Picard Groups
Picard groups¶
Example: Why do we need the separateness condition? Picard group of an affine line with double origins¶
Let \(k\) be a field and let \(X\) be the affine line with double origins over \(k\), constructed by gluing two copies \(U_1 = \operatorname{Spec}(k[t])\) and \(U_2 = \operatorname{Spec}(k[t])\) along \(U_1 \setminus \{0\} \cong U_2 \setminus \{0\} = \operatorname{Spec}(k[t, t^{-1}])\) via the identity map. We compute
showing that the separatedness hypothesis is not needed for these identifications in this particular case.
Structural properties of \(X\). The scheme \(X\) is Noetherian, integral, and locally factorial: each local ring is either a field (at the generic point) or a DVR (at each closed point, including both origins). In particular, \(X\) is normal and regular in codimension 1. Since \(X\) is locally factorial, every Weil divisor is locally principal, so \(\operatorname{CaCl}(X) \cong \operatorname{Cl}(X)\). Since \(X\) is integral, the natural map \(\operatorname{CaCl}(X) \to \operatorname{Pic}(X)\) is an isomorphism: every line bundle embeds into the constant sheaf of rational functions, hence arises from a Cartier divisor.
Computation via Cech cohomology. The open cover \(\{U_1, U_2\}\) is acyclic for \(\mathcal{O}_X^*\) (both \(U_i\) are affine with \(\operatorname{Pic}(\mathbb{A}^1_k) = 0\) since \(k[t]\) is a PID). The Cech complex gives
Now \(\mathcal{O}_X^*(U_i) = k[t]^* = k^*\) and \(\mathcal{O}_X^*(U_1 \cap U_2) = k[t, t^{-1}]^* = \{ c \cdot t^n : c \in k^*,\, n \in \mathbb{Z}\} \cong k^* \times \mathbb{Z}\), where the isomorphism records the unit part and the power of \(t\). The image of the difference map is \(k^*\) (ratios of constants), so the quotient is \(\mathbb{Z}\), generated by the class of \(t\).
Remark: Relations between \(\operatorname{Cl}, \operatorname{CaCl}, \operatorname{Pic}\)¶
For a Noetherian integral scheme \(X\) that is regular in codimension 1, all three divisor class groups \(\operatorname{Cl}(X)\), \(\operatorname{CaCl}(X)\), and \(\operatorname{Pic}(X)\) are defined. The maps between them are as follows.
\(\operatorname{CaCl}(X) \to \operatorname{Cl}(X)\). This sends a Cartier divisor to its associated Weil divisor. It is always injective: if a Cartier divisor has trivial Weil divisor, then locally its defining equation has no zeros or poles, hence is a unit. This map is an isomorphism when \(X\) is factorial (every local ring is a UFD), because in a UFD every height-1 prime is principal, making every Weil divisor locally principal, hence Cartier.
\(\operatorname{CaCl}(X) \to \operatorname{Pic}(X)\). This sends the class of a Cartier divisor \(D\) to the line bundle \(\mathcal{O}_X(D)\). Injectivity holds without the Noetherian hypothesis. For surjectivity, one needs every line bundle \(\mathcal{L}\) to embed into the sheaf \(\mathcal{M}\) of total quotient rings. This holds whenever \(X\) is integral, since then every nonzero section is a non-zero-divisor in the function field, so the natural map \(\mathcal{L} \hookrightarrow \mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{M} \cong \mathcal{M}\) is injective on each stalk.
In summary: for \(X\) Noetherian, integral, and locally factorial, all three groups coincide. Dropping the locally factorial hypothesis, \(\operatorname{CaCl}(X)\) and \(\operatorname{Pic}(X)\) still agree (provided \(X\) is integral), but \(\operatorname{CaCl}(X) \to \operatorname{Cl}(X)\) may be a proper injection.
Example: Another way to compute the Picard group of an affine line with double origins¶
We compute \(\operatorname{Pic}(X) \cong \mathbb{Z}\) for the affine line with double origins \(X\) using the Mayer--Vietoris sequence, and exhibit a generator as an explicit line bundle.
The Mayer--Vietoris sequence. Write \(X = U_1 \cup U_2\) with \(U_i \cong \mathbb{A}^1_k\) and \(U_1 \cap U_2 \cong \mathbb{A}^1_k \setminus \{0\} = \operatorname{Spec}(k[t, t^{-1}])\). Since \(\operatorname{Pic}(\mathbb{A}^1_k) = 0\) (the polynomial ring is a PID), the exact sequence for \(\mathcal{O}_X^*\) on this cover gives
The map \(\delta\) sends \((a, b)\) to \(a \cdot b^{-1}\). We have \(\mathcal{O}_X^*(U_i) = k^*\) and \(\mathcal{O}_X^*(U_1 \cap U_2) = k[t, t^{-1}]^* = k^* \times t^{\mathbb{Z}}\), so \(\operatorname{im}(\delta) = k^*\) and
The generator. The class of \(t \in k[t, t^{-1}]^*\) corresponds to a line bundle \(\mathcal{L}\) that is trivial on each \(U_i\) with transition function \(t\) on \(U_1 \cap U_2\): sections \(s\) on \(U_1\) glue to \(t \cdot s\) on \(U_2\). This bundle is nontrivial because \(t\) is not a unit in either copy of \(k[t]\), so \(\mathcal{L}\) is not isomorphic to \(\mathcal{O}_X\).
Remark¶
The groups \(\operatorname{CaCl}(X)\) and \(\operatorname{Pic}(X)\) can be defined for any scheme \(X\) without Noetherian hypotheses: \(\operatorname{Pic}(X) = H^1(X, \mathcal{O}_X^*)\) is a sheaf cohomology group, and \(\operatorname{CaCl}(X) = H^0(X, \mathcal{M}^*/\mathcal{O}_X^*) / H^0(X, \mathcal{M}^*)\) uses only the sheaf of total quotient rings. In contrast, the Weil divisor class group \(\operatorname{Cl}(X)\) requires \(X\) to be at least locally Noetherian, integral, and regular in codimension 1, since its definition involves codimension-1 points whose local rings must be DVRs. This makes \(\operatorname{Cl}(X)\) the most restrictive of the three notions.
Example: Hartshorne, Example \(\mathrm{II}.6.11.4\), \(\operatorname{CaCl}^{0}\)¶
Let \(C\) be a nonsingular projective curve of genus \(g\) over an algebraically closed field \(k\). Since \(C\) is smooth, it is locally factorial, so \(\operatorname{CaCl}(C) \cong \operatorname{Cl}(C) \cong \operatorname{Pic}(C)\). The degree map \(\deg : \operatorname{Pic}(C) \to \mathbb{Z}\) is a surjective homomorphism (choosing any closed point \(p\), the divisor \([p]\) has degree 1), giving the exact sequence
The kernel \(\operatorname{Pic}^0(C)\) is the group of \(k\)-rational points of the Jacobian variety \(J(C)\), an abelian variety of dimension \(g\).
Genus 0. When \(C \cong \mathbb{P}^1_k\), we have \(\operatorname{Pic}(\mathbb{P}^1) \cong \mathbb{Z}\) generated by \(\mathcal{O}(1)\), so \(\operatorname{Pic}^0(\mathbb{P}^1) = 0\). The Jacobian is trivial.
Genus 1. When \(C\) is an elliptic curve with a chosen base point \(O\), the Abel--Jacobi map \(P \mapsto [P] - [O]\) gives an isomorphism \(C(k) \xrightarrow{\sim} \operatorname{Pic}^0(C)\), identifying \(\operatorname{Pic}^0(C)\) with \(C\) itself (as a group under its elliptic curve law).
General genus. For \(g \geq 1\), the Abel--Jacobi theorem gives \(\operatorname{Pic}^0(C) \cong J(C)(k)\), where \(J(C)\) is an abelian variety of dimension \(g\). Over \(\mathbb{C}\), analytically \(J(C) \cong \mathbb{C}^g / \Lambda\) for a lattice \(\Lambda \cong \mathbb{Z}^{2g}\) determined by the period matrix of \(C\).
Example: Hartshorne, \(\mathrm{II}.6.9\), Picard groups of singular curves¶
Let \(C\) be a reduced irreducible curve over an algebraically closed field \(k\), with normalization \(f : \widetilde{C} \to C\). There is an exact sequence
where \(\widetilde{\mathcal{O}}_{C,p}\) is the integral closure of \(\mathcal{O}_{C,p}\) in its fraction field.
Derivation. The normalization \(f\) is a finite birational morphism, inducing a short exact sequence of sheaves \(0 \to \mathcal{O}_C^* \to f_* \mathcal{O}_{\widetilde{C}}^* \to f_* \mathcal{O}_{\widetilde{C}}^* / \mathcal{O}_C^* \to 0\). The quotient sheaf is supported on \(C^{\mathrm{sing}}\) (since \(f\) is an isomorphism over the smooth locus). The associated long exact sequence in cohomology gives the result, using \(H^1(C, f_* \mathcal{O}_{\widetilde{C}}^*) \cong H^1(\widetilde{C}, \mathcal{O}_{\widetilde{C}}^*) = \operatorname{Pic}(\widetilde{C})\) by the Leray spectral sequence (which degenerates because \(f\) is affine, so \(R^i f_* = 0\) for \(i \geq 1\)).
Cuspidal cubic. Let \(C : y^2 = x^3\) in \(\mathbb{A}^2_k\). The normalization is \(\widetilde{C} \cong \mathbb{A}^1_k\) via \(t \mapsto (t^2, t^3)\). The unique singular point is the origin \(p\), with \(\mathcal{O}_{C,p} = k[t^2, t^3]_{(t^2, t^3)}\) and \(\widetilde{\mathcal{O}}_{C,p} = k[t]_{(t)}\). An element of \(\widetilde{\mathcal{O}}_{C,p}^*\) has the form \(u = a_0 + a_1 t + a_2 t^2 + \cdots\) with \(a_0 \neq 0\); modulo \(\mathcal{O}_{C,p}^*\) (units involving only \(t^2, t^3, \ldots\)), the quotient is parameterized by \(a_1 / a_0 \in k\). Thus \(\widetilde{\mathcal{O}}_{C,p}^* / \mathcal{O}_{C,p}^* \cong \mathbb{G}_a = (k, +)\), and since \(\operatorname{Pic}(\mathbb{A}^1_k) = 0\) we get \(\operatorname{Pic}(C) \cong \mathbb{G}_a\).
Nodal cubic. Let \(C : y^2 = x^2(x+1)\) in \(\mathbb{A}^2_k\). Again \(\widetilde{C} \cong \mathbb{A}^1_k\). At the node \(p = (0,0)\), the integral closure has two maximal ideals (corresponding to the two branches), and \(\widetilde{\mathcal{O}}_{C,p}^* / \mathcal{O}_{C,p}^* \cong \mathbb{G}_m = (k^*, \cdot)\): the quotient detects the ratio of the leading coefficients along the two branches. Hence \(\operatorname{Pic}(C) \cong \mathbb{G}_m\).
The dichotomy is instructive: cusps contribute additive local terms \(\mathbb{G}_a\), while nodes contribute multiplicative local terms \(\mathbb{G}_m\).
Example: Hartshorne \(\mathrm{II}.7.13\), a complete nonprojective surface¶
Let \(k\) be an algebraically closed field with \(\operatorname{char}(k) \neq 2\) and let \(C : y^2z = x^3 \subset \mathbb{P}^2_k\) be the projective cuspidal cubic. We construct a complete surface \(X\) fibered over \(\mathbb{P}^1_k\) with \(\operatorname{Pic}(X) \cong \mathbb{G}_a\), and conclude that \(X\) is not projective.
Construction. Take two copies \(V_1 = C \times \mathbb{A}^1_k\) and \(V_2 = C \times \mathbb{A}^1_k\), and glue them along \(C \times (\mathbb{A}^1_k \setminus \{0\})\) via a morphism \((x, t) \sim (\phi_t(x), t^{-1})\) where \(\phi_t\) is an automorphism of \(C\) parameterized by \(t\). This yields a surface \(X\) with a proper morphism \(\pi : X \to \mathbb{P}^1_k\).
Completeness. Over each standard affine \(\mathbb{A}^1 \subset \mathbb{P}^1\), the map \(\pi\) is isomorphic to \(C \times \mathbb{A}^1 \to \mathbb{A}^1\), a base change of the proper morphism \(C \to \operatorname{Spec}(k)\). Since properness is local on the base and \(\mathbb{P}^1_k\) is proper over \(k\), the composition \(X \to \operatorname{Spec}(k)\) is proper.
Picard group of \(C \times \mathbb{A}^1_k\). Apply the normalization exact sequence: the normalization of \(C \times \mathbb{A}^1_k\) is \(\mathbb{P}^1_k \times \mathbb{A}^1_k\), with singular locus \(\{0\} \times \mathbb{A}^1_k\). The local contribution at the cusp gives \(\widetilde{\mathcal{O}}_p^* / \mathcal{O}_p^* \cong \mathbb{G}_a\) (exactly as for the cuspidal cubic alone, since the extra parameter \(z\) is a unit near the singular locus). Since \(\operatorname{Pic}(\mathbb{P}^1_k \times \mathbb{A}^1_k) \cong \mathbb{Z}\) (generated by the pullback of \(\mathcal{O}_{\mathbb{P}^1}(1)\)), the normalization sequence splits to give
Picard group of \(C \times (\mathbb{A}^1_k \setminus \{0\})\). Replacing \(\mathbb{A}^1_k\) by \(\mathbb{A}^1_k \setminus \{0\}\) introduces an additional \(\mathbb{Z}\)-factor from the Laurent units \(k[z, z^{-1}]^* \cong k^* \times \mathbb{Z}\):
Restriction map. The restriction \(r : \operatorname{Pic}(C \times \mathbb{A}^1_k) \to \operatorname{Pic}(C \times (\mathbb{A}^1_k \setminus \{0\}))\) is the identity on the \(\mathbb{G}_a\)-factor and injects \(\mathbb{Z}\) into the first \(\mathbb{Z}\)-factor:
Mayer--Vietoris computation. Writing \(X = V_1 \cup V_2\) with \(V_1 \cap V_2 \cong C \times (\mathbb{A}^1_k \setminus \{0\})\), the Mayer--Vietoris sequence for \(\mathcal{O}_X^*\) gives
Computing the cokernel of \(r_1 - r_2\) yields \(\operatorname{Pic}(X) \cong \mathbb{G}_a\).
Nonprojectivity. Since \(\mathbb{G}_a \cong (k, +)\) admits no nontrivial homomorphism to \(\mathbb{Z}\), there is no ample line bundle on \(X\): any such bundle would generate a copy of \(\mathbb{Z}\) inside \(\operatorname{Pic}(X)\), but \(\mathbb{G}_a\) is torsion in positive characteristic and divisible in characteristic zero, and in neither case contains \(\mathbb{Z}\). Therefore \(X\) is complete but not projective.
Remark: Why does \(\operatorname{Pic}(X) \cong \mathbb{G}_{a}\) imply \(X\) is not projective?¶
If \(X\) were projective over \(k\), the Neron--Severi theorem would give an exact sequence
where \(\operatorname{NS}(X)\) is a finitely generated abelian group and \(\operatorname{Pic}^0(X)\) is an abelian variety (the Picard variety). However, \(\mathbb{G}_a\) is a unipotent algebraic group: it is affine, connected, of dimension 1, and admits no nontrivial algebraic quotient that is an abelian variety. (An abelian variety is projective and has no unipotent subgroups, while \(\mathbb{G}_a\) is itself unipotent.) So \(\mathbb{G}_a\) cannot appear as \(\operatorname{Pic}(X)\) for a projective \(X\).
More concretely, a projective variety carries an ample line bundle \(\mathcal{L}\), and the powers \(\mathcal{L}^{\otimes n}\) for \(n \geq 1\) generate a free subgroup \(\mathbb{Z} \hookrightarrow \operatorname{Pic}(X)\). But \(\mathbb{G}_a \cong (k, +)\) has no element of infinite order when \(\operatorname{char}(k) > 0\) (every element is killed by \(p\)), and even in characteristic zero, the abstract group \((k, +)\) contains copies of \(\mathbb{Z}\) as subgroups, but these cannot arise as the image of the powers of a single ample class in a way compatible with the algebraic group structure: the subgroup \(\mathbb{Z} \subset \mathbb{G}_a\) is not Zariski closed, contradicting the requirement that the Neron--Severi group be a finitely generated discrete quotient.
Remark¶
Since the cuspidal cubic \(C\) is singular at the cusp, the scheme \(C \times \mathbb{A}^1_k\) is not regular in codimension 1 along \(\{0\} \times \mathbb{A}^1_k\), and the excision exact sequence for Weil divisors
does not apply. This sequence requires \(X\) to be normal, and \(C\) (hence \(C \times \mathbb{A}^1_k\)) has a non-normal singular point. One must therefore work directly with \(\operatorname{Pic}\) (i.e., with line bundles or Cartier divisors) and use the normalization exact sequence instead of Weil divisor arguments.
Example: Hironaka varieties, smooth complete nonprojective \(3\)-folds¶
Hironaka constructed smooth complete (proper over \(\mathbb{C}\)) threefolds that are not projective. The idea is to perform birational modifications on a smooth projective threefold, choosing different orders of blow-up at two points to create exceptional curves that obstruct the existence of an ample divisor.
Setup. Start with a smooth projective threefold \(Y\) over \(\mathbb{C}\) containing two smooth curves \(C_1, C_2 \subset Y\) that meet transversally at exactly two points \(p\) and \(q\), with \(C_1\) and \(C_2\) disjoint away from \(\{p, q\}\).
Local modification. In a neighborhood of \(p\), first blow up along \(C_1\), then blow up the strict transform of \(C_2\). In a neighborhood of \(q\), reverse the order: first blow up along \(C_2\), then blow up the strict transform of \(C_1\). Away from \(p\) and \(q\) (where the curves are disjoint), perform ordinary blow-ups along each curve. These local constructions glue to give a smooth complete threefold \(X\).
Exceptional curves. The fibers over \(p\) and \(q\) each contain a copy of \(\mathbb{P}^1\), call them \(\ell_p\) and \(\ell_q\). The curve \(\ell_p\) arises as the exceptional fiber of the blow-up of \(C_1\) at \(p\) (where \(C_1\) is blown up first), and \(\ell_q\) arises from the blow-up of \(C_1\) at \(q\) (where \(C_2\) is blown up first, so \(C_1\) is blown up second). The reversed blow-up order at \(p\) and \(q\) is the essential asymmetry.
Nonprojectivity. The curves \(\ell_p\) and \(\ell_q\) satisfy \(\ell_p + \ell_q \sim 0\) as a 1-cycle (they bound a 2-chain coming from the strict transform of \(C_1\)). Any ample divisor \(H\) would require \(H \cdot \ell_p > 0\) and \(H \cdot \ell_q > 0\) (since \(\ell_p\) and \(\ell_q\) are complete curves), but then \(H \cdot (\ell_p + \ell_q) > 0\), contradicting \(\ell_p + \ell_q \sim 0\). More precisely, one shows that \(\ell_p\) and \(\ell_q\) are algebraically equivalent to zero, and the Nakai--Moishezon criterion (a divisor \(D\) is ample if and only if \(D^{\dim V} \cdot V > 0\) for every positive-dimensional subvariety \(V\)) then implies no divisor can be ample.
Example: \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) is not necessarily ample¶
Let \(S\) be a scheme and \(\mathscr{E}\) a locally free sheaf on \(S\). The tautological line bundle \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) on the projective bundle \(\pi : \mathbb{P}\mathscr{E} \to S\) is always \(\pi\)-ample (ample relative to \(\pi\)), since by construction \(\mathbb{P}\mathscr{E} = \operatorname{Proj}(\operatorname{Sym}(\mathscr{E}))\) and \(\mathcal{O}(1)\) corresponds to the Serre twist on fibers. However, \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) need not be ample in the absolute sense.
Counterexample on \(\mathbb{F}_2\). Let \(S = \mathbb{P}^1_k\) and \(\mathscr{E} = \mathcal{O}(1) \oplus \mathcal{O}(-1)\). Then \(\mathbb{P}\mathscr{E}\) is the Hirzebruch surface \(\mathbb{F}_2\). The surjection \(\mathscr{E} \twoheadrightarrow \mathcal{O}(-1)\) (projection onto the second factor) determines a section \(\sigma_0 : \mathbb{P}^1 \hookrightarrow \mathbb{F}_2\). One computes \(\sigma_0^2 = -2\) from the general formula: for a section of \(\mathbb{P}(\mathcal{O}(a) \oplus \mathcal{O}(b))\) corresponding to the quotient \(\mathcal{O}(b)\), the self-intersection is \(b - a\), which gives \(-1 - 1 = -2\).
The intersection of \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) with \(\sigma_0\) is the degree of \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)|_{\sigma_0}\). Since \(\sigma_0\) corresponds to the quotient \(\mathscr{E} \twoheadrightarrow \mathcal{O}(-1)\), the pullback \(\sigma_0^* \mathcal{O}_{\mathbb{P}\mathscr{E}}(1) \cong \mathcal{O}_{\mathbb{P}^1}(-1)\), so
By the Nakai--Moishezon criterion, \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) is not ample.
General criterion. The bundle \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) is ample (absolutely) if and only if \(\mathscr{E}\) is an ample vector bundle on \(S\). For a curve \(S\), this means every quotient line bundle of \(\mathscr{E}\) has positive degree. In the example above, \(\mathcal{O}(-1)\) is a quotient of \(\mathscr{E}\) with negative degree, so ampleness fails.
Remark: Degree of \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\)¶
Let \(\mathscr{E}\) be a locally free sheaf of rank \(r\) on a smooth projective curve \(C\). The projective bundle \(\pi : \mathbb{P}\mathscr{E} \to C\) is a smooth projective variety of dimension \(r\), and the top self-intersection of \(\mathcal{O}_{\mathbb{P}\mathscr{E}}(1)\) is
This follows from the tautological relation in \(A^*(\mathbb{P}\mathscr{E})\): writing \(\xi = c_1(\mathcal{O}(1))\), the identity \(\xi^r - \pi^* c_1(\mathscr{E}) \cdot \xi^{r-1} + \pi^* c_2(\mathscr{E}) \cdot \xi^{r-2} - \cdots = 0\) together with the push-forward formula \(\pi_*(\xi^{r-1}) = 1\) yields \(\pi_*(\xi^r) = c_1(\mathscr{E}) = \deg(\mathscr{E})\) on a curve.
For rank 2 on a curve, this gives \(c_1(\mathcal{O}(1))^2 = \deg(\mathscr{E})\). When \(\mathscr{E} = \mathcal{L}\) is a line bundle, \(\mathbb{P}\mathcal{L} = C\) and \(\mathcal{O}_{\mathbb{P}\mathcal{L}}(1) = \mathcal{L}\), so the formula recovers \(\deg(\mathcal{L})\). Thus \(c_1(\mathcal{O}(1))^r = \deg(\mathscr{E})\) is the natural generalization of the degree of a line bundle to higher-rank bundles on curves.
Example: \(\operatorname{Cl}(X)\) of quadric hypersurfaces, Hartshorne \(\mathrm{II}.6.5\)¶
Let \(k\) be a field with \(\operatorname{char}(k) \neq 2\). We compute the divisor class group of both affine and projective quadrics.
Affine quadric cones. Let \(X = \operatorname{Spec}(k[x_0, \ldots, x_n] / (x_0^2 + \cdots + x_r^2))\), which is normal for \(r \geq 2\).
| \(r\) | \(\operatorname{Cl}(X)\) | Method |
|---|---|---|
| \(2\) | \(\mathbb{Z}/2\mathbb{Z}\) | Excision + non-principality |
| \(3\) | \(\mathbb{Z}\) | Cone over \(\mathbb{P}^1 \times \mathbb{P}^1\) |
| \(\geq 4\) | \(0\) | Grothendieck--Lefschetz |
Projective quadrics. Let \(Q = \operatorname{Proj}(k[x_0, \ldots, x_n]/(x_0^2 + \cdots + x_r^2))\).
| \(r\) | \(Q\) | \(\operatorname{Cl}(Q)\) |
|---|---|---|
| \(2\) | \(\cong \mathbb{P}^1\) | \(\mathbb{Z}\) |
| \(3\) | \(\cong \mathbb{P}^1 \times \mathbb{P}^1\) (Segre) | \(\mathbb{Z}^2\) |
| \(\geq 4\) | smooth quadric, \(\dim \geq 3\) | \(\mathbb{Z}\) (Lefschetz) |
Case \(r = 2\) in detail. Take the isomorphic form \(X = \operatorname{Spec}(k[x,y,z]/(xy - z^2))\). Let \(Y\) be the prime divisor defined by the ideal \((y, z)\). At the generic point of \(Y\), the local ring is a DVR with uniformizer \(z\); since \(y = z^2/x\) and \(x\) is a unit at this point, we get \(\operatorname{div}(y) = 2Y\), so \(2[Y] = 0\) in \(\operatorname{Cl}(X)\).
The excision sequence for the closed subset \(Y\) gives \(\mathbb{Z} \to \operatorname{Cl}(X) \to \operatorname{Cl}(X \setminus Y) \to 0\). The complement \(X \setminus Y = \operatorname{Spec}(k[x, y, z, y^{-1}]/(xy - z^2)) \cong \operatorname{Spec}(k[y, y^{-1}, z])\) is the spectrum of a Laurent polynomial ring, hence a UFD with trivial class group. So \(\operatorname{Cl}(X)\) is generated by \([Y]\).
It remains to show \([Y] \neq 0\), i.e., the ideal \((y, z)\) is not principal. At the closed point \(\mathfrak{m} = (x, y, z)\), the cotangent space \(\mathfrak{m}/\mathfrak{m}^2\) is spanned by \(\bar{x}, \bar{y}, \bar{z}\) (the relation \(xy = z^2\) vanishes in \(\mathfrak{m}^2\)), so \(\dim_k \mathfrak{m}/\mathfrak{m}^2 = 3\) while \(\dim X = 2\), confirming the vertex is singular. The ideal \((y, z)\) requires two generators at this point, so it cannot be principal. Hence \(\operatorname{Cl}(X) \cong \mathbb{Z}/2\mathbb{Z}\).
Case \(r = 3\). The affine quadric \(X\) is the cone over \(V = \operatorname{Proj}(k[x,y,z,w]/(xy - z^2 - w^2)) \cong \mathbb{P}^1 \times \mathbb{P}^1\) (Segre embedding). By Hartshorne II.6.3, the exact sequence \(\mathbb{Z} \xrightarrow{\alpha} \operatorname{Cl}(V) \to \operatorname{Cl}(X) \to 0\) sends \(1 \mapsto (1,1) \in \mathbb{Z}^2 = \operatorname{Cl}(\mathbb{P}^1 \times \mathbb{P}^1)\) (the hyperplane class under Segre), so
Case \(r \geq 4\). The Grothendieck--Lefschetz hyperplane theorem gives \(\operatorname{Cl}(Q) \cong \operatorname{Cl}(\mathbb{P}^n) \cong \mathbb{Z}\) for smooth complete intersections of dimension \(\geq 3\). Then \(\operatorname{Cl}(X) \cong \mathbb{Z} / \mathbb{Z} = 0\), and by Hartshorne II.6.2, the coordinate ring is a UFD.
Remark: Hartshorne \(\mathrm{II}.6.3\), relations between \(\operatorname{Cl}(V)\) and \(\operatorname{Cl}(C(V))\)¶
Let \(V \subset \mathbb{P}^n_k\) be a normal projective variety with homogeneous coordinate ring \(S(V)\) and affine cone \(C(V) = \operatorname{Spec}(S(V))\). There is an exact sequence
where \(\alpha(1) = [H \cap V]\), the class of a hyperplane section.
The surjectivity of \(\beta\) comes from excision: the punctured cone \(C(V) \setminus \{\text{vertex}\}\) is an \(\mathbb{A}^1 \setminus \{0\}\)-bundle over \(V\), so \(\operatorname{Cl}(C(V) \setminus \{\text{vertex}\}) \cong \operatorname{Cl}(V)\), and removing a point of codimension \(\geq 2\) does not change the class group. The kernel of \(\beta\) is the subgroup generated by the hyperplane class, because two Weil divisors on \(V\) that differ by a hyperplane section become linearly equivalent on \(C(V)\) (the hyperplane corresponds to a principal divisor of a degree-1 element of \(S(V)\)).
In particular, \(\operatorname{Cl}(C(V)) = 0\) if and only if \(\operatorname{Cl}(V)\) is generated by the hyperplane class. When \(\operatorname{Cl}(C(V)) = 0\), the ring \(S(V)\) is a UFD by Hartshorne II.6.2.
Example. For \(V = \mathbb{P}^1 \times \mathbb{P}^1\) under the Segre embedding, \(\operatorname{Cl}(V) \cong \mathbb{Z}^2\) and the hyperplane class is \((1,1)\). Then \(\operatorname{Cl}(C(V)) \cong \mathbb{Z}^2 / \mathbb{Z}(1,1) \cong \mathbb{Z}\), generated by the class \((1,0) - (0,1)\).
Remark: Lefschetz hyperplane theorem¶
The Grothendieck--Lefschetz theorem (SGA 2, Expose XII) states that for a smooth complete intersection \(V \subset \mathbb{P}^n_k\) of dimension \(\geq 3\), the restriction \(\operatorname{Cl}(\mathbb{P}^n) \to \operatorname{Cl}(V)\) is an isomorphism. In particular, \(\operatorname{Cl}(V) \cong \mathbb{Z}\), generated by the hyperplane class.
Application to quadrics. For a smooth quadric \(Q \subset \mathbb{P}^n_k\) with \(\dim Q \geq 3\) (i.e., \(n \geq 4\)), we get \(\operatorname{Cl}(Q) \cong \mathbb{Z}\). The exact sequence of Hartshorne II.6.3 then gives \(\operatorname{Cl}(C(Q)) = 0\) for the affine cone.
Klein's theorem. As a corollary, every irreducible codimension-1 subvariety \(Y\) of a smooth quadric \(Q\) of dimension \(\geq 3\) is a complete intersection \(Y = Q \cap H\) for some hypersurface \(H \subset \mathbb{P}^n\). Indeed, \([Y] = d \cdot [H \cap Q]\) in \(\operatorname{Cl}(Q) \cong \mathbb{Z}\) for some \(d \geq 1\). The restriction map \(H^0(\mathcal{O}_{\mathbb{P}^n}(d)) \to H^0(\mathcal{O}_Q(d))\) is surjective: from the exact sequence
the cokernel vanishes because \(H^1(\mathcal{O}_{\mathbb{P}^n}(d-2)) = 0\) for \(n \geq 3\). Hence \(Y\) is cut out by a degree-\(d\) hypersurface.
Since \(\operatorname{Cl}(C(Q)) = 0\) for \(n \geq 4\), Hartshorne II.6.2 gives that \(k[x_0, \ldots, x_n]/(x_0^2 + \cdots + x_n^2)\) is a UFD.
Example: Smooth cubic surface \(X \subset \mathbb{P}^3\), \(\operatorname{Cl}(X) \cong \mathbb{Z}^7\)¶
Every smooth cubic surface \(X \subset \mathbb{P}^3_k\) over an algebraically closed field \(k\) is isomorphic to \(\operatorname{Bl}_{p_1, \ldots, p_6} \mathbb{P}^2\), the blow-up of \(\mathbb{P}^2\) at 6 points in general position (no 3 collinear, no 6 on a conic). This classical fact immediately gives the Picard group.
Picard group. Let \(\sigma : X \to \mathbb{P}^2\) denote the blow-up with exceptional divisors \(E_i = \sigma^{-1}(p_i)\). Since each blow-up adds one \(\mathbb{Z}\)-summand,
with basis \(\{H, E_1, \ldots, E_6\}\), where \(H = \sigma^* \mathcal{O}_{\mathbb{P}^2}(1)\). Since \(X\) is smooth and projective, \(\operatorname{Cl}(X) \cong \operatorname{Pic}(X) \cong \mathbb{Z}^7\).
Intersection form. The intersection pairing is determined by \(H^2 = 1\), \(E_i^2 = -1\), \(H \cdot E_i = 0\), \(E_i \cdot E_j = 0\) for \(i \neq j\). The resulting lattice is \(I_{1,6} = \langle 1 \rangle \oplus \langle -1 \rangle^6\).
Anticanonical class. The canonical class of \(\mathbb{P}^2\) is \(K_{\mathbb{P}^2} = -3H\). The blow-up formula gives \(K_X = \sigma^* K_{\mathbb{P}^2} + \sum E_i = -3H + E_1 + \cdots + E_6\), so
This is the hyperplane class of the cubic embedding \(X \hookrightarrow \mathbb{P}^3\), confirming \((-K_X)^2 = 9 - 6 = 3 = \deg(X)\).
The 27 lines. The Cayley--Salmon theorem states that \(X\) contains exactly 27 lines. In terms of the basis \(\{H, E_i\}\), they decompose as:
| Type | Class | Count |
|---|---|---|
| Exceptional curves | \(E_i\) | 6 |
| Strict transforms of lines \(\overline{p_i p_j}\) | \(H - E_i - E_j\) | \(\binom{6}{2} = 15\) |
| Strict transforms of conics through 5 points | \(2H - \sum_{j \neq i} E_j\) | 6 |
| Total | 27 |
Each of these 27 classes \(\ell\) satisfies \(\ell^2 = -1\) and \(\ell \cdot (-K_X) = 1\), i.e., every line is a \((-1)\)-curve of degree 1 in \(\mathbb{P}^3\).
The following examples use a result of V. L. Popov ("Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles"): if \(G\) is a connected linear algebraic group with \(\operatorname{Pic}(G) = 0\) (e.g., \(G\) simply connected semisimple) and \(H \subset G\) is a closed subgroup, there is an exact sequence
where \(\operatorname{X}(H) = \operatorname{Hom}(H, \mathbb{G}_m)\) is the character group of \(H\) and \(\operatorname{X}_G(H)\) consists of those characters of \(H\) that extend to \(G\). We work over \(k = \mathbb{C}\) in what follows.
Example: \(\operatorname{Pic}(X)\) of flag varieties¶
Let \(F_n = SL_n / B_+\) be the complete flag variety, where \(B_+ \subset SL_n\) is the Borel subgroup of upper triangular matrices. We compute \(\operatorname{Pic}(F_n) \cong \mathbb{Z}^{n-1}\).
Character group of \(B_+\). Every character of \(B_+\) factors through \(B_+ / [B_+, B_+] \cong T\), the maximal torus of diagonal matrices. Since \(T = \{(t_1, \ldots, t_n) : \prod t_i = 1\} \cong \mathbb{G}_m^{n-1}\), we have \(\operatorname{X}(B_+) \cong \operatorname{X}(T) \cong \mathbb{Z}^{n-1}\).
Extendable characters. Since \(SL_n\) is semisimple, \(\operatorname{X}(SL_n) = 0\). A character of \(B_+\) extends to \(SL_n\) only if it is trivial, so \(\operatorname{X}_{SL_n}(B_+) = 0\).
Popov's sequence. Since \(\operatorname{Pic}(SL_n) = 0\) (\(SL_n\) is simply connected semisimple), we obtain
The generators are the line bundles \(\mathcal{L}(\omega_i)\) associated to the fundamental weights \(\omega_1, \ldots, \omega_{n-1}\). Concretely, \(\mathcal{L}(\omega_i) = SL_n \times^{B_+} k_{\omega_i}\) is the line bundle whose global sections give the \(i\)-th exterior power representation, and these line bundles pull back to \(\mathcal{O}(1)\) under the Plucker embeddings.
Remark: \(\operatorname{Pic}(G/B)\)¶
For a connected reductive group \(G\) with Borel subgroup \(B\) and simple roots \(\Delta\), the Bruhat decomposition gives a cell decomposition of \(G/B\) indexed by the Weyl group \(W\). Each Schubert cell \(C_w = BwB/B\) has codimension \(\ell(w)\) in \(G/B\). The codimension-1 Schubert varieties (Schubert divisors) correspond to simple reflections \(s_\alpha\) for \(\alpha \in \Delta\), giving
where \(|\Delta|\) is the number of simple roots, equivalently the semisimple rank of \(G\). Note that \(|\Delta|\) is the number of simple roots, not the number of positive roots \(|\Phi_+|\).
The generators are line bundles \(\mathcal{L}(\omega_\alpha)\) associated to the fundamental weights. The corresponding Schubert divisors are \(\overline{Bw_0 s_\alpha B / B}\), where \(w_0 \in W\) is the longest element. The Chow group \(A^1(G/B)\) is freely generated by these Schubert divisor classes.
Example: Computing \(\operatorname{Cl}(X)\) of quadrics from a different point of view¶
The Picard groups of smooth projective quadrics can also be computed using sporadic isogenies between spin groups and products of \(SL_2\), combined with Popov's exact sequence.
Case \(Q_2\) (smooth conic). The group \(SO(3, \mathbb{C})\) acts transitively on \(Q_2\). We lift to the simply connected cover \(\operatorname{Spin}(3, \mathbb{C}) \cong SL(2, \mathbb{C})\), which acts on \(\mathfrak{sl}(2, \mathbb{C}) \cong \mathbb{C}^3\) via the adjoint representation, preserving the Killing form. This realizes the double cover \(SL_2 \to SO_3\). The stabilizer of a point on \(Q_2\) in \(SO_3\) is \(SO_2 \cong \mathbb{G}_m\); its preimage in \(SL_2\) is the maximal torus \(T \cong \mathbb{G}_m\) (equivalently, the Borel subgroup \(B_+\), since the stabilizer of a highest weight line in the adjoint representation is \(B_+\)). By Popov's exact sequence with \(\operatorname{Pic}(SL_2) = 0\) and \(\operatorname{X}_{SL_2}(B_+) = 0\):
Case \(Q_3\) (smooth quadric surface). Here \(\operatorname{Spin}(4, \mathbb{C}) \cong SL_2 \times SL_2\), acting on \(M_2(\mathbb{C}) \cong \mathbb{C}^4\) by \((g, h) \cdot A = gAh^{-1}\), with the quadratic form \(\det : M_2 \to \mathbb{C}\) giving \(Q_3 = \{A : \det(A) = 0\} \setminus \{0\}\). The stabilizer of a rank-1 matrix is \(B_+ \times B_-\) (a product of opposite Borel subgroups). Since \(\operatorname{Pic}(SL_2 \times SL_2) = 0\):
This recovers the well-known fact \(Q_3 \cong \mathbb{P}^1 \times \mathbb{P}^1\).
Remark: Picard groups of higher-dimensional quadrics via spin groups¶
For \(r \geq 4\), the stabilizer in \(\operatorname{Spin}(r+1, \mathbb{C})\) of a point on \(Q_r\) is \(\operatorname{Spin}(r, \mathbb{C})\). Since \(\operatorname{Spin}(r, \mathbb{C})\) is simply connected for all \(r \geq 3\), its character group \(\operatorname{X}(\operatorname{Spin}(r, \mathbb{C})) = 0\), and Popov's exact sequence gives \(\operatorname{Pic}(Q_r) \cong \mathbb{Z}\) for \(r \geq 3\), using \(\operatorname{Pic}(\operatorname{Spin}(r+1)) = 0\) and \(\operatorname{X}_{\operatorname{Spin}(r+1)}(\operatorname{Spin}(r)) = 0\).
For \(r = 4\), one can alternatively use the sporadic isogeny \(\operatorname{Spin}(5, \mathbb{C}) \cong Sp(4, \mathbb{C})\) to identify the stabilizer explicitly. A uniform treatment via the Dynkin diagram inclusions \(B_{\lfloor r/2 \rfloor} \subset B_{\lfloor (r+1)/2 \rfloor}\) (or \(D_{r/2} \subset B_{(r+1)/2}\) for even \(r\)) recovers \(\operatorname{Pic}(Q_r) \cong \mathbb{Z}\) for all \(r \geq 3\).
Remark: When do we have \(\operatorname{X}(G) = 0\)?¶
For a connected linear algebraic group \(G\), the character group \(\operatorname{X}(G) = \operatorname{Hom}(G, \mathbb{G}_m)\) vanishes if and only if \(G = [G, G]\) (the derived subgroup equals \(G\)).
For a connected reductive group \(G\) with maximal torus \(T\), the restriction \(\operatorname{X}(G) \hookrightarrow \operatorname{X}(T)\) identifies \(\operatorname{X}(G)\) with the sublattice of characters trivial on \([G, G] \cap T\). Thus \(\operatorname{X}(G) = 0\) if and only if \(\operatorname{X}(T)\) equals the root lattice (equivalently, \(G\) is semisimple). Concretely:
- \(G = SL_n\): semisimple and simply connected, so \(\operatorname{X}(SL_n) = 0\).
- \(G = Sp_{2n}\), \(\operatorname{Spin}(n)\): also simply connected semisimple, so \(\operatorname{X}(G) = 0\).
- \(G = GL_n\): the commutator subgroup is \([GL_n, GL_n] = SL_n\), so \(\operatorname{X}(GL_n) \cong \operatorname{X}(GL_n / SL_n) = \operatorname{X}(\mathbb{G}_m) \cong \mathbb{Z}\), generated by the determinant.
- In general, \(\operatorname{X}(G) \cong \operatorname{X}(G/[G,G])\), where \(G/[G,G]\) is a torus of rank equal to the dimension of the center modulo the semisimple part.
Example: \(\operatorname{Cl}(X)\) of Grassmannian¶
Let \(V\) be an \(n\)-dimensional vector space over \(\mathbb{C}\) and \(1 \leq m \leq n-1\). We show \(\operatorname{Pic}(\operatorname{Gr}(V, m)) \cong \mathbb{Z}\).
Homogeneous space description. The group \(SL_n\) acts transitively on \(\operatorname{Gr}(V, m)\). The stabilizer of the coordinate \(m\)-plane \(\langle e_1, \ldots, e_m \rangle\) is the maximal parabolic subgroup
so \(\operatorname{Gr}(V, m) \cong SL_n / P\).
Popov's sequence. Since \(\operatorname{Pic}(SL_n) = 0\) and \(\operatorname{X}_{SL_n}(P) = 0\) (no nontrivial character of \(P\) extends to \(SL_n\)), we get \(\operatorname{Pic}(\operatorname{Gr}(V, m)) \cong \operatorname{X}(P)\).
Character group of \(P\). The derived subgroup \([P, P]\) consists of block upper triangular matrices with \(\det(A) = 1\) and \(\det(D) = 1\) (subject to the constraint \(\det(A)\det(D) = 1\) already in \(P\)). The abelianization is \(P / [P, P] \cong \mathbb{G}_m\) via \(h \mapsto \det(A)\). Hence \(\operatorname{X}(P) \cong \mathbb{Z}\), and
The generator corresponds to the character \(\det(A)\), which gives the line bundle \(\det(\mathcal{S}^*)\) on the Grassmannian, where \(\mathcal{S}\) is the tautological rank-\(m\) subbundle. Under the Plucker embedding \(\operatorname{Gr}(V, m) \hookrightarrow \mathbb{P}(\bigwedge^m V)\), this generator pulls back to \(\mathcal{O}(1)\).
Remark: Chow ring of Grassmannian¶
The codimension-1 part of \(A^*(\operatorname{Gr}(V, m))\) can be computed from the Schubert cell decomposition. The cells are indexed by Young diagrams \(\lambda\) fitting inside an \(m \times (n-m)\) rectangle, with \(\operatorname{codim}(\Omega_\lambda) = |\lambda|\). There is a unique partition of size 1, namely \(\lambda = (1)\) (a single box), so \(A^1(\operatorname{Gr}(V, m)) \cong \mathbb{Z}\), generated by the special Schubert variety \(\Omega_{(1)} = \{W \in \operatorname{Gr}(V, m) : W \cap H \neq 0\}\) for a fixed hyperplane \(H \subset V\). This class corresponds to \(\det(\mathcal{S}^*)\) under the identification \(\operatorname{Pic} \cong A^1\), confirming the group-theoretic computation.
Example: Relations between \(\operatorname{K}(X)\) and \(\operatorname{Cl}(X)\)¶
For a Noetherian scheme \(X\), there are natural maps connecting the Grothendieck group \(K_0(X)\), the Picard group \(\operatorname{Pic}(X)\), and the Chow groups \(A^*(X)\).
The inclusion \(\operatorname{Pic}(X) \hookrightarrow K_0(X)\). The Grothendieck group \(K_0(X)\) is the free abelian group on isomorphism classes of locally free sheaves modulo the relation \([\mathscr{E}] = [\mathscr{E}'] + [\mathscr{E}'']\) for each short exact sequence \(0 \to \mathscr{E}' \to \mathscr{E} \to \mathscr{E}'' \to 0\). The map \(\mathcal{L} \mapsto [\mathcal{L}] - [\mathcal{O}_X]\) is a group homomorphism \(\operatorname{Pic}(X) \to K_0(X)\).
The determinant map. The map \(\det : K_0(X) \to \operatorname{Pic}(X)\) sends \([\mathscr{E}] \mapsto [\det(\mathscr{E})] = [\bigwedge^{\operatorname{rk}\mathscr{E}} \mathscr{E}]\). Composing with the inclusion, \(\det([\mathcal{L}] - [\mathcal{O}_X]) = \mathcal{L} \otimes \mathcal{O}_X^{-1} = \mathcal{L}\), so \(\det\) is a left inverse. In particular, \(\operatorname{Pic}(X)\) is a direct summand of \(K_0(X)\).
Chern character for smooth varieties. When \(X\) is smooth over a field, the Chern character gives an isomorphism \(\operatorname{ch} : K_0(X) \otimes \mathbb{Q} \xrightarrow{\sim} \bigoplus_p A^p(X) \otimes \mathbb{Q}\), and the first Chern class provides an isomorphism \(c_1 : \operatorname{Pic}(X) \xrightarrow{\sim} A^1(X) = \operatorname{Cl}(X)\).
Example: \(\mathbb{P}^n\). The ring \(K_0(\mathbb{P}^n) \cong \mathbb{Z}[h]/(h^{n+1})\) where \(h = [\mathcal{O}(1)] - 1\), and \(\operatorname{Cl}(\mathbb{P}^n) \cong \mathbb{Z}\) with generator \([H]\). The first Chern class sends \(c_1([\mathcal{O}(d)]) = d \cdot [H]\), recovering the degree.
Example: \(\operatorname{Cl}(X)\) of connected algebraic groups¶
Let \(G\) be a connected linear algebraic group over an algebraically closed field \(k\), with maximal torus \(T\) and Borel subgroup \(B\). The fibration \(G \to G/B\) with solvable fiber \(B\) (so \(\operatorname{Pic}(B) = 0\)) yields, via the Leray spectral sequence, the exact sequence
The map \(\operatorname{X}(T) \to \operatorname{Pic}(G/B)\) sends a character \(\chi\) to the line bundle \(\mathcal{L}(\chi) = G \times^B k_\chi\).
Simply connected semisimple groups. If \(G\) is simply connected semisimple, then \(\operatorname{X}(G) = 0\) and \(\operatorname{X}(T) \cong \Lambda_W\) (the weight lattice) maps isomorphically onto \(\operatorname{Pic}(G/B) \cong \Lambda_W\), so \(\operatorname{Pic}(G) = 0\).
Adjoint groups. If \(G\) is of adjoint type, then \(\operatorname{X}(T) = \Lambda_R\) (the root lattice) and \(\operatorname{Pic}(G/B) \cong \Lambda_W\). The map \(\Lambda_R \to \Lambda_W\) expresses simple roots in terms of fundamental weights, which is the Cartan matrix \(C\). The cokernel gives \(\operatorname{Pic}(G) \cong \Lambda_W / C \cdot \Lambda_R\).
Examples.
| \(G\) | Type | \(\operatorname{Pic}(G)\) |
|---|---|---|
| \(SL_n\) | \(A_{n-1}\), simply connected | \(0\) |
| \(PGL_n\) | \(A_{n-1}\), adjoint | \(\mathbb{Z}/n\mathbb{Z}\) |
| \(SO_{2n+1}\) | \(B_n\), adjoint | \(\mathbb{Z}/2\mathbb{Z}\) |
| \(Sp_{2n}\) | \(C_n\), simply connected | \(0\) |
For \(PGL_n\): the Cartan matrix of type \(A_{n-1}\) has determinant \(n\), and its Smith normal form reveals a single invariant factor \(n\), so \(\operatorname{coker}(C) \cong \mathbb{Z}/n\mathbb{Z}\).
Example: \(\operatorname{Pic}(X)\) of determinantal varieties \(R_{m \times n, s}\)¶
Let \(R_{m \times n, s}\) denote the variety of \(m \times n\) matrices over \(\mathbb{C}\) with rank \(\leq s\), where \(s \leq m \leq n\). If \(s = m\), then \(R_{m \times n, m} = \mathbb{A}^{mn}\) and \(\operatorname{Pic} = 0\). For \(s < m\), we show \(\operatorname{Pic}(R_{m \times n, s}) \cong \mathbb{Z}\).
The open stratum. Let \(U_s = R_{m \times n, s} \setminus R_{m \times n, s-1}\) be the locus of matrices with rank exactly \(s\). The group \(G = SL_m \times SL_n\) acts transitively on \(U_s\) by \((g, h) \cdot M = gMh^{-1}\), with every rank-\(s\) matrix conjugate to the standard form
Stabilizer computation. The stabilizer \(H \subset G\) of \(E_s\) consists of pairs of block matrices
where \(A \in GL_s\), \(D \in GL_{m-s}\), \(K \in GL_{n-s}\), with \(\det(A)\det(D) = 1\) and \(\det(A)\det(K) = 1\). The constraint \(gE_s h^{-1} = E_s\) forces the upper-left \(s \times s\) blocks to match and the lower-left block of \(g\) to vanish.
Dimension. From \(\dim(U_s) = \dim(G) - \dim(H)\), a direct count gives \(\dim(R_{m \times n, s}) = (m + n - s)s\).
Codimension of the singular locus. The codimension of \(R_{m \times n, s-1}\) in \(R_{m \times n, s}\) is
Since \(s < m \leq n\), we have \(m + n - 2s + 1 \geq 3\). Therefore \(R_{m \times n, s-1}\) has codimension \(\geq 3\), and excision gives \(\operatorname{Pic}(R_{m \times n, s}) \cong \operatorname{Pic}(U_s)\) (removing a closed subset of codimension \(\geq 2\) does not change the Picard group).
Character group. By Popov's theorem, \(\operatorname{Pic}(U_s) \cong \operatorname{X}(H)\). The abelianization \(H / [H, H] \cong \mathbb{G}_m\) is generated by the character
(the determinant of the common upper-left \(s \times s\) block). Hence \(\operatorname{X}(H) \cong \mathbb{Z}\) and
Remark¶
The rank stratification \(\{0\} = R_{m \times n, 0} \subset R_{m \times n, 1} \subset \cdots \subset R_{m \times n, s}\) cannot be used with Kleiman's theorem (which computes Chow groups via cellular decompositions) because the strata \(U_j\) are homogeneous spaces for \(SL_m \times SL_n\), not affine spaces \(\mathbb{A}^N\). The strata are rationally isomorphic to affine spaces but lack the cellular structure required by Kleiman's theorem. Nevertheless, the codimension/excision approach combined with Popov's sequence yields \(\operatorname{Pic}(R_{m \times n, s}) \cong \mathbb{Z}\) by a rigorous method.
Example: \(\operatorname{Pic}(X)\) of the variety \(S_{n,m}\)¶
Let \(S_{n,m}\) denote the variety of unordered \(m\)-frames in \(\mathbb{C}^n\), i.e., unordered \(m\)-tuples of linearly independent vectors (formally, the quotient of the Stiefel variety by the symmetric group \(S_m\) acting on the ordering). The group \(GL_n\) acts transitively on \(S_{n,m}\).
General framework. Since \(\operatorname{Pic}(GL_n) = 0\) (the only character is \(\det\), and \([GL_n, GL_n] = SL_n\) gives \(\operatorname{X}(GL_n) \cong \mathbb{Z}\), but \(\operatorname{Pic}(GL_n) = 0\) because \(GL_n\) is a special group), Popov's exact sequence gives
where \(H\) is the stabilizer of the standard frame \(\{e_1, \ldots, e_m\}\).
Case \(m = 1\). The stabilizer is \(H \cong GL_{n-1}\) (matrices fixing \(e_1\), acting freely on the complement), so \(H' = SL_n \cap H \cong SL_{n-1}\) and \(\operatorname{X}(SL_{n-1}) = 0\). Hence \(\operatorname{Pic}(S_{n,1}) = 0\).
Case \(n = m\). The stabilizer is \(H = S_n\) (permutation matrices), so \(H' = H \cap SL_n = A_n\) (even permutations). The Picard group depends on the abelianization of \(A_n\):
| \(n\) | \(A_n\) | \(A_n^{\mathrm{ab}} = A_n/[A_n, A_n]\) | \(\operatorname{Pic}(S_{n,n})\) |
|---|---|---|---|
| \(1, 2\) | trivial | \(0\) | \(0\) |
| \(3\) | \(\mathbb{Z}/3\mathbb{Z}\) (cyclic) | \(\mathbb{Z}/3\mathbb{Z}\) | \(\mathbb{Z}/3\mathbb{Z}\) |
| \(4\) | order 12; \([A_4, A_4] = V_4\) | \(\mathbb{Z}/3\mathbb{Z}\) | \(\mathbb{Z}/3\mathbb{Z}\) |
| \(\geq 5\) | simple | \(0\) | \(0\) |
For \(n = 4\): the commutator subgroup \([A_4, A_4]\) is the Klein four-group \(V_4 \cong (\mathbb{Z}/2\mathbb{Z})^2\), so \(A_4^{\mathrm{ab}} \cong A_4/V_4 \cong \mathbb{Z}/3\mathbb{Z}\). For \(n \geq 5\): \(A_n\) is simple, so \([A_n, A_n] = A_n\) and \(\operatorname{X}(A_n) = 0\).
Case \(n > m > 1\). The stabilizer \(H\) consists of block matrices that permute the first \(m\) basis vectors and act invertibly on the remaining \(n - m\) coordinates. Setting \(H' = H \cap SL_n\), one shows \(H'/[H', H'] \cong \mathbb{Z}/2\mathbb{Z}\): the only nontrivial character detects the sign of the permutation part (modulo the determinant-1 constraint). Hence