Multigraded Regularity of the Complete Flag Variety

Technical Summary: Multigraded Regularity of the Complete Flag Variety

Problem Statement
This paper addresses the computation of the multigraded regularity of the complete flag variety, $Fl(\mathbb{C}^n)$, under the Plücker embedding. While Maclagan and Smith introduced multigraded regularity to generalize Castelnuovo-Mumford regularity to products of projective spaces, explicit computations for specific families of varieties remain scarce. The authors focus on the complete flag variety, a class of varieties possessing rich combinatorial structure and a natural multigrading induced by the Plücker embedding into a product of projective spaces $\mathbb{P}^{\binom{n}{1}-1} \times \cdots \times \mathbb{P}^{\binom{n}{n-1}-1}$. The central question is to determine the set of multidegrees $d \in \mathbb{Z}^{n-1}$ for which the coherent sheaf $\iota^*\mathcal{O}_{Fl(\mathbb{C}^n)}$ is $d$-regular, where $\iota$ denotes the Plücker embedding.

Methodology
The study leverages the interplay between combinatorial data and multigraded commutative algebra. The primary theoretical tool is the Borel-Weil-Bott theorem, which provides a complete combinatorial description of the cohomology of line bundles on complete flag varieties. Specifically, the authors utilize Bott's algorithm (and its inversion-counting variant) to determine the vanishing of cohomology groups $H^i(F, \mathcal{O}_F(a))$.

The definition of multigraded regularity requires the vanishing of cohomology groups $H^i(P, \mathcal{G}(d-c))$ for all $i > 0$ and all $c \in \mathbb{N}^{n-1}$ with $|c|=i$. The authors establish that for the complete flag variety, the regularity of the defining ideal $S/I$ coincides with the regularity of the associated coherent sheaf $\iota^*\mathcal{O}_{Fl(\mathbb{C}^n)}$.

To manage the computational complexity as $n$ increases, the paper develops inductive relationships between the regularity regions of $Fl(\mathbb{C}^n)$ and $Fl(\mathbb{C}^{n-1})$. These relationships allow the authors to derive inner and outer bounds for the regularity region of $Fl(\mathbb{C}^{n+1})$ based on the computed region for $Fl(\mathbb{C}^n)$. For small values of $n$, the authors perform explicit computations using the software Macaulay2, verifying cohomology vanishing conditions via the inversion-counting criteria derived from Bott's theorem.

Key Contributions and Results

1. Inductive Bounds (Theorems A and B):
   The paper proves two critical inductive propositions regarding the propagation of regularity:

   • Propagation of Non-Regularity: If a multidegree $a$ is not in the regularity region of $Fl(\mathbb{C}^n)$, then for any integer $b$, the extended degrees $(a, b)$ and $(b, a)$ are not in the regularity region of $Fl(\mathbb{C}^{n+1})$. This provides a method for establishing outer bounds.
   • Propagation of Regularity: If $a$ is in the regularity region of $Fl(\mathbb{C}^n)$, then $(a, b)$ and $(b, a)$ are in the regularity region of $Fl(\mathbb{C}^{n+1})$, provided $b$ is sufficiently large (specifically $b \geq \dim(Fl(\mathbb{C}^{n+1}))$). This provides inner bounds.

2. Explicit Computations for Low $n$ (Theorem A):
   The authors provide a complete description of the minimal elements of the regularity region for $n \leq 6$:

   • $n=3$: The regularity region is $\mathbb{N}^2$ (minimal element $(0,0)$).
   • $n=4$: The minimal elements are $(1,0,0), (0,1,0), (0,0,1)$. Verifying these requires checking 83 cohomology conditions.
   • $n=5$: There are 19 minimal elements.
   • $n=6$: There are 179 minimal elements.
     The number of cohomology conditions required to verify a single element grows rapidly (e.g., 15,503 conditions for $n=6$), necessitating computational assistance.

3. Asymptotic Behavior (Theorem C):
   By analyzing the minimal total degree $m(n) = \min \{|d| : d \in \text{reg}(Fl(\mathbb{C}^n))\}$, the authors establish bounds on the growth of the regularity region:

   • Lower Bound: $m(n) \geq \frac{6}{5}n - 6$ for $n \geq 7$.
   • Upper Bound: $m(n) \leq \frac{n^3 - n - 24}{6}$.
     This demonstrates that the growth of the minimal total degree is at least linear and at most cubic in $n$.

4. Standard Graded Regularity:
   The paper also computes the standard graded regularity of the flag variety when embedded in a single projective space via the Segre embedding of the Plücker image. The regularity is shown to be $d-1$, where $d = \dim(Fl(\mathbb{C}^n)) = \binom{n}{2}$.

Significance and Claims
The paper claims to provide the first complete description of the multigraded regularity for the complete flag variety for a range of $n$, moving beyond theoretical definitions to explicit computational results. It highlights the difficulty of the problem, noting that the number of conditions to verify grows factorially with $n$.

The significance lies in bridging combinatorial algebraic geometry and multigraded commutative algebra. By utilizing the combinatorial structure of the flag variety (specifically the Bruhat order and the Borel-Weil-Bott theorem), the authors demonstrate that multigraded regularity is indeed encoded in combinatorial data, similar to how Castelnuovo-Mumford regularity is encoded for Grassmannians and Schubert varieties.

The authors remain modest regarding the general case for arbitrary $n$. While they conjecture that the growth of $m(n)$ is quadratic (specifically $m(n) = \binom{n-2}{2}$ for $n \geq 4$) and that the regularity region along the diagonal $(a, \dots, a)$ is determined by $a \geq \frac{n-3}{2}$, these remain open questions. The paper does not claim to have solved the problem for all $n$, but rather establishes a framework of inductive bounds and provides a concrete dataset for low dimensions that suggests the asymptotic behavior.