Show that \(\Pic(\PP^1_\CC) \cong \ZZ\).
Compute \(H^0(X, \mcf)\) for \(X = \PP^1\) and \(\mcf = \OO_X\) or \(\OO_X(m)\); for \(X = \PP^n\) and \(\mcf = \OO_X(m)\) or \(\Omega_{X/k}\); for the same sheaves on \(X = \AA^n\); and for \(X\) a toric variety and \(\mcf = \OO(D)\) with \(D\) a torus-invariant divisor.
Describe the morphisms \(\PP^n \to \AA^m\).
For each of the following, describe its consequences and give an example and a counterexample: separated varieties and morphisms, proper varieties and morphisms, complete varieties, flat morphisms, reduced schemes, normal varieties.
Show that \(\Bl_0 \AA^{n+1}\) is the total space of \(\OO_{\PP^n}(-1)\).
Show that \(\omega_{\PP^1} = \OO_{\PP^1}(-2)\).
Define the tautological bundle on \(\PP^n\), and find the degree of the canonical line bundle on \(\PP^1\).
Show that \(\CaCl(X) \cong H^1(X, \OO_X^*)\) for an integral scheme \(X\).
Produce a Weil divisor that is not Cartier.
How are line bundles related to Cartier divisors?
Part (g) is stated in the source for \(\CC\PP^n\) with the answer \(-2\), which is the degree only for \(n = 1\). Part (h) is stated in the source as \(\operatorname{CaDiv}(X) \cong H^1(X, \OO_X^*)\); it is the group of Cartier divisor classes, not of Cartier divisors, that is isomorphic to \(\Pic(X) \cong H^1(X, \OO_X^*)\), and the isomorphism with \(\Pic\) uses integrality.
Solution
1. (a) The degree map gives\[ \boxed{\Pic(\PP^1_{\CC})\cong\ZZ}, \]with \(1\in\ZZ\) corresponding to \(\OO_{\PP^1}(1)\).
The curve \(\PP^1_\CC\) is smooth, so Cartier divisors, Weil divisors, and line bundles have the same divisor-class group. Fix the point \(\infty=[1:0]\) and the affine coordinate \(t\) on \[ \PP^1\sm\{\infty\}\cong\AA^1. \] Let \[ D=\sum_{p\in\PP^1}n_p[p] \] be a divisor and put \[ d=\deg D=\sum_p n_p. \] For each finite point \(p=a\in\CC\), the rational function \(t-a\) has divisor \[ \div(t-a)=[a]-[\infty]. \] Therefore \[ D-d[\infty] = \sum_{a\in\CC}n_a\bigl([a]-[\infty]\bigr) \] is principal. Hence every divisor class is represented by \[ d[\infty]. \] Degree vanishes on principal divisors, so two such representatives are linearly equivalent only when their integers agree. Thus degree is an isomorphism \[ \CaCl(\PP^1_\CC)\xrightarrow{\sim}\ZZ. \] Under the divisor-line-bundle correspondence PR-DIVLB, \[ \OO_{\PP^1}([\infty])\cong\OO_{\PP^1}(1), \] which gives the claimed identification with \(\Pic(\PP^1_\CC)\).
2. (b) On projective space,\[ \boxed{ H^0(\PP_k^n,\OO(m)) = \begin{cases} k[x_0,\ldots,x_n]_m,&m\ge0,\\ 0,&m<0, \end{cases}} \]and, for \(n\ge1\),\[ \boxed{H^0(\PP_k^n,\Omega^1_{\PP^n/k})=0}. \]
The twisting-sheaf calculation is T-IJW1K: \[ H^0(\PP_k^n,\OO(m))=k[x_0,\ldots,x_n]_m \] for \(m\ge0\) and is zero for \(m<0\). In particular, \[ H^0(\PP^1,\OO)=k, \] and \[ \dim_k H^0(\PP^n,\OO(m)) = \binom{n+m}{n} \] when \(m\ge0\).
The Euler sequence T-MODEULER \[ 0\longrightarrow\Omega^1_{\PP^n/k} \longrightarrow \OO(-1)^{\oplus(n+1)} \longrightarrow \OO \longrightarrow0 \] gives an injection \[ H^0(\PP^n,\Omega^1_{\PP^n/k}) \injects H^0(\PP^n,\OO(-1))^{\oplus(n+1)} =0. \] Thus the differential sheaf has no nonzero global sections.
3. (b) On affine space,\[ \boxed{H^0(\AA_k^n,\OO)=k[x_1,\ldots,x_n]} \]and\[ \boxed{ H^0(\AA_k^n,\Omega^1_{\AA^n/k}) = \bigoplus_{i=1}^n k[x_1,\ldots,x_n]\,dx_i. } \]If \(\OO(m)\) means the restriction of the projective twisting sheaf to thestandard affine chart \(\AA^n\subseteq\PP^n\), then it is trivial there andhas the same global sections as \(\OO\).
For \[ A=k[x_1,\ldots,x_n], \] the affine scheme \(\AA^n=\Spec A\) satisfies \[ \Gamma(\AA^n,\OO)=A. \] Its module of Kähler differentials is the free \(A\)-module \[ \Omega^1_{A/k} = \bigoplus_{i=1}^n A\,dx_i, \] and the associated quasicoherent sheaf has these same global sections.
On the standard chart \(D_+(x_0)\cong\AA^n\), the projective sheaf \(\OO(m)\) is trivialized by \(x_0^m\), so its restriction has global sections \(A\).
4. (b) For a toric variety \(X_\Sigma\) and a torus-invariant divisor\[ D=\sum_{\rho\in\Sigma(1)}a_\rho D_\rho, \]one has\[ \boxed{ H^0(X_\Sigma,\OO(D)) = \bigoplus_{m\in P_D\cap M}k\chi^m, } \]where\[ P_D = \left\{ m\in M_\RR: \langle m,u_\rho\rangle\ge-a_\rho \text{ for every }\rho \right\}. \]
The character \(\chi^m\) is a section of \(\OO(D)\) exactly when \[ \div(\chi^m)+D\ge0. \] Since the coefficient of \(D_\rho\) in \(\div(\chi^m)\) is \[ \langle m,u_\rho\rangle, \] this is equivalent to the inequalities defining \(P_D\). Distinct characters are linearly independent on the dense torus, and every section decomposes into torus weights. This is the standard toric section formula D-TORQD.
5. (c) Every morphism\[ \PP_k^n\longrightarrow\AA_k^m \]is constant.
A morphism to \[ \AA^m=\Spec k[t_1,\ldots,t_m] \] is determined by the pullbacks of the coordinate functions \(t_i\), hence by \(m\) global regular functions on \(\PP^n\). Step 2 with \(m=0\) gives \[ \Gamma(\PP^n,\OO)=k. \] Therefore every coordinate pullback is constant, so the morphism is constant.
6. (d) The requested properties, consequences, examples, andnonexamples are as follows.
6.1. Separatedness is the closed-diagonal condition.
A morphism \(f:X\to Y\) is separated if \[ \Delta_f:X\longrightarrow X\times_YX \] is a closed immersion. Consequently, if \(Y\) is separated over the base, the graph of every morphism \(X\to Y\) is closed. Every affine scheme over a field is separated. The affine line with doubled origin is not separated: the pair \((0_1,0_2)\) of its two origins lies in the closure of the diagonal but not in the diagonal.
6.2. Properness is finite type, separatedness, and universal closedness.
A proper morphism remains closed after every base change; properness is also stable under composition and base change. Every projective morphism is proper. The structure morphism \[ \AA^1_k\longrightarrow\Spec k \] is not proper. Indeed, the morphism \[ \Spec k((t))\longrightarrow\AA^1, \qquad x\longmapsto t^{-1}, \] does not extend to \(\Spec k[[t]]\), because \(t^{-1}\notin k[[t]]\). This violates the valuative criterion for properness.
6.3. A complete variety is a variety proper over its base field.
If \(X\) is complete and \(Y\) is a separated variety, every morphism \(f:X\to Y\) is proper: its graph is a closed immersion into \(X\times Y\), and the projection \(X\times Y\to Y\) is the base change of the proper structure morphism \(X\to\Spec k\). Hence \(f\) has closed image. Projective varieties are complete; in particular \(\PP^n\) is complete. The affine line \(\AA^1\) is not complete, as in step 2.
6.4. Flatness is flatness of the local rings over the base.
A morphism \(f:X\to Y\) is flat at \(x\) when \[ \OO_{X,x} \] is a flat \(\OO_{Y,f(x)}\)-module D-MORFLAT. Consequently, for \(f\) flat the pullback \(f^*\) is exact on quasi-coherent sheaves; for flat projective families over a connected Noetherian base, the Hilbert polynomial of the fibres is constant. A projection \[ Y\times\AA^r\longrightarrow Y \] is flat. The blowup \[ \Bl_0\AA^2\longrightarrow\AA^2 \] is not flat: its fibres away from the origin are points while the fibre over the origin is \(\PP^1\) FE-MORNOTFLAT.
6.5. Reducedness means that no local ring has a nonzero nilpotent.
Equivalently, an affine scheme \(\Spec A\) is reduced exactly when the nilradical of \(A\) is zero. Affine space is reduced because its polynomial ring is a domain. The scheme \[ \Spec k[\varepsilon]/(\varepsilon^2) \] is not reduced because the nonzero class of \(\varepsilon\) is nilpotent. Passing from any scheme to its reduced subscheme removes precisely these nilpotents without changing the underlying topological space.
6.6. Normality means that the local rings are integrally closed domains.
Every local ring of a normal scheme is an integrally closed domain D-QJ5M9. Regular schemes are normal. Thus \(\AA^n\) is normal. The cuspidal cubic \[ \Spec k[t^2,t^3] \] is not normal because \(t\) is integral over \(k[t^2,t^3]\) but does not belong to it; its normalization is \(\Spec k[t]\). In dimension one, normality forces the local rings to be DVRs, hence regularity.
6.7. Q.E.D.
Steps <2>1–<2>6 give a definition, a standard consequence, an example, and a nonexample for each property requested in part (d).
7. (e) There is an isomorphism over \(\AA^{n+1}\)\[ \boxed{ \Bl_0\AA^{n+1} \cong \operatorname{Tot}\bigl(\OO_{\PP^n}(-1)\bigr). } \]
The blowup of the origin is the incidence variety \[ B = \left\{ (x,[u])\in\AA^{n+1}\times\PP^n: x_i u_j=x_j u_i \text{ for all }i,j \right\}. \] For a point \([u]\in\PP^n\), these equations say exactly that the vector \(x\in k^{n+1}\) lies on the line represented by \([u]\).
But the tautological line bundle \(\OO_{\PP^n}(-1)\) has fibre over \([u]\) equal to that very line: \[ \operatorname{Tot}(\OO(-1)) = \{([u],x):x\in k u\} \subseteq \PP^n\times\AA^{n+1}. \] Thus the two incidence subschemes have the same defining equations. On the chart \(u_i\ne0\), normalize \(u_i=1\) and put \[ v_j=u_j/u_i \] and \(\lambda=x_i\); then the equations become \[ x_j=\lambda v_j, \] which identifies both schemes with \[ \Spec k[v_0,\ldots,\widehat{v_i},\ldots,v_n,\lambda]. \] These chart identifications have the transition functions of \(\OO(-1)\), so they glue to the asserted isomorphism. Projection to \(x\) is the blowdown map.
8. (f) The canonical line bundle of the projective line is\[ \boxed{\omega_{\PP^1}\cong\OO_{\PP^1}(-2)}. \]
The Euler sequence on \(\PP^1\) is \[ 0 \longrightarrow \Omega^1_{\PP^1/k} \longrightarrow \OO(-1)^{\oplus2} \longrightarrow \OO \longrightarrow0. \] Taking determinants gives \[ \det\Omega^1_{\PP^1/k} \cong \det(\OO(-1)^{\oplus2}) \cong \OO(-2). \] Since \(\PP^1\) has dimension one, \[ \omega_{\PP^1}=\Omega^1_{\PP^1/k}. \] This is also the \(n=1\) case of T-MODEULER.
9. (g) The tautological line bundle on \(\PP^n\) is \(\OO_{\PP^n}(-1)\),whose fibre at a point \([\ell]\in\PP^n\) is the line\[ \ell\subseteq k^{n+1}. \]Moreover,\[ \boxed{\deg\omega_{\PP^1}=-2}. \]
The fibre description is the definition of the tautological bundle D-CB9XS. By step 8, \[ \omega_{\PP^1}\cong\OO_{\PP^1}(-2). \] Since \[ \deg\OO_{\PP^1}(d)=d, \] its degree is \(-2\).
10. (h) For an integral scheme \(X\),\[ \boxed{ \CaCl(X) \cong \Pic(X) \cong H^1(X,\OO_X^\times). } \]
Let \(\mck\) be the constant sheaf of the function field \(K(X)\). A Cartier divisor is represented by local rational functions \(f_i\in K(X)^\times\) whose ratios \[ f_i/f_j \] are units on overlaps. These ratios are transition functions for the invertible sheaf \(\OO_X(D)\). A principal divisor gives a trivial line bundle, so \[ [D]\longmapsto[\OO_X(D)] \] defines an injection \[ \CaCl(X)\injects\Pic(X). \]
Conversely, if \(\mcl\) is an invertible sheaf, its generic fibre is a one-dimensional \(K(X)\)-vector space. Choose a nonzero rational section \(s\). On a trivializing cover, write \[ s=f_i e_i \] with \(f_i\in K(X)^\times\). On overlaps \(f_i/f_j\) is a unit, so the \(f_i\) define a Cartier divisor whose associated line bundle is \(\mcl\). Thus \(\CaCl(X)\cong\Pic(X)\); this is PR-DIVLB.
Finally, a line bundle trivialized on an open cover has transition functions \[ g_{ij}\in\OO_X^\times(U_i\cap U_j) \] satisfying the cocycle identity. Changing trivializations changes the cocycle by a coboundary, and every such cocycle glues trivial line bundles to an invertible sheaf. Hence \[ \Pic(X)\cong H^1(X,\OO_X^\times), \] as proved in P-AGH345PICH1.
11. (i) The ruling on the quadric cone gives a Weil divisor that isnot Cartier.
Let \[ A=k[x,y,z]/(xy-z^2), \qquad X=\Spec A, \] and let \[ D=V(x,z). \] The ring \(A\) is a normal two-dimensional domain D-5PQ5W. The quotient \[ A/(x,z)\cong k[y] \] is a domain of dimension one, so \((x,z)\) is a height-one prime. Hence \(D\) is a prime Weil divisor.
At the vertex \[ \mfm=(x,y,z), \] the ideal of \(D\) is \[ I=(x,z)A_\mfm. \] The relation \(xy-z^2\) has degree two, so the classes of \(x,y,z\) are linearly independent in \[ \mfm A_\mfm/\mfm^2A_\mfm. \] The image of \(I\) in this vector space therefore has dimension two, spanned by the classes of \(x\) and \(z\). A principal ideal contained in \(\mfm A_\mfm\) has image of dimension at most one. Hence \(I\) is not principal, so \(D\) is not Cartier at the vertex. This is the example in D-5PQ5W.
12. (j) On an integral scheme, Cartier divisors modulo principaldivisors are exactly line bundles up to isomorphism; more precisely aCartier divisor is equivalent to a line bundle equipped with a nonzerorational section.
For a Cartier divisor represented by rational functions \(f_i\), the transition functions \[ f_i/f_j\in\OO_X^\times \] glue the local trivial bundles to \(\OO_X(D)\). The rational functions \(f_i\) simultaneously determine a distinguished nonzero rational section.
Conversely, if \(\mcl\) is a line bundle with nonzero rational section \(s\), choose local frames \(e_i\) and write \[ s=f_i e_i. \] The ratios \(f_i/f_j\) are units, so the \(f_i\) define a Cartier divisor. Tensor product of line bundles corresponds to addition of divisors, duals correspond to negatives, and changing \(s\) by a global rational function changes the divisor by a principal divisor. This is precisely the correspondence PR-DIVLB.
13. Q.E.D.
Steps <1>1–<1>12 answer parts (a)–(j), with part (b) divided into the projective, affine, and toric cases and part (d) expanded into its six requested properties.