Let \(f \in k[x,y,z]\) be a homogeneous polynomial of positive degree, let \(Y = Z(f) \subseteq \PP^2\) be the algebraic set defined by \(f\), and suppose that for every \(P \in Y\) at least one of \[ \frac{\partial f}{\partial x}(P), \qquad \frac{\partial f}{\partial y}(P), \qquad \frac{\partial f}{\partial z}(P) \] is nonzero. Show that \(f\) is irreducible, and hence that \(Y\) is a nonsingular variety.
Hint: Use (Ex. 3.7).
(The positive-degree hypothesis)
The source states the exercise for an arbitrary homogeneous polynomial. If \(f\in k^\times\) is a nonzero constant, then \(Z(f)=\varnothing\), so the derivative hypothesis holds vacuously, while a unit is not an irreducible polynomial. Thus positive degree, equivalently nonemptiness in the intended hypersurface situation, is necessary for the stated conclusion.
Solution
The ground field \(k\) is algebraically closed, as throughout the chapter.
1. If \(f\) has a factorization\[ f=gh \]with \(g,h\) homogeneous of positive degree, then there is a point\[ P\in Z(g)\cap Z(h). \]
Each of \(Z(g)\) and \(Z(h)\) is a nonempty projective plane curve. Indeed, a nonconstant homogeneous polynomial in three variables defines a positive-dimensional projective hypersurface. Exercise I.3.7, proved on P-AGH37HYPMEETS, says that any two projective plane curves meet. Hence their intersection contains a point \(P\).
2. At every point \(P\in Z(g)\cap Z(h)\), all three first partial derivatives of \(f=gh\) vanish.
For each coordinate \(x_i\in\{x,y,z\}\), the product rule gives \[ \frac{\partial f}{\partial x_i} = g\frac{\partial h}{\partial x_i} +h\frac{\partial g}{\partial x_i}. \] At a point where \(g(P)=h(P)=0\), both summands vanish. Thus \[ f_x(P)=f_y(P)=f_z(P)=0. \] Since \(f(P)=0\) as well, such a point is singular on the hypersurface.
3. The polynomial \(f\) is irreducible.
Suppose instead that \(f\) is reducible. Because \(f\) is homogeneous, it admits a factorization \[ f=gh \] with \(g,h\) homogeneous of positive degree. To see that the factors may be taken homogeneous, factor \(f\) into irreducibles in the graded UFD \(k[x,y,z]\); the least- and greatest-degree terms of a product show that every irreducible factor of a homogeneous element is homogeneous.
By step 1, choose \[ P\in Z(g)\cap Z(h). \] Step 2 makes all three first partial derivatives of \(f\) vanish at \(P\). This contradicts the hypothesis of the problem. Therefore no such factorization exists and \(f\) is irreducible.
4. The algebraic set \(Y=Z(f)\) is a nonsingular projective variety.
Step 3 makes the principal homogeneous ideal \((f)\) prime. Thus \(Y\) is irreducible by the projective ideal correspondence, and it is nonempty because \(f\) has positive degree. Hence \(Y\) is a projective variety.
For a hypersurface in \(\PP^2\), the projective Jacobian criterion P-AGH58JACOBIANRANK says that a point is nonsingular exactly when the gradient has rank one, i.e. when not all of \[ f_x,\quad f_y,\quad f_z \] vanish there. This is the assumed condition at every \(P\in Y\). Thus \(Y\) is nonsingular.
5. Q.E.D.
Steps <1>1–<1>3 prove irreducibility, and step 4 gives the asserted nonsingular-variety conclusion.