Cohomological aspects of power ideals

1. Problem Statement and Context

The paper addresses the structural and combinatorial properties of zonotopal algebras and their generalizations, specifically power ideals associated with hyperplane arrangements.

• Background: Zonotopal algebras (introduced by Holtz and Ron, and generalized by Ardila and Postnikov) are quotients of polynomial rings defined by ideals generated by powers of linear forms. They arise in approximation theory, combinatorics, and algebraic geometry.
• The Gap: While the Hilbert series of standard zonotopal algebras (for parameters $k=0, -1, -2$) are known to be combinatorially determined by the Tutte polynomial of the arrangement, the behavior for other parameters (particularly $k \le -2$ and $k \ge 1$) is less understood.
• Specific Challenges:
  • Proving the right-exactness of deletion-contraction sequences for internal zonotopal algebras ($k=-2$) has historically been difficult and required ad-hoc combinatorial arguments.
  • The "superspace" generalization of zonotopal algebras (involving differential forms) introduced by Rhoades, Tewari, and Wilson (RTW) lacked a unified geometric interpretation for all parameters $k$.
  • Conjectures regarding the Hilbert series of superspace zonotopal algebras remained unproven.

2. Methodology

The authors employ a geometric approach using augmented wonderful varieties ($W_L$) associated with a hyperplane arrangement $L$.

• Augmented Wonderful Varieties: These are smooth, projective compactifications of the vector space $L$, constructed by iteratively blowing up the projective completion $\mathbb{P}(L \oplus \mathbb{C})$ along strata corresponding to flats of the arrangement.
• Cohomological Interpretation: The core methodology involves identifying the sections of specific line bundles on $W_L$ with the inverse systems (Macaulay duals) of power ideals.
  • They establish that the space of global sections $H^0(W_L, \mathcal{O}(D))$ forms a coalgebra under the action of the additive group $L$.
  • They relate the inverse system of a power ideal to the space of sections of line bundles $\mathcal{O}(1, \dots, 1)(k\alpha)$ and their twists by the log-cotangent bundle $\Omega_L$.
• Vanishing Theorems: A crucial technical component is the proof of cohomology vanishing results ($H^i(W_L, \mathcal{F}) = 0$ for $i > 0$) for specific twists of line bundles and log-cotangent bundles. This allows the authors to convert long exact sequences in cohomology into short exact sequences of global sections.
• Superspace and Logarithmic Forms: For the superspace case, they utilize the log-cotangent bundle $\Omega_L$ (forms with logarithmic poles along the boundary) to interpret sections as differential forms on $L$, linking them to the ring of polynomial differential forms (superspace).

3. Key Contributions and Results

A. Geometric Characterization of Power Ideals

• Theorem 1.1: The authors prove that the inverse system of a general power ideal (defined by exponents $a_F$ for each flat $F$) is isomorphic to the space of global sections $H^0(W_L, \mathcal{O}(\sum a_F D_F))$. This generalizes previous results restricted to specific cases.
• Corollary 1.2: They identify the inverse systems of standard zonotopal algebras ($Z_{L,k}$) with sections of line bundles $\mathcal{O}(1, \dots, 1)(k\alpha)$ on $W_L$.

B. Resolution of Exactness and Recursion

• Deletion-Contraction Sequences: The paper provides a unified proof for the right-exactness of the deletion-contraction short exact sequence for zonotopal algebras for all $k \ge -2$.
  • Previously, the case $k=-2$ (internal zonotopal algebra) was difficult; the authors prove it using the vanishing of $H^1(W_L, \mathcal{O}(1, \dots, 1)(-2\alpha))$ via induction and properties of the log-cotangent bundle.
  • This confirms that the Hilbert series of $Z_{L,k}$ satisfies a deletion-contraction recursion for $k \ge -2$, proving it is combinatorially determined by the matroid.
• Hierarchical Zonotopal Algebras: They extend these results to hierarchical zonotopal algebras (introduced by Lenz), recovering dimension formulas via cohomological vanishing.

C. Superspace Zonotopal Algebras

• Geometric Interpretation (Theorem 1.5): For $k \le 0$, the inverse system of the superspace zonotopal algebra $Z_{L,k}$ is identified with the direct sum of sections of line bundles twisted by exterior powers of the log-cotangent bundle:
  $$C_{L,k} = \bigoplus_j H^0(W_L, \mathcal{O}(1, \dots, 1)(k\alpha) \otimes \wedge^j \Omega_L)$$
• Exact Sequences (Theorem 1.6): They establish an exact sequence for superspace algebras involving deletion and contraction, which includes a splitting term dependent on the choice of a vector.
• Proof of Conjecture (Corollary 1.7): Using the exact sequence and vanishing results, they prove a conjecture by Rhoades, Tewari, and Wilson regarding the bigraded Hilbert series of the superspace central zonotopal algebra ($k=-1$):
  $$\text{Hilb}(Z_{L,-1}) = (1+t)^r q^{n-r} T_L\left(\frac{1}{1+t}, \frac{1}{q}\right)$$
  where $T_L$ is the Tutte polynomial.

D. Exactness of the Differential Complex

• Theorem 1.8: They prove that the exterior derivative $d$ acts as an exact differential on the inverse system of the superspace algebra (for $k \ge -1$), except in bidegree $(0,0)$.
• Categorification of Brylawski's Relations: This exactness implies linear relations among the coefficients of the Tutte polynomial for matroids realizable over $\mathbb{C}$. The authors show these relations categorify a classification by Brylawski, providing a geometric proof for these combinatorial identities.

E. Extensions to $k \ge 1$

• For $k \ge 1$, the standard differential closure of the ideal does not behave well. The authors define a new quotient algebra $W_{L,k}$ via the sections of the line bundles. They show this new algebra satisfies the deletion-contraction recursion and has a combinatorially determined Hilbert series (a specialization of the "umbral Tutte polynomial"), even though it differs from the standard superspace quotient.

4. Significance

1. Unification: The paper unifies the study of various zonotopal algebras (external, central, internal, hierarchical, and superspace) under a single geometric framework involving augmented wonderful varieties.
2. Proof of Conjectures: It resolves open conjectures regarding the Hilbert series of superspace zonotopal algebras and the exactness of deletion-contraction sequences for the internal case ($k=-2$).
3. New Tools: The introduction of cohomological vanishing theorems for log-cotangent bundles on augmented wonderful varieties provides a powerful new tool for studying hyperplane arrangement invariants, potentially applicable beyond zonotopal algebras.
4. Combinatorial-Geometric Bridge: By linking the Hilbert series of these algebras directly to the cohomology of line bundles, the authors provide a "categorification" of classical combinatorial identities (Brylawski's relations) involving the Tutte polynomial.
5. Generalization: The results extend the understanding of power ideals to cases where the Hilbert series was previously unknown or suspected to be non-combinatorial, showing that while the standard algebraic construction fails for $k \ge 1$, a geometrically defined alternative retains the desirable combinatorial properties.

In summary, Crowley and Larson successfully translate algebraic problems concerning power ideals and zonotopal algebras into the language of algebraic geometry (line bundles and cohomology on wonderful varieties), leveraging vanishing theorems to prove deep structural results and resolve long-standing conjectures.