Picard Stacks in Algebraic Geometry

• Picard stacks are stacks in groupoids with a strictly commutative group law that serve as universal moduli objects for families of line bundles in various geometric frameworks.
• Variants such as universal, holomorphic, and logarithmic Picard stacks parameterize moduli of curves and line bundles, incorporating smooth, gerbe, and log structures.
• Tropical subdivisions and logarithmic modifications resolve representability issues in Picard stacks, linking tautological classes, cohomological invariants, and mapping class group actions.

A Picard stack is a stack in groupoids equipped with a strictly commutative, associative group law up to specified natural isomorphisms, serving as a fundamental moduli object in algebraic geometry and related fields. Variants such as logarithmic Picard stacks, holomorphic Picard stacks, and universal Picard stacks, arise as moduli of line bundles under various geometric, topological, and logarithmic enhancements, providing extensive applications to the theory of moduli spaces, abelian varieties, tautological classes, and tropical and logarithmic geometry.

1. Structure and Algebraicity of Picard Stacks

A Picard stack $P$ over a site $S$ is defined as a stack in groupoids $(\mathcal{C}_S,\mathcal{J}_S)$ equipped with:

• a group law $+ : P \times_S P \to P$,
• associator $$a: + \circ (+ \times id) \Rightarrow + \circ (id \times +)$$,
• commutator $c: + \circ s \Rightarrow +$ (where $s(X,Y) = (Y,X)$, the swap).

Strict commutativity entails that $c_{X,X}$ is the identity for every $X \in P(U)$, for $U$ in the site.

The 2-stack of strictly commutative Picard stacks admits an algebraic 2-stack structure, via equivalence with the 2-stack of 2-algebras over a certain algebraic 2-stack theory $T_1$, where:

• Objects: trivial Picard stacks $1^n$ for $n \geq 1$.
• 1-arrows: given by integer matrices $P: 1^n \to 1^k$.
• 2-arrows: unique identities.

A 2-algebra for $T_1$ is a 2-functor $A: T_1 \to S$ with $A(P)$ realizing the group law via the matrix $P$. This encodes the entire Picard category structure, including associativity and commutativity constraints, in terms of formal matrix identities, ensuring algebraicity of the 2-stack of strictly commutative Picard stacks (Bertolin et al., 2023).

Examples include:

• The trivial Picard stack.
• The stack of invertible sheaves with tensor product.
• Discrete Picard stacks from an abelian sheaf.

2. Universal, Holomorphic, and Logarithmic Picard Stacks

Different geometric frameworks give rise to refinements of the Picard stack.

(a) Universal Picard Stack

The universal Picard stack $\mathfrak{Pic}_{g,n,d}$ parametrizes tuples $(C \to S, p_1, \ldots, p_n, L)$, where $C \to S$ is a family of prestable genus-$g$ $n$-marked curves and $L$ is a line bundle of degree $d$ on $C$. It is a smooth Artin stack, locally of finite type over the base, and is a $\mathbb{G}_m$-gerbe over the relative Picard algebraic space, whose fibers are isomorphism classes of line bundles (Bae et al., 2020).

(b) Holomorphic Picard Stack

For families of Riemann surfaces of genus $g$, the holomorphic Picard stack $\mathrm{Hol}_g^{(k)}$ parametrizes holomorphic families together with a holomorphic line bundle of fiberwise degree $k$. There is a morphism $$\Phi_g^{(k)}: \mathrm{Hol}_g^{(k)} \to \mathrm{Pic}_g^{(k)}$$ (the moduli stack of families with a choice of line bundle class), making $\mathrm{Hol}_g^{(k)}$ into a $\mathrm{C}^\times$-gerbe over $\mathrm{Pic}_g^{(k)}$ (Ebert et al., 2010).

(c) Logarithmic Picard Stack

Given a family of logarithmic curves $X \to S$ over a fine, saturated log scheme, the logarithmic Picard stack

$\LogPic_{X/S} = \mathbf{B}G^\mathrm{bounded}$

classifies $G$-torsors (with $G = M_X^{\mathrm{gp}}$) equipped with bounded monodromy. It is a commutative group stack in the log-étale topology, fitting into the exact sequence

$0 \longrightarrow \Jac_{X/S} = \mathrm{Pic}^{[0]}(X/S) \longrightarrow \LogPic_{X/S} \longrightarrow \TropJac_{X/S} \longrightarrow 0,$

where $\Jac$ is the multidegree 0 algebraic Jacobian and $\TropJac$ is the tropical Picard sheaf (Molcho et al., 2018).

3. Representability, Logarithmic Modifications, and Tropicalization

The representability properties of Picard stacks encode their geometric complexity:

• The logarithmic Picard stack $\LogPic_{X/S}$ is a proper, logarithmically smooth group stack with finite diagonal, and its connected component of the identity, $\LogJac_{X/S} = \LogPic^0_{X/S}$, is a log abelian variety in the sense of Kajiwara–Kato–Nakayama.
• There is an obstruction to representability by an algebraic stack because the log multiplicative group $\mathbf{G}_m^{\log}(T)$ is not algebraic; since $G$ acts by automorphisms on log line bundles, no underlying algebraic stack structure exists for $\LogPic$.

To address non-representability, one employs logarithmic modifications via tropical subdivisions:

• Tropicalization associates to $X \to S$ a dual graph $\Gamma_s$ with edge-lengths in $\overline{M}_{S,s}$.
• The tropical Picard group $\TropJac_{X/S}$ arises as $R^1\pi_*L$, where $L$ is the sheaf of linear (balancing) functions on $\Gamma_s$.
• Tropical subdivisions refine cone decompositions, creating ind-algebraic covers by cone stacks.
• Pullbacks along such subdivisions yield logarithmic modifications of $\LogPic$ that are representable by proper, toroidal (or toricoidal) algebraic spaces with log structure.

Every tropical subdivision produces a proper, log-étale, schematic cover,

$\bigsqcup_{\sigma} \mathrm{Spec}(\text{monoid algebra for subdivision cell } \sigma) \longrightarrow \LogPic_{X/S},$

so tropical combinatorics resolves the representability failure (Molcho et al., 2018).

4. Exact Sequences and Tautological Class Calculus

Key exact sequences capture the relation between the various objects:

• For the logarithmic Picard stack: $0 \longrightarrow \mathrm{Pic}^{[0]}(X/S) \longrightarrow \LogPic_{X/S} \longrightarrow \TropJac_{X/S} \longrightarrow 0.$
• For the tropical Abel exact sequence (on a fiber), with $\Gamma$ the dual graph and $L$ the sheaf of balancing functions: $$0 \longrightarrow H_1(\Gamma) \xrightarrow{\partial} \mathrm{Hom}(H_1(\Gamma), \overline{M}_S^{\mathrm{gp}}) \longrightarrow H^1(\Gamma, L) \xrightarrow{\deg} \mathbb{Z}^{V} \longrightarrow 0.$$

Mumford’s tautological formula, and various tautological class constructions on the universal Picard stack, arise in subsequent developments (Chiodo et al., 2023, Bae et al., 2020); for example, Chern character formulas for the derived pushforward of $r$th power roots of the universal line bundle are central to connecting the stack-theoretic invariants to explicit cycle classes.

5. Abel–Jacobi Theory, Double Ramification Cycles, and Tautological Relations

The interplay between Abel–Jacobi theory and Picard stacks is realized through universal cycle constructions:

• The universal double ramification cycle $\mathsf{DR}^{\mathsf{op}}_{g,A}$ is defined as the operational pushforward $(AJ)_*[Div_{g,A}]$ on the universal Picard stack $\mathfrak{Pic}_{g,n,d}$, serving as the closure of the Abel–Jacobi locus under logarithmic or b-Chow resolutions (Bae et al., 2020).
• The cycle is computed by Pixton’s tautological formula, which sums over decorated graphs, mod $r$ weightings, and generates the tautological ring’s relations.
• Vanishing theorems for Pixton’s cycles in codimension $>g$ produce infinite families of tautological relations on the universal Picard stack, pulling back to moduli of curves and Gromov–Witten theory.
• For $r$-spin and logarithmic generalizations, the universal Chern character formulas allow the expression of log double-ramification cycles in terms of tautological generators, elucidating the structure of Chiodo’s classes and their relation to DR cycles (Chiodo et al., 2023).

6. Cohomology, Gerbe Structures, and Mapping Class Groups

Cohomological analysis of Picard stacks reveals rich topological behavior:

• The holomorphic Picard stack $\mathrm{Hol}_g^{(k)}$ is a $\mathrm{C}^\times$-gerbe over $\mathrm{Pic}_g^{(k)}$, with a principal $K(\mathbb{Z},2)$ fibration on classifying spaces. The obstruction to a universal Poincaré bundle is measured by the Dixmier–Douady class $\theta \in H^3(|\mathrm{Pic}_g^{(k)}|;\mathbb{Z})$.
• The holomorphic Picard group is isomorphic to the topological Picard group $H^2$, with explicit generators given by the Hodge class $\lambda$ and a further generator $\eta$. The rational cohomology in the stable range is generated by universal Miller–Morita–Mumford classes (Ebert et al., 2010).
• The structure of $\mathrm{Hol}_g^{(k)}$ reflects the geometry of the extended mapping class group, via central extensions and homological stability theorems (Madsen–Weiss, Cohen–Madsen).
• In low degrees: $$H^2(|\mathrm{Pic}_g^{(k)}|;\mathbb{Z}) \cong \mathbb{Z}^2 \cong \langle \lambda, \eta \rangle$$, and torsion in $H^3$ corresponds to the gerbe class. The topological and algebro-geometric models exhibit full correspondence via classifying spaces.

7. Summary and Implications

The Picard stack, in its various incarnations, provides a universal moduli object encoding families of line bundles, abelian group structures, and intricate relationships between algebraic, topological, and logarithmic geometry. The logarithmic Picard stack achieves properness and log smoothness but is non-representable algebraically; tropical and log modifications restore representability locally via cone stacks and toroidal models. Universal cycle classes, such as the double ramification cycle, emerge from operational Chow theory and have explicit computational models via tautological generators and Pixton’s formula. Cohomological structures, gerbe phenomena, and connections with mapping class groups further illustrate the depth of the subject and its centrality to modern moduli theory.