* Let \(X\) be an irreducible nonsingular curve in \(\PP^3\). Then for each \(m \gg 0\), there is a nonsingular surface \(F\) of degree \(m\) containing \(X\).
Hint: Let \(\pi: \tilde{\PP} \to \PP^3\) be the blowing-up of \(X\) and let \(Y=\pi^{-1}(X)\). Apply Bertini’s theorem to the projective embedding of \(\tilde{\PP}\) corresponding to \(\mci_Y \tensor \pi^* \OO_{\PP^3}(m)\).
Solution
Let \[ \pi:\widetilde{\PP^3}=\Bl_X\PP^3\longrightarrow\PP^3 \] be the blowup of \(X\), and write \[ E=\pi^{-1}(X) \] for its exceptional divisor.
1. The threefold \(\widetilde{\PP^3}\) is nonsingular, and\[ E\cong\PP_X(\mci_X/\mci_X^2). \]For every \(P\in X\), the fibre\[ E_P=\pi^{-1}(P) \]is a projective line and\[ \OO_{\widetilde{\PP^3}}(-E)|_{E_P}\cong\OO_{\PP^1}(1). \]
The center \(X\) is nonsingular of codimension \(2\) in the nonsingular threefold \(\PP^3\). The blowup theorem for a nonsingular center, used in Exercise II.8.5, gives \[ \widetilde{\PP^3}\text{ nonsingular}, \qquad E\cong\PP_X(\mci_X/\mci_X^2), \] and \[ \OO_{\widetilde{\PP^3}}(-E)|_E\cong\OO_E(1). \] Since the conormal bundle \(\mci_X/\mci_X^2\) has rank \(2\), every fibre of \(E\to X\) is \(\PP^1\), and restriction of the last identity gives the claimed \(\OO_{\PP^1}(1)\).
2. For every sufficiently large integer \(m\), the invertible sheaf\[ L_m = \OO_{\widetilde{\PP^3}}(-E) \tensor \pi^*\OO_{\PP^3}(m) \]is very ample.
By the Rees-algebra construction, \[ \widetilde{\PP^3} = \Proj_{\PP^3}\bigoplus_{q\ge0}\mci_X^q, \] and its relative twisting sheaf is \[ \OO_{\widetilde{\PP^3}}(1) \cong \mci_X\OO_{\widetilde{\PP^3}} \cong \OO_{\widetilde{\PP^3}}(-E). \] Apply Exercise II.7.14(b) to this relative \(\Proj\), with the ample sheaf \(\OO_{\PP^3}(1)\) on the base. It says that \[ \OO_{\widetilde{\PP^3}}(1) \tensor \pi^*\OO_{\PP^3}(m) \] is very ample for every \(m\gg0\). This is exactly \(L_m\).
3. Fix such an \(m\) and let\[ \iota_m:\widetilde{\PP^3}\hookrightarrow\PP^N \]be the embedding defined by \(L_m\).Each exceptional fibre \(E_P\) is mapped isomorphically to a line in \(\PP^N\).
On the fibre \(E_P\), the pullback \[ \pi^*\OO_{\PP^3}(m)|_{E_P} \] is trivial. Step 1 therefore gives \[ L_m|_{E_P}\cong\OO_{\PP^1}(1). \] The restriction of a projective embedding remains a closed immersion. Its image therefore has degree \[ \deg L_m|_{E_P}=1, \] so it is a line in \(\PP^N\).
4. The hyperplanes in \(\PP^N\) containing at least one exceptional fibre form a proper closed subset of \(\dualof{(\PP^N)}\).
Consider the incidence variety \[ \mathcal B = \{(P,H)\in X\dualof{\times(\PP^N)}: \iota_m(E_P)\subseteq H\}. \] By step 3, \(\iota_m(E_P)\) is a line. Hyperplanes containing a fixed line form a projective subspace of codimension \(2\) in \(\dualof{(\PP^N)}\). Consequently \[ \dim\mathcal B = 1+(N-2) = N-1. \] The projection \[ \mathcal B\dualof{\longrightarrow(\PP^N)} \] is proper because \(X\) is projective, so its image is closed. Its dimension is at most \(N-1\), strictly smaller than the dimension \(N\) of the dual projective space. Hence the image is proper.
5. There is a hyperplane \(H\subseteq\PP^N\) such that\[ \widetilde F = \iota_m^{-1}(H) \]is nonsingular and \(H\) contains no exceptional fibre.
By step 1, \(\widetilde{\PP^3}\) is nonsingular. Bertini’s hyperplane theorem T-BERTINI says that the hyperplanes whose inverse image under \(\iota_m\) is nonsingular form a dense open subset of \(\dualof{(\PP^N)}\).
Step 4 says that the hyperplanes containing an exceptional fibre form a proper closed subset. The complement of that subset is therefore another nonempty open subset. Since \(\dualof{(\PP^N)}\) is irreducible, the two open sets meet. Choose \(H\) in their intersection.
6. The divisor \(\widetilde F\) of step 5 is the strict transform of a degree-\(m\) surface\[ F\subseteq\PP^3 \]containing \(X\).
The line bundle of \(\widetilde F\) is \[ L_m = \OO_{\widetilde{\PP^3}}(-E) \tensor \pi^*\OO_{\PP^3}(m). \] For a blowup, one has \[ \pi_*\OO_{\widetilde{\PP^3}}(-E)=\mci_X. \] The projection formula therefore identifies \[ H^0(\widetilde{\PP^3},L_m) \cong H^0(\PP^3,\mci_X(m)). \] Thus the section cutting out \(\widetilde F\) corresponds to a degree-\(m\) homogeneous form vanishing on \(X\). Let \(F\) be its zero surface in \(\PP^3\). Then \[ X\subseteq F, \] and away from \(E\) the blowup is an isomorphism. Step 5 says that \(\widetilde F\) contains no exceptional fibre, so in particular it has no component contained in \(E\). It is therefore the closure of \[ \pi^{-1}(F\setminus X). \] Hence it is the strict transform of \(F\).
7. The surface \(F\) is nonsingular away from \(X\).
The morphism \[ \pi:\widetilde{\PP^3}\setminus E \xrightarrow{\sim} \PP^3\setminus X \] is an isomorphism. By step 6 it identifies \[ \widetilde F\setminus E \] with \[ F\setminus X. \] Step 5 says that \(\widetilde F\) is nonsingular, so the open subset \(F\setminus X\) is nonsingular.
8. The surface \(F\) is nonsingular at every point of \(X\).
Fix a point \(P\in X\). Because \(X\) is a nonsingular codimension-two subvariety of the nonsingular threefold \(\PP^3\), choose regular local parameters \[ x,y,z \] at \(P\) such that \[ \mci_{X,P}=(x,y). \] Let \(f\) be a local equation for \(F\). Since \(X\subseteq F\), write \[ f=ax+by \] for some \[ a,b\in\OO_{\PP^3,P}. \]
On the blowup of the ideal \((x,y)\), the exceptional fibre over \(P\) is the projective line of normal directions \([u:v]\). The strict transform of \(F\) meets this fibre in the zero locus of the linear form \[ a(P)u+b(P)v. \] If \[ a(P)=b(P)=0, \] that linear form vanishes identically, so the whole exceptional fibre \(E_P\) is contained in \(\widetilde F\). Step 5 excludes this. Therefore \[ (a(P),b(P))\ne(0,0). \]
Modulo the maximal ideal at \(P\), differentiation of \[ f=ax+by \] gives \[ df(P)=a(P)\,dx+b(P)\,dy\ne0. \] By the Jacobian criterion, the hypersurface \(F\) is nonsingular at \(P\). Since \(P\) was arbitrary, \(F\) is nonsingular along all of \(X\).
9. For every sufficiently large \(m\), there is a nonsingular degree-\(m\) surface containing \(X\).
Step 2 gives one threshold beyond which \(L_m\) is very ample for every \(m\). For each such \(m\), steps <1>3–<1>6 construct a degree-\(m\) surface \(F\) containing \(X\). Steps <1>7–<1>8 prove that \(F\) is nonsingular both away from and along \(X\). Therefore \[ \boxed{ \text{for every }m\gg0\text{ there is a nonsingular degree-}m \text{ surface }F\supseteq X. } \]
10. Q.E.D.
Step 9 is exactly the required assertion.