Let \(k\) be an algebraically closed field of characteristic \(\neq 2\). Let \(C \subseteq \PP^2_k\) be the nodal cubic curve \(y^2 z = x^3 + x^2 z\). If \(P_0 = (0, 0, 1)\) is the singular point, then \(C - P_0\) is isomorphic to the multiplicative group \(\GG_m = \Spec k[t, \inverseof{t}]\) (Ex. 6.7). For each \(a \in k\), \(a \neq 0\), consider the translation of \(\GG_m\) given by \(t \mapsto at\). This induces an automorphism of \(C\) which we denote \(\varphi_a\). Now consider \(C \times (\PP^1 - \ts{0})\) and \(C \times (\PP^1 - \ts{\infty})\). We glue their open subsets \(C \times (\PP^1 - \ts{0, \infty})\) by the isomorphism \[ \varphi: \gens{P, u} \mapsto \gens{\varphi_u(P), u}, \qquad P \in C,\ u \in \GG_m = \PP^1 - \ts{0, \infty} . \] Thus we obtain a scheme \(X\), which is our example. The projections to the second factor are compatible with \(\varphi\), so there is a natural morphism \(\pi: X \to \PP^1\).
Show that \(\pi\) is a proper morphism, and hence that \(X\) is a complete variety over \(k\).
Use the method of (Ex. 6.9) to show that \[ \Pic(C \times \AA^1) \cong \GG_m \times \ZZ \quad\text{and}\quad \Pic(C \times (\AA^1 - \ts{0})) \cong \GG_m \times \ZZ \times \ZZ . \]
Now show that the restriction map \(\Pic(C \times \AA^1) \to \Pic(C \times (\AA^1 - \ts{0}))\) is of the form \(\gens{t, n} \mapsto \gens{t, 0, n}\), and that the automorphism \(\varphi\) of \(C \times (\AA^1 - \ts{0})\) induces a map of the form \(\gens{t, d, n} \mapsto \gens{t, d + n, n}\) on its Picard group.
Conclude that the image of the restriction map \(\Pic X \to \Pic(C \times \ts{0})\) consists entirely of divisors of degree \(0\) on \(C\). Hence \(X\) is not projective over \(k\), and \(\pi\) is not a projective morphism.
Hint
For (b), if \(A\) is a domain and \(*\) denotes the group of units, then \((A[u])^*\cong A^*\) and \((A[u,u^{-1}])^*\cong A^*\times\ZZ\).
Solution
The symbol \(\GG_m\) in these Picard-group formulas denotes the group \(k^\times\). Let \(V_0=\Spec k[u]\) and \(V_\infty=\Spec k[u^{-1}]\) be the two standard opens of the base \(\PP^1\). Write \(X_0=C\times V_0\) and \(X_\infty=C\times V_\infty\).
1. The normalization of \(C\) is \(\PP_t^1\), with the two points \(0,\infty\) identified at the node, and the action \(t\mapsto ut\) descends to the asserted overlap automorphism.
The normalization calculation uses \(v=y/x\), with \(x=v^2-1\) and \(y=v(v^2-1)\) on \(z=1\). The two branches are \(v=1,-1\). The fractional linear coordinate \[ t=\frac{v-1}{v+1} \] carries these branches to \(0,\infty\), respectively, and identifies the smooth locus of \(C\) with \(\GG_m\).
For \(R=k[u]\) or \(k[u,u^{-1}]\), put \(C_R=C\times\Spec R\) and let \(\nu_R:\PP_R^1\to C_R\) be the base-changed normalization. On the nodal affine chart the coordinate rings are \[ A=R+(v^2-1)R[v]\subseteq B=R[v]. \] Indeed, the polynomial identity proved in P-AGH267NODALCUBIC, step 2, remains true with coefficients in \(R\). Writing \(J=(v^2-1)B\), one obtains \[ A/J=R,\qquad B/J=R\oplus R, \qquad A=B\times_{R\oplus R}R, \] where the last map is the diagonal and the map from \(B\) is evaluation at the two branches. Away from the node the normalization is an isomorphism. Thus the structure sheaf of \(C_R\) consists of functions on its normalization whose values on the two branch sections agree.
The \(R\)-automorphism \(t\mapsto ut\) for \(R=k[u,u^{-1}]\) fixes the two branch sections and preserves this equality of values. It therefore descends to \(C_R\): the continuous map descends through the finite closed surjection identifying the branches, and its pullback on functions preserves the displayed subalgebra. The same construction with \(u^{-1}\) is its inverse. This gives the regular overlap automorphism, not only separate automorphisms of its closed fibres.
2. The morphism \(\pi:X\to\PP^1\) is proper, and \(X\) is a complete integral variety.
Over each of \(V_0,V_\infty\), the morphism is the projection \(C\times V_i\to V_i\). It is a base change of the projective curve \(C\to\Spec k\), so is proper. Properness is local on the target, hence \(\pi\) is proper [@Har10a, Chapter II, §4]. Since \(\PP^1\) is proper over \(k\), their composite is proper.
Each \(X_i\) is integral: an affine coordinate ring of \(C\) is a domain, and adjoining one polynomial variable preserves this property. The two opens have a nonempty overlap, so their union is irreducible and reduced. They are of finite type over \(k\), and properness gives separatedness. Thus \(X\) is an integral variety proper over \(k\), which is completeness.
3. For \(R=k[u]\) or \(k[u,u^{-1}]\), an invertible sheaf on \(C_R\) is equivalent to an invertible sheaf \(L\) on \(\PP_R^1\) together with an isomorphism\[ \theta:L|_{0_R}\xrightarrow{\cong}L|_{\infty_R}. \]
Pulling back an invertible sheaf \(M\) from \(C_R\) gives \(L=\nu_R^*M\). The two restrictions are canonically identified with the restriction of \(M\) to the nodal section, giving \(\theta\). Conversely, from \((L,\theta)\) form the subsheaf of \(\nu_{R*}L\) consisting of sections whose two branch values match under \(\theta\). It is coherent, being the kernel of the branch-difference map between coherent sheaves for the finite morphism \(\nu_R\). It agrees with \(L\) away from the nodal section.
1. It is consequently free of rank one, and its pullback is the original \(L\).
Starting with \(M\), the matching construction recovers \(M\) by tensoring the equalizer description of \(\OO_{C_R}\) with this locally free sheaf. Starting with \((L,\theta)\), the local generator \(h\) shows that both \(L\) and the matching isomorphism are recovered. The same equalizer description identifies the morphisms between these objects. This proves the claimed equivalence, including isomorphism classes.
4. There is a group isomorphism\[ \Pic(C_R)\cong R^\times\times\ZZ. \]The integer is the degree on a normalized fibre.
Both rings \(R\) are PIDs, so \(\Pic(\Spec R)=0\). The projective-bundle Picard calculation gives \(\Pic(\PP_R^1)=\ZZ\), generated by \(\OO(1)\). Thus the normalization sheaf in step 3 is uniquely \(\OO(n)\) up to isomorphism. In homogeneous coordinates \([T_0:T_1]\) with \(t=T_1/T_0\), use the frames \[ e_0=T_0^n\text{ at }0_R,\qquad e_\infty=T_1^n\text{ at }\infty_R. \] For negative \(n\) these are the corresponding dual tensor-power frames. The matching isomorphism is uniquely of the form \(\theta(e_0)=\lambda e_\infty\) for \(\lambda\in R^\times\).
3. Tensor products multiply \(\lambda\) and add \(n\), giving the group isomorphism.
A unit of \(k[u]\) is a nonzero constant, since degrees add under multiplication. A unit of \(k[u,u^{-1}]\) is uniquely \(cu^d\) with \(c\in k^\times\) and \(d\in\ZZ\): the largest and smallest exponents both add under multiplication, so an invertible Laurent polynomial has only one term. Hence (b) gives \[ \boxed{\Pic(C\times\AA^1)\cong k^\times\times\ZZ,\qquad \Pic(C\times\GG_m)\cong k^\times\times\ZZ\times\ZZ.} \] On the latter group use coordinates \((c,d,n)\) corresponding to \((cu^d,n)\).
5. In these coordinates, restriction is \((c,n)\mapsto(c,0,n)\) and pullback by \(\varphi\) is \((c,d,n)\mapsto(c,d+n,n)\).
Restriction from \(k[u]\) to \(k[u,u^{-1}]\) preserves \(\OO(n)\) and its chosen branch frames and sends the constant matching unit \(c\) to the same constant. This proves the restriction formula; the chart at infinity, with coordinate \(u^{-1}\), has the same image.
On the normalization, \(\varphi\) is \[ \tau:[T_0:T_1]\longmapsto[T_0:uT_1]. \] The induced isomorphism \(\tau^*\OO(n)\cong\OO(n)\) sends the pulled-back frame at zero to \(e_0\), and the pulled-back frame at infinity to \(u^n e_\infty\). Thus pulling back the matching map \(e_0\mapsto\lambda e_\infty\) and making this identification changes its coefficient to \(u^n\lambda\). For \(\lambda=cu^d\), this is \(cu^{d+n}\). The degree remains \(n\), proving the stated pullback formula and its sign.
6. Every invertible sheaf on \(X\) has degree zero on \(C\times\{0\}\).
5. Equality of the middle coordinates gives \(n_0=0\), and equality of the last coordinates also gives \(n_\infty=0\).The restriction to the fibre over zero therefore has normalization degree zero.This is the degree on the nodal curve, as in P-AGH267NODALCUBIC, step 3, and P-AGH269SINGCURVEPIC, step
8.
7. Neither \(X\) nor the morphism \(\pi\) is projective.
If \(X\) had a projective embedding over \(k\), pulling back \(\OO(1)\) and restricting to the closed fibre over zero would give a very ample invertible sheaf on \(C\). A projective embedding of \(\pi\) would give the same conclusion after base change to zero. But step 6 says that either invertible sheaf would pull back to \(\OO_{\PP^1}(0)\) on the normalization of \(C\). The coordinate sections of such an embedding would then become global sections of \(\OO_{\PP^1}\), all of which are constants. They would define a constant morphism from \(\PP^1\), whereas the composite of the surjective normalization \(\PP^1\to C\) with an embedding of the curve \(C\) is nonconstant. This contradiction proves both nonprojectivity assertions.
8. Q.E.D.
Step 2 proves (a), steps <1>3–<1>4 prove (b), step 5 proves (c), and steps <1>6–<1>7 prove (d).