For any closed subscheme \(X \subseteq \PP^n\), we denote by \(C(X) \subseteq \PP^{n+1}\) the projective cone over \(X\) (I, Ex. 2.10). If \(I \subseteq k[x_0, \ldots, x_n]\) is the largest homogeneous ideal of \(X\), then \(C(X)\) is defined by the ideal generated by \(I\) in \(k[x_0, \ldots, x_{n+1}]\).
Give an example to show that if \(\ts{X_t}\) is a flat family of closed subschemes of \(\PP^n\), then \(\ts{C(X_t)}\) need not be a flat family in \(\PP^{n+1}\).
To remedy this situation, we make the following definition. Let \(X \subseteq \PP_T^n\) be a closed subscheme, where \(T\) is a noetherian integral scheme. For each \(t \in T\), let \(I_t \subseteq S_t = k(t)[x_0, \ldots, x_n]\) be the homogeneous ideal of \(X_t\) in \(\PP_{k(t)}^n\). We say that the family \(\ts{X_t}\) is very flat if for all \(d \geq 0\) the dimension \[ \dim_{k(t)} (S_t / I_t)_d \] is independent of \(t\). Here \((\cdot)_d\) means the homogeneous part of degree \(d\).
If \(\ts{X_t}\) is a very flat family in \(\PP^n\), show that it is flat. Show also that \(\ts{C(X_t)}\) is a very flat family in \(\PP^{n+1}\), and hence flat.
If \(\ts{X_{(t)}}\) is an algebraic family of projectively normal varieties in \(\PP_k^n\), parametrized by a nonsingular curve \(T\) over an algebraically closed field \(k\), then \(\ts{X_{(t)}}\) is a very flat family of schemes.
Solution
For a fibre \(X_t\subseteq\PP^n_{k(t)}\), put \[ h_t(d)=\dim_{k(t)}(S_t/I_t)_d. \]
1. There is a flat family of three reduced points in \(\PP^2\) whose general fibre is noncollinear and whose special fibre is collinear.
Take \(T=\AA^1\) with coordinate \(t\) and define three sections of \[ \PP^2\times T\longrightarrow T \] by \[ P_1(t)=[1:0:0],\qquad P_2(t)=[0:1:0],\qquad P_3(t)=[1:1:t]. \] They are pairwise disjoint for every \(t\). Let \(\mathcal X\) be their union. Then \[ \mathcal X\cong T\amalg T\amalg T \] over \(T\), so \(\mathcal X\to T\) is flat.
For \(t\ne0\), the point \(P_3(t)\) does not lie on the line \(z=0\) through \(P_1,P_2\), so the three points are noncollinear. For \(t=0\), all three distinct points lie on \(z=0\).
2. The Hilbert functions of the fibres in step 1 are\[ h_t(d)= \begin{cases} 1,&d=0,\\ 3,&d\ge1, \end{cases} \qquad(t\ne0), \]whereas\[ h_0(d)= \begin{cases} 1,&d=0,\\ 2,&d=1,\\ 3,&d\ge2. \end{cases} \]
For three noncollinear points, linear forms already give arbitrary values at the three points, so the degree-one coordinate quotient has dimension three. Multiplying separating linear forms by powers of a linear form nonvanishing at all three points shows that the restriction map remains surjective in every degree \(d\ge1\). Thus the first formula holds.
For the special fibre, all points lie on the line \(z=0\). The restrictions of linear forms therefore have dimension two. Quadratic polynomials on that line interpolate arbitrary values at three distinct points, so the degree-two quotient has dimension three; multiplying by a nonvanishing linear form gives dimension three in every higher degree. This gives the second formula.
3. The cones over the flat family of step 1 do not form a flat family.
If \(R_t=S_t/I_t\) is the homogeneous coordinate ring of \(X_t\), then the homogeneous coordinate ring of the projective cone is \[ R_t[u], \] where \(u\) is the new degree-one cone coordinate. Hence its degree-\(d\) Hilbert function is \[ H_{C(X_t)}(d)=\sum_{j=0}^d h_t(j). \] By step 2, for \(t\ne0\) and \(d\ge1\) this equals \[ 1+3d, \] so the cone has Hilbert polynomial \(3d+1\). For \(t=0\) and \(d\ge2\) it equals \[ 1+2+3(d-1)=3d, \] so the special cone has Hilbert polynomial \(3d\). The Hilbert polynomials are different. By Theorem III.9.9, fibres of a flat projective family over an integral base have constant Hilbert polynomial. Therefore the cones cannot form a flat family. This proves part (a).
4. Every very flat family is flat.
Very flatness says that for every \(d\ge0\) the integer \[ h_t(d)=\dim_{k(t)}(S_t/I_t)_d \] is independent of \(t\). For \(d\gg0\), this Hilbert function agrees with the Hilbert polynomial \(P_t(d)\) of the fibre. Thus all fibres have the same Hilbert polynomial. Theorem III.9.9 then implies that the closed subscheme \[ X\subseteq\PP_T^n \] is flat over the integral noetherian base \(T\).
5. If \(\{X_t\}\) is very flat, then the family of cones \(\{C(X_t)\}\) is very flat.
For each fibre, the cone coordinate ring is \[ (S_t/I_t)[u]. \] Therefore \[ \dim_{k(t)}\bigl((S_t/I_t)[u]\bigr)_d =\sum_{j=0}^d h_t(j). \] Every summand on the right is independent of \(t\) by very flatness. Hence the degree-\(d\) dimension of the cone coordinate ring is independent of \(t\) for every \(d\ge0\). Thus the cone family is very flat. By step 4 it is therefore flat as well. This proves part (c).
6. Let \(\eta\) be the generic point of the nonsingular parameter curve \(T\) in part (d), and let \(t\in T\) be closed. The algebraic family is a flat family of schemes near \(t\).
Every fibre \(X_{(s)}\) is projectively normal, hence normal. Theorem III.9.11 says that an algebraic family of normal varieties parametrized by a nonsingular curve is flat as a family of schemes. Thus, after replacing \(T\) by an affine neighborhood of \(t\), we have a flat closed family \[ \mathcal X\subseteq\PP_T^n. \] Since the assertion is local on the parameter curve, we may further replace the base by \[ \Spec R, \qquad R=\OO_{T,t}, \] a discrete valuation ring with fraction field \(K=k(\eta)\) and residue field \(k(t)\).
7. For each \(d\ge0\), one has\[ h_t(d)\le h_\eta(d). \]
Let \(\mathcal I\) be the ideal sheaf of \[ \mathcal X\subseteq\PP_R^n \] and put \[ M_d=H^0(\PP_R^n,\mathcal I(d)). \] This is a finite \(R\)-module, being the kernel of a map between finite modules of global sections on a projective \(R\)-scheme. It sits inside \[ H^0(\PP_R^n,\OO(d))\cong R^{N_d}, \qquad N_d=\binom{n+d}{n}. \]
The quotient \[ R^{N_d}/M_d \] injects into \[ H^0(\mathcal X,\OO_{\mathcal X}(d)). \] Because \(\mathcal X\) is flat over the DVR \(R\), the invertible sheaf \(\OO_{\mathcal X}(d)\) is \(R\)-flat and therefore torsion-free; its module of global sections is torsion-free as well. Thus \(R^{N_d}/M_d\) is torsion-free, so \(M_d\) is a saturated submodule of the free module \(R^{N_d}\). Consequently reduction modulo the uniformizer gives an injection \[ M_d\otimes_R k(t) \hookrightarrow H^0(\PP_{k(t)}^n,\OO(d)). \] Its image consists of equations vanishing on the special fibre, hence lies in \((I_t)_d\).
After tensoring with the fraction field \(K\), cohomology commutes with this flat base change, so \[ M_d\otimes_R K=(I_\eta)_d. \] Since \(M_d\) is finite torsion-free over the DVR, it is free; therefore \[ \dim_{k(t)}(M_d\otimes k(t)) =\dim_K(I_\eta)_d. \] We obtain \[ \dim_K(I_\eta)_d\le\dim_{k(t)}(I_t)_d. \] Subtracting from \(N_d\) gives \[ h_t(d)\le h_\eta(d). \]
8. Projective normality and upper semicontinuity give the reverse inequality\[ h_t(d)\ge h_\eta(d). \]
For a projectively normal variety \(X_s\subseteq\PP_{k(s)}^n\), the restriction map \[ H^0(\PP_{k(s)}^n,\OO(d)) \longrightarrow H^0(X_s,\OO_{X_s}(d)) \] is surjective for every \(d\ge0\). Equivalently, \[ h_s(d)=\dim_{k(s)}H^0(X_s,\OO_{X_s}(d)). \]
For the proper flat family \(\mathcal X\to\Spec R\) and the coherent sheaf \(\OO_{\mathcal X}(d)\), upper semicontinuity of fibre cohomology gives \[ \dim_{k(t)}H^0(X_t,\OO(d)) \ge \dim_KH^0(X_\eta,\OO(d)). \] Using projective normality on both fibres gives \[ h_t(d)\ge h_\eta(d). \]
9. The algebraic family of projectively normal varieties in part (d) is very flat.
Steps <1>7–<1>8 give \[ h_t(d)=h_\eta(d) \] for every closed point \(t\in T\) and every \(d\ge0\). Thus, for each fixed degree \(d\), \(h_t(d)\) is independent of \(t\). This is exactly very flatness.
10. Q.E.D.
Steps <1>1–<1>3 prove part (a), steps <1>4–<1>5 prove part (c), and steps <1>6–<1>9 prove part (d). Part (b) is the definition used in those arguments.