# 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.

## Related Questions

### If I’m given the distances of some point $p$ from $n$ coplanar points, can I tell whether $p$ is also coplanar?

2  Asked on February 2, 2021

### Reference request: The area of the modular curve $mathcal{H}/SL(2,mathbb{Z})$ is $pi/3$.

0  Asked on February 2, 2021 by stupid_question_bot

### What is the connection between the Radon-Nikodym derivative and the fundamental theorem of calculus in $mathbb{R}^n$?

0  Asked on February 2, 2021 by 5fec

### Find the General solution $u(t,x)$ to the partial differential equation $u_x+tu=0$.

3  Asked on February 1, 2021 by c-web

### Does the set $left{ left(begin{array}{c} x\ y end{array}right)inmathbb{R}^{2}|xleq yright}$ span all of $mathbb{R}^2$

3  Asked on February 1, 2021 by user75453

### Proof regarding continuity and Dirichlet function.

1  Asked on February 1, 2021 by mathcurious

### variation of Vitali in $mathbb{R}^2$

1  Asked on February 1, 2021 by samantha-wyler

### Seifert-Van Kampen Theorem Application

0  Asked on February 1, 2021

### Show whether the sequence converges (sequence given as a sum)

2  Asked on February 1, 2021

### Why this inequality is correct

2  Asked on February 1, 2021 by nizar

### Show that the number of faces in a planar connected graph on $n$ vertices is bounded from above by $2n-4$

1  Asked on February 1, 2021 by carl-j

### Definition of commutative and non-commutative algebra and algebra isomorphism

1  Asked on February 1, 2021 by theoreticalphysics

### Derivative of Inverse of sum of matrices

1  Asked on February 1, 2021 by ronaldinho

### The time gap between the two instants, one before and one after 12:00 noon, when the angle between the hour hand and the minute hand is 66°

1  Asked on February 1, 2021 by nmasanta

### How can I approximate $left(1+frac{2}{4x-1}right)^{x}$

4  Asked on February 1, 2021 by no-one-important

### Simulating Brownian motion in R. Is this correct?

1  Asked on February 1, 2021 by parseval

### Factoristaion of $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$

2  Asked on January 31, 2021 by a-level-student

### self intersection of two Lorentzian manifolds also Lorentzian in this case?

0  Asked on January 31, 2021 by jack-zimmerman

### Prove that if G is isomorphic to H, then $alpha(G) = alpha(H)$

1  Asked on January 31, 2021 by marconian