Let \(Y \subseteq \PP^n\) be a nonsingular variety of dimension \(\geq 2\) over an algebraically closed field \(k\). Suppose \(\PP^{n-1}\) is a hyperplane in \(\PP^n\) which does not contain \(Y\), and such that the scheme \(Y' = Y \intersect \PP^{n-1}\) is also nonsingular. Prove that \(Y\) is a complete intersection in \(\PP^n\) if and only if \(Y'\) is a complete intersection in \(\PP^{n-1}\).
Hint: see (II, Ex. 8.4) and use (9.12) applied to the affine cones over \(Y\) and \(Y'\).
Solution
Put \[ c=\operatorname{codim}_{\PP^n}Y=n-\dim Y. \] Then \[ \operatorname{codim}_{\PP^{n-1}}Y'=c. \]
1. If \(Y\) is a complete intersection in \(\PP^n\), then \(Y'\) is a complete intersection in the hyperplane \(\PP^{n-1}\).
Choose coordinates with \(H=V_+(\ell)\) and write \[ S=k[x_0,\ldots,x_n]. \] Since \(Y\) is a complete intersection of codimension \(c\), its homogeneous ideal is generated by \(c\) forms \[ I(Y)=(F_1,\ldots,F_c). \] The scheme-theoretic hyperplane section inside \[ H=\Proj S/(\ell) \] is defined by the images \[ \bar F_1,\ldots,\bar F_c\in S/(\ell). \] Its codimension in \(H\) is \(c\) because \(H\) does not contain the \(r\)-dimensional variety \(Y\) and \(Y'\) has dimension \(r-1\). Thus \(Y'\) is a codimension-\(c\) closed subscheme of \(H\) defined scheme-theoretically by \(c\) hypersurface equations. By II.8.4(a), it is a complete intersection in \(H\cong\PP^{n-1}\).
2. For the converse, choose coordinates so that\[ H=V_+(\ell),\qquad \ell=x_n. \]Let\[ S=k[x_0,\ldots,x_n],\qquad I=I(Y),\qquad A=S/I. \]Let \(\mathfrak p\subseteq A\) be the homogeneous prime ideal defining the affine cone over \(Y'\) inside the affine cone over \(Y\).
The affine cone over \(Y\) is \[ C(Y)=\Spec A. \] The scheme-theoretic hyperplane section is cut out projectively by \(\ell\), but the affine scheme \(V(\ell)\subseteq C(Y)\) may a priori have extra structure at the cone vertex. Its unique projective component is the cone over the irreducible nonsingular variety \(Y'\). The corresponding homogeneous prime in \(A\) is denoted \(\mathfrak p\). It is the saturation of the principal homogeneous ideal \(\ell A\) away from the irrelevant maximal ideal.
3. The ideal \(\ell A\) has the unique minimal prime \(\mathfrak p\), and\[ \ell A_{\mathfrak p}=\mathfrak pA_{\mathfrak p}. \]
Every minimal prime over the principal ideal \(\ell A\) has height at most one by the principal ideal theorem. The cone vertex has codimension \[ \dim C(Y)=\dim Y+1\ge3, \] so no component supported only at the vertex can be a minimal prime of \(\ell A\). Away from the vertex, \(V(\ell)\) is the \(\GG_m\)-bundle over the projective hyperplane section \(Y'\), which is irreducible. Thus \(\mathfrak p\) is the unique minimal prime over \(\ell A\).
At the generic point of \(Y'\), the scheme-theoretic intersection \(Y\cap H\) is nonsingular and hence reduced. Therefore the hyperplane equation vanishes there to order one. On the cone this says exactly that the one-dimensional local ring \(A_{\mathfrak p}\) has maximal ideal generated by \(\ell\): \[ \mathfrak pA_{\mathfrak p}=\ell A_{\mathfrak p}. \]
4. The quotient \(A/\mathfrak p\), the homogeneous coordinate ring of \(Y'\), is a normal domain.
By assumption \(Y'\) is a complete intersection and is nonsingular. Its dimension is \[ \dim Y'=\dim Y-1\ge1. \] Thus \(Y'\) is normal. Exercise II.8.4(b) says that a normal complete intersection of positive dimension is projectively normal. By definition, its homogeneous coordinate ring is therefore integrally closed. That homogeneous coordinate ring is precisely \[ A/\mathfrak p. \] Hence \(A/\mathfrak p\) is normal.
5. One has\[ \boxed{\mathfrak p=\ell A}, \]and the affine cone ring \(A\) is normal.
Let \(\mathfrak q\subseteq A\) be a prime containing \(\ell\). Because \(\mathfrak p\) is the unique minimal prime over \(\ell A\), we have \[ \mathfrak p\subseteq\mathfrak q. \] Apply Proposition III.9.12 to the noetherian local domain \[ A_{\mathfrak q} \] with \(t=\ell\) and prime \(\mathfrak pA_{\mathfrak q}\). Step 3 gives hypotheses (1) and (2) of that proposition, and step 4 gives hypothesis (3), since \[ A_{\mathfrak q}/\mathfrak pA_{\mathfrak q} \] is a localization of the normal domain \(A/\mathfrak p\). Therefore III.9.12 gives \[ \mathfrak pA_{\mathfrak q}=\ell A_{\mathfrak q} \] and also shows that \(A_{\mathfrak q}\) is normal.
If \(\mathfrak q\) does not contain \(\ell\), both ideals localize to the unit ideal. Hence \(\mathfrak p\) and \(\ell A\) have equal localizations at every prime of \(A\), so \[ \mathfrak p=\ell A. \]
For completeness, \(A\) is also normal at primes not containing \(\ell\). Indeed, \[ A_\ell\cong B[\ell,\ell^{-1}], \] where \[ B=(A_\ell)_0 \] is the affine coordinate ring of the open subset \[ Y\setminus Y'. \] This open subset is nonsingular and therefore normal, so the Laurent polynomial ring \(A_\ell\) is normal. Together with the normality supplied by III.9.12 at primes containing \(\ell\), every local ring of \(A\) is normal.
6. Let \(f_1,\ldots,f_c\in k[x_0,\ldots,x_{n-1}]\) generate the homogeneous ideal of \(Y'\) in \(H\cong\PP^{n-1}\). Then there exist homogeneous elements\[ g_1,\ldots,g_c\in I(Y) \]whose restrictions to \(H\) are \(f_1,\ldots,f_c\).
Because \(Y'\) is a complete intersection of codimension \(c\), choose homogeneous generators \[ I_H(Y')=(f_1,\ldots,f_c). \] Viewed in \(S\), the homogeneous ideal of the affine cone over \(Y'\) is \[ (f_1,\ldots,f_c,\ell). \] Step 5 says that its image in \(A=S/I\) is the principal ideal \(\ell A\). Equivalently, \[ I+(\ell)=(f_1,\ldots,f_c,\ell). \] Thus for each \(i\) there are homogeneous \(g_i\in I\) and \(h_i\in S\) with \[ f_i=g_i+\ell h_i. \] After choosing homogeneous components of the same degree, each \(g_i\) is homogeneous and restricts to \(f_i\) on \(H\).
7. The ideal \(I(Y)\) is generated by \(g_1,\ldots,g_c\).
Put \[ J=(g_1,\ldots,g_c)\subseteq I. \] Step 6 gives \[ I+(\ell)=J+(\ell). \] Let \(F\in I\) be homogeneous. Then \[ F=G+\ell H \] for some \(G\in J\) and \(H\in S\). Since \(F,G\in I\), we have \[ \ell H\in I. \] The prime ideal \(I\) does not contain \(\ell\), because the hyperplane does not contain \(Y\). Therefore \(H\in I\). Thus \[ I=J+\ell I. \]
Pass to the finitely generated graded \(S\)-module \[ M=I/J. \] The equality above says \[ M=\ell M. \] If \(M\ne0\), choose a nonzero homogeneous element of least degree. Writing it as \(\ell\) times another element of \(M\) lowers the degree by one, a contradiction. Hence \(M=0\), so \[ I=J=(g_1,\ldots,g_c). \]
8. The variety \(Y\) is a complete intersection in \(\PP^n\).
Step 7 shows that the homogeneous ideal of \(Y\) is generated by exactly \[ c=\operatorname{codim}_{\PP^n}Y \] homogeneous elements. This is the definition of a complete intersection. Thus the converse implication holds.
9. Q.E.D.
Step 1 proves the forward implication, and steps <1>2–<1>8 prove the converse.