Let \(Y \subseteq \PP^n\) be a projective variety of dimension \(r\). Let \(f_1, \ldots, f_t \in S = k[x_0, \ldots, x_n]\) be homogeneous polynomials generating the ideal of \(Y\). Let \(P \in Y\) be a point with homogeneous coordinates \(P = (a_0, \ldots, a_n)\). Show that \(P\) is nonsingular on \(Y\) if and only if the matrix \[ \left[ \frac{\partial f_i}{\partial x_j}(a_0, \ldots, a_n) \right] \] has rank \(n - r\).
Hint: Show that this rank does not depend on the homogeneous coordinates chosen for \(P\); pass to an open affine \(U \subseteq \PP^n\) containing \(P\) and use the affine Jacobian matrix; and use Euler’s lemma, which says that a homogeneous polynomial \(f\) of degree \(d\) satisfies \(\sum_i x_i \frac{\partial f}{\partial x_i} = d \cdot f\).
Solution
Let \(d_i=\deg f_i\) and write \[ J(P)= \left[ \frac{\partial f_i}{\partial x_j}(a_0,\ldots,a_n) \right]_{\substack{1\le i\le t\\0\le j\le n}}. \]
1. The rank of \(J(P)\) is independent of the chosen nonzero homogeneous-coordinate representative of \(P\).
Replace \[ (a_0,\ldots,a_n) \] by \[ (\lambda a_0,\ldots,\lambda a_n), \qquad \lambda\in k^\times. \] Because \(f_i\) is homogeneous of degree \(d_i\), each first partial derivative is homogeneous of degree \(d_i-1\). Thus every entry in row \(i\) is multiplied by the same nonzero scalar \[ \lambda^{d_i-1}. \] Multiplying individual rows by nonzero scalars does not change matrix rank. Hence the rank depends only on the projective point \(P\).
2. Choose an index \(\ell\) with \(a_\ell\ne0\) and normalize the coordinates so that \(a_\ell=1\).On the standard affine chart \(U_\ell=D_+(x_\ell)\cong\AA^n\), the affine equations of \(Y\cap U_\ell\) are\[ g_i(u_0,\ldots,\widehat{u_\ell},\ldots,u_n) = f_i(u_0,\ldots,u_{\ell-1},1,u_{\ell+1},\ldots,u_n). \]
This is the standard dehomogenization of homogeneous equations on the chart \(x_\ell\ne0\). The point \(P\) corresponds to the affine point \[ p=(a_0,\ldots,\widehat{a_\ell},\ldots,a_n). \] Since localization and dehomogenization preserve the vanishing ideal on the chart, these \(g_i\) generate the ideal of \(Y\cap U_\ell\) there. The open subvariety \(Y\cap U_\ell\) has the same dimension \(r\) as \(Y\), because it is a nonempty open subset of the irreducible variety \(Y\).
3. The affine Jacobian matrix of the \(g_i\) at \(p\) is obtained from \(J(P)\) by deleting column \(\ell\).
For \(j\ne\ell\), differentiating the dehomogenized polynomial gives \[ \frac{\partial g_i}{\partial u_j}(p) = \frac{\partial f_i}{\partial x_j}(a_0,\ldots,a_n). \] Thus the \(t\times n\) affine Jacobian consists exactly of the columns of \(J(P)\) indexed by \(j\ne\ell\).
4. The deleted column \(\ell\) lies in the span of the remaining columns, so the projective and affine Jacobian matrices have the same rank.
Euler’s identity for the homogeneous polynomial \(f_i\) is \[ \sum_{j=0}^n x_j\frac{\partial f_i}{\partial x_j}=d_i f_i. \] Evaluate it at the representative \(a=(a_0,\ldots,a_n)\) of the point \(P\). Since \(P\in Y\), one has \(f_i(a)=0\), hence \[ \sum_{j=0}^n a_j\frac{\partial f_i}{\partial x_j}(a)=0. \] This holds for every row \(i\), even when the characteristic divides \(d_i\). Because \(a_\ell=1\), it can be rewritten simultaneously for all rows as the column relation \[ C_\ell=-\sum_{j\ne\ell}a_j C_j, \] where \(C_j\) denotes column \(j\) of \(J(P)\). Thus adjoining column \(\ell\) to the affine Jacobian does not increase its rank. By step 3, the two ranks are equal.
5. One has\[ \boxed{P\text{ nonsingular on }Y \quad\Longleftrightarrow\quad \operatorname{rank}J(P)=n-r.} \]
2.The affine Jacobian criterion says that \(P\) is nonsingular on this affine open exactly when its affine Jacobian has rank\[ n-r \][@Har10a, Chapter I, §5].Nonsingularity is local, so this is equivalent to \(P\) being nonsingular on \(Y\).Step 4 identifies the affine Jacobian rank with \(\operatorname{rank}J(P)\).The displayed equivalence follows.
6. Q.E.D.
Step 1 verifies that the stated rank is intrinsic to the projective point. Steps <1>2–<1>4 compare it with the affine Jacobian rank, and step 5 applies the affine criterion to prove the result.