $f_i$ are homogeneous of degree $i$; is it because any polynomial can be decomposed into a sum of homogenoues components? And lastly, how does then the intersection become the hypersurface given by the polynomial $f_0$, where the $x_1$ coordinate become 0; is this because the tangent hyperplane is given by coordinate $x_1 = 0$?

If no, then how do we solve this problem?

The definition of the tangent space to an affine hypersurface I am given is: The tangent space $T_PV$ to an affine hypersurface $V = (f) subseteq mathbb{A}^{n}$ at $P = (a_1, ldots, a_n) in V$ is the linear subvariety

$$V(frac{partial f}{partial x_1}(P)(X_1 – a_1) + ldots + frac{partial f}{partial x_n}(P)(X_n – a_n)).$$

And the point $P$ is non-singular if $T_P V$ is a hyperplane, i.e. if $df(P) = (frac{partial f}{partial x_1}(P), ldots, frac{partial f}{partial x_n}(P)) neq 0$.

What I know: $p$ is non-singular, i.e. $frac{partial F}{partial X_j} neq 0$ for some $j$.

What I have to show: That the polynomial $f$ defining the intersection is singular at $p$, i.e. $frac{partial f}{partial X_i} = 0$ for all $i$.

BTW: the question is taken from Reid’s Undergraduate algebraic geometry.

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