---
title: 'Lower-Upper Varieties: Schemes, Lattices, Nef Cones'
url: https://www.emergentmind.com/topics/lower-upper-varieties
type: topic
---

# Lower-Upper Varieties: Schemes, Lattices, Nef Cones

“Lower-Upper Varieties” is not a single standardized notion. In the literature, the phrase and its close variants occur in several distinct settings united by a common pattern: a geometric, combinatorial, or lattice-theoretic object is constrained simultaneously by a “lower” condition and an “upper” condition. The most literal usage is the matrix-geometric one, where the lower-upper scheme
\[
E=\{(X,Y): XY \text{ lower triangular},\ YX \text{ upper triangular}\}
\]
has irreducible components \(E_\pi\) called lower-upper varieties [2509.01857]. In lattice theory, the phrase naturally refers to varieties that are both lower-modular and upper-modular, a class completely determined in the lattice \(\mathbf{EPI}\) of epigroup varieties [1408.0356]. In birational and toric geometry, one also encounters a “lower-upper” structure on invariants such as nef cones, where explicit polyhedral lower bounds and upper bounds are extracted from toric embeddings [1009.0220].

## 1. Terminological scope

In the available literature, “lower-upper” has at least three mathematically precise uses.

| Setting | Object | Characteristic statement |
|---|---|---|
| Matrix geometry | Lower-upper varieties \(E_\pi\) | \(XY\) lower triangular and \(YX\) upper triangular |
| Lattice theory | Varieties both lower-modular and upper-modular | In \(\mathbf{EPI}\), these are exactly four neutral varieties |
| Nef-cone approximation | Lower-upper structure on \(\Nef(Y)\) | Polyhedral lower and upper bounds from a toric embedding |

The matrix-geometric usage is the most literal. For \(m,n\in\mathbb N\), the rectangular lower-upper scheme is
\[
E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\  X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},
\]
and its irreducible components \(E_\pi\), indexed by partial permutations \(\pi\), are the lower-upper varieties in the strict geometric sense [2509.01857]. In the square case \(m=n\), later work recalls that this scheme was introduced in \([{\rm Knu05}]\), while the rectangular case is needed for induction and for the hybrid-pipe-dream formalism [2509.01857].

The lattice-theoretic usage is different. There, “lower-upper varieties” are varieties that are simultaneously lower-modular and upper-modular as elements of a lattice of varieties. In \(\mathbf{EPI}\), the interaction is especially rigid: an epigroup variety is neutral if and only if it is both lower-modular and upper-modular [1408.0356]. A broader survey shows analogous questions for \(\mathbf{SEM}\), \(\mathbf{COM}\), and related sublattices, with complete results in some cases and open problems in others [1309.0228].

A third usage is best read as a “lower-upper structure” attached to a variety rather than a new class of varieties. Given a projective variety \(Y\) embedded in a toric variety \(X_\Delta\), explicit cones
\[
\mathcal G_\Delta,\qquad \mathcal L_\Delta,\qquad \mathcal F_{\Delta,\mathbf w}
\]
provide polyhedral lower bounds and upper bounds for \(\Nef(Y)\), yielding a systematic lower-upper approximation framework [1009.0220].

## 2. Lower-upper schemes and their irreducible components

The geometric theory begins with the scheme
\[
E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\  X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},
\]
whose defining equations are
\[
(XY)_{ii'}=0 \quad (i<i'),\qquad (YX)_{jj'}=0 \quad (j>j').
\]
Thus \(XY\) is forced to lie in the lower-triangular sector and \(YX\) in the upper-triangular sector [2509.01857]. The square case \(m=n\) was introduced earlier, but the rectangular generalization is structurally important because many inductive arguments pass through it [2509.01857].

The irreducible components are indexed by maximal-rank \(m\times n\) partial permutation matrices, equivalently by injective maps \(\pi\in {}_{m,n}\). For such \(\pi\),
\[
E_\pi := \overline{ \left\{(X,Y) \in E\colon (YX)_{\pi(i)\pi(i)} = (XY)_{ii}\text{ distinct nonzero}\right\} }
\]
and equivalently
\[
E_\pi = \overline{(B_-\times B_+)\cdot \{(\pi,\pi^{T} s)\colon s\ \text{\(m\times m\) diagonal} \} }.
\]
Here \(B_-\) and \(B_+\) are the invertible lower- and upper-triangular groups of the appropriate sizes [2509.01857]. Geometrically, \(E_\pi\) is the closure of the locus where the nonzero diagonal entries of \(XY\) and \(YX\) are matched by \(\pi\).

A central reformulation is the flux equation
\[
(YX)_{jj} = \begin{cases} (XY)_{ii}&\text{if }j=\pi(i),\qquad i=1,\ldots,m,\\ 0&\text{else.} \end{cases}
\]
This identifies the diagonal entry \((XY)_{ii}\) with \((YX)_{\pi(i)\pi(i)}\) and is the algebraic origin of the pipe-dream connectivity data [2509.01857]. In the square case, the same components are indexed by permutations \(w\in S_n\), and one has
\[
E_w := \overline{\{(X,Y)\in E:\ \operatorname{diag}(XY)= w\cdot \operatorname{diag}(YX)\ \text{ nonrepeating}\}}
\]
as well as
\[
E_w = \overline{(B_-\times B_+)\cdot \{(w,\; w^{-1}D)\colon D \text{ diagonal}\}},
\]
which makes the \(B_-\times B_+\)-orbit geometry completely explicit [2411.11208].

The distinction between the global scheme \(E\) and its irreducible varieties \(E_\pi\) is essential. The scheme structure matters in degeneration arguments, and later work repeatedly notes that flat limits may have embedded components, even though those lower-dimensional embedded pieces do not affect the top-dimensional equivariant class formula [2509.01857].

## 3. Pipe dreams, equivariant classes, and degeneration theory

The modern structure theory of lower-upper varieties is combinatorial. In the square case, generic pipe dreams produce a polynomial
\[
G_w := \sum_{\delta\in GPDs(w)} wt(\delta),
\]
and the fundamental class formula is
\[
[E_w] = (A+B)^{-n}\, G_w.
\]
Equivalently, the \((B_-\times B_+\times (\mathbb C^\times)^2)\)-equivariant cohomology class of \(E_w\) is computed by a sum over generic pipe dreams with tile weights \(A+x_i-y_j\), \(B-x_i+y_j\), and \(A+B\), according to tile type [2411.11208].

The rectangular theory sharpens this. For each hybridization \(\beta\), hybrid generic pipe dreams yield a polynomial \(G_\pi^\beta\), and Theorem \(\ref{thm:beta}\) shows that it is independent of \(\beta\), so the notation becomes \(G_\pi\). The central class formula is
\[
G_\pi=(A+B)^m\,[E_\pi].
\]
Here the torus weights are
\[
wt_T(X_{ij}) = A+x_i-y_j,\qquad wt_T(Y_{ji})  = B-x_i+y_j,
\]
so \([E_\pi]\) lies in
\[
H_T^*(pt)\cong \mathbb Z[x_1,\ldots,x_m,y_1,\ldots,y_n,A,B].
\]
The same work proves a divided-difference type recurrence for both \(G_\pi\) and \([E_\pi]\), thereby identifying the combinatorial and geometric quantities exactly [2509.01857].

The degeneration theorem explains why the combinatorics is not ad hoc. For each hybridization \(\beta\), there is a degeneration of \(E_\pi\) to a union of complete intersections \(V_\delta\), indexed by hybrid generic pipe dreams \(\delta\), and the contribution of \(\delta\) to \(G_\pi\) is \((A+B)^m[V_\delta]\) [2509.01857]. In the square case, an analogous degeneration yields a union of quadratic complete intersections \(F_\delta\), possibly together with lower-dimensional embedded components, and the class of each \(F_\delta\) is the corresponding summand in \((A+B)^{-n}G_w\) [2411.11208].

One of the conceptual innovations is the flux formalism. Fluxes \(\Phi_e\) are attached to edges of the matrix grid, and local equalities among these fluxes reconstruct the pipe-dream conditions. In the rectangular theory, pipe dreams are derived from equalities among edge fluxes rather than postulated independently; in the square theory, the defining ideal
\[
\mathcal I=\langle (XY)_>,\ (YX)_<\rangle
\]
degenerates in a way that preserves flux equations and yields local relations such as
\[
X_{ij}(\Phi_S-\Phi_E)=0,
\]
forcing the admissible tile patterns [2509.01857] [2411.11208].

These formulas place lower-upper varieties between matrix Schubert geometry and commuting-variety geometry. The projection \((X,Y)\mapsto X\) maps \(E_\pi\) onto the rectangular matrix Schubert variety \(X_\pi\) [2509.01857], while in the square case \(\deg C_n=\deg E_1\) for the commuting variety \(C_n\), and
\[
\deg E_w=\sum_{\delta\in GPDs(w)} 2^{\#(\text{tiles with turns})-n}
\]
gives a concrete degree formula [2411.11208]. The \(B\to\infty\) or \(A\to\infty\) limits recover classic and bumpless pipe dream formulas for double Schubert polynomials, showing that lower-upper geometry is strictly richer than either specialization [2509.01857] [2411.11208].

## 4. Lower-upper varieties in lattices of algebraic varieties

In lattice theory, “lower-upper varieties” refers to varieties that are both lower-modular and upper-modular as elements of a lattice of varieties. The cleanest complete result is in \(\mathbf{EPI}\), the lattice of all epigroup varieties. There, Theorem 1.1 states that for an epigroup variety \(\mathcal V\), the following are equivalent: \(\mathcal V\) is neutral; \(\mathcal V\) is costandard; \(\mathcal V\) is simultaneously lower-modular and upper-modular; and
\[
\mathcal V\in\{\mathcal T,\mathcal{SL},\mathcal{ZM},\mathcal{SL}\vee\mathcal{ZM}\}.
\]
Thus the lower-upper epigroup varieties are exactly four neutral varieties [1408.0356].

This theorem is rigid because the separate lower-modular and upper-modular classifications are much larger. Lower-modular epigroup varieties are completely classified by
\[
\mathcal V=\mathcal M\vee \mathcal N
\]
with \(\mathcal M\in\{\mathcal T,\mathcal{SL}\}\) and \(\mathcal N\) \(0\)-reduced [1408.0356]. By contrast, upper-modular epigroup varieties are completely classified only in the strongly permutative case: either
\[
\mathcal V=\mathcal M\vee \mathcal N
\]
with \(\mathcal M\in\{\mathcal T,\mathcal{SL}\}\) and \(\mathcal N\) commutative satisfying
\[
x^2y=xy^2,
\]
or
\[
\mathcal V=\mathcal G\vee \mathcal C_m\vee \mathcal N,\qquad 0\le m\le 2,
\]
with \(\mathcal G\) Abelian and \(\mathcal N\) commutative satisfying
\[
x^2y=0.
\]
The simultaneous lower-upper condition collapses this much larger landscape to four points [1408.0356].

The semigroup-variety survey shows that analogous phenomena occur in \(\mathbf{SEM}\) and \(\mathbf{COM}\), but with different final lists. In \(\mathbf{SEM}\), a semigroup variety is both lower-modular and upper-modular if and only if it is both distributive and codistributive, equivalently costandard, equivalently neutral, and this happens exactly for
\[
\mathcal T,\ \mathcal{SL},\ \mathcal{ZM},\ \mathcal{SL}\vee \mathcal{ZM},\ \mathbf{SEM}.
\]
In \(\mathbf{COM}\), simultaneous COM-lower-modularity and COM-upper-modularity is equivalent to COM-neutrality and is characterized by \(\mathbf{COM}\) or by varieties of the form \(\mathcal M\vee\mathcal N\) with \(\mathcal M\in\{\mathcal T,\mathcal{SL}\}\) and \(\mathcal N\models x^2y\approx 0\) [1309.0228].

For commutative semigroup varieties, the “upper” side is itself unusually rigid. A commutative semigroup variety \(\mathcal V\) is upper-modular if and only if it is codistributive, and this happens exactly in one of three cases:
\[
\mathcal V=\mathcal{COM},
\]
or
\[
\mathcal V=\mathcal M\vee \mathcal N,\qquad \mathcal M\in\{\mathcal T,\mathcal{SL}\},
\]
with \(\mathcal N\) satisfying
\[
x^2yz=0,\qquad x^2y=xy^2,
\]
or
\[
\mathcal V=\mathcal G\vee \mathcal M\vee \mathcal N,
\]
with \(\mathcal G\) an Abelian periodic group variety, \(\mathcal M\in\{\mathcal T,\mathcal{SL},\mathcal C_2\}\), and \(\mathcal N\models x^2y=0\). Moreover, costandard is exactly the modular plus upper-modular case [1501.02650].

A common misconception is that lower-modular and upper-modular classifications are known in equal generality. The literature says otherwise. Lower-modular elements are completely classified in several settings, including \(\mathbf{EPI}\), \(\mathbf{SEM}\), and \(\mathbf{COM}\), while upper-modular classifications are often complete only under additional hypotheses such as strong permutativity or commutativity [1408.0356] [1309.0228] [1501.02650].

## 5. Lower-upper structures on nef cones

A different but conceptually related use of “lower-upper” appears in the study of nef cones. Given an embedding
\[
i:Y\hookrightarrow X_\Delta
\]
of a \(d\)-dimensional projective variety into an \(n\)-dimensional normal toric variety, one constructs three cones in \(\Pic(X_\Delta)_\mathbb R\):
\[
\mathcal G_\Delta,\qquad \mathcal L_\Delta,\qquad \mathcal F_{\Delta,\mathbf w},
\]
together with
\[
\mathcal F_\Delta:=\bigcap_{\mathbf w\in W}\mathcal F_{\Delta,\mathbf w}.
\]
Pulling back via \(i^*\), one obtains explicit lower and upper bounds for \(\Nef(Y)\) [1009.0220].

The lower cones have distinct geometric meanings. The cone
\[
\mathcal G_\Delta = \pos\big([D]\in \Pic(X_\Delta): D \text{ is globally generated}\big)
\]
is the strongest lower bound and coincides with the toric nef cone in the projective toric case. The cone
\[
\mathcal L_\Delta = \{[D]\in \Pic(X_\Delta)_\mathbb R:\ i_\sigma^*([D])\text{ is effective for all }\sigma\in \Delta\}
\]
is larger and tests effectivity on all orbit closures. The upper cone \(\mathcal F_{\Delta,\mathbf w}\) is defined by codimension-one orbit tests, and for pure \(\Delta\) one has
\[
\mathcal G_\Delta \subseteq \mathcal L_\Delta \subseteq \mathcal F_\Delta.
\]
If \(X_\Delta\) is projective toric, then all three coincide and equal \(\Nef(X_\Delta)\) [1009.0220].

The main theorem states that for any projective \(Y\),
\[
i^*(\mathcal G_\Delta)\subseteq i^*(\mathcal L_\Delta)\subseteq \Nef(Y).
\]
If in addition \(Y^0:=Y\cap T\) satisfies
\[
\trop(Y^0)=|\Delta|
\]
and
\[
i^*:\Pic(X_\Delta)_\mathbb R\to N^1(Y)_\mathbb R
\]
is surjective, then
\[
\Nef(Y)\subseteq i^*(\mathcal F_{\Delta,\mathbf w}),
\]
where \(\mathbf w\) is induced by the tropical multiplicities. Hence
\[
i^*(\mathcal G_\Delta)\subseteq i^*(\mathcal L_\Delta)\subseteq \Nef(Y)\subseteq i^*(\mathcal F_{\Delta,\mathbf w}).
\]
This is a genuine lower-upper sandwich for the nef cone [1009.0220].

The framework generalizes two established exact descriptions. For Mori dream spaces,
\[
\Nef(Y)=i^*(\mathcal G_\Delta)=i^*(\mathcal L_\Delta),
\]
so the lower bounds are exact. For \(\overline M_{0,n}\), the upper cone \(\mathcal F_\Delta(M)\) is exactly the cone of \(F\)-divisors, making the inclusion
\[
\Nef(M)\subseteq \mathcal F_\Delta(M)
\]
the “only if” direction of the \(F\)-conjecture [1009.0220]. In this usage, “lower-upper” describes a computational approximation package attached to a projective variety via toric and tropical geometry.

## 6. Upper-motive and upper-cluster perspectives

Two adjacent strands of the literature extend the lower-upper vocabulary beyond the preceding explicit usages. The first is motivic. For a suitable homology theory \(H\) and a complete variety \(X\), one defines
\[
H_X(k):=\operatorname{im}\bigl(x_*:H(X)\to H(k)\bigr)\subset H(k).
\]
For complete regular varieties, this invariant is preserved under \(p\)-equivalence; equivalently, when \(p\)-equivalence is detected by upper Chow motives with \(\mathbb F_p\)-coefficients, the invariant depends only on the upper motive [1208.2586]. In particular, the classical invariants
\[
\mathrm{CH}_X(k)=n_X\mathbb Z,\qquad (K_0')_X(k)=d_X\mathbb Z
\]
fit this scheme, and one obtains the lower bound
\[
\operatorname{cdim}_p(X)\ge (p-1)\bigl(n_p(X)-d_p(X)\bigr)
\]
when \(\operatorname{char}(k)\neq p\) [1208.2586]. This suggests a broader interpretation in which upper structures control compressibility and splitting behavior, especially for projective homogeneous varieties.

The second strand is cluster-theoretic. For any symmetrizable Kac–Moody type, the coordinate ring of an open Richardson variety \(\mathring{\mathcal B}_{v,w}\) is an upper cluster algebra:
\[
\mathbb{C}[\mathring{\mathcal B}_{v,w}] \cong \mathcal U(v,\mathbf w).
\]
Likewise, the coordinate ring of the generalized open Richardson variety \(\mathring{\mathcal Z}_{v,\mathbf w}\) in a twisted product of flag varieties is an upper cluster algebra, covering reduced double Bruhat cells, Bott–Samelson varieties, and braid varieties as special cases [2506.10382]. The lower-versus-upper cluster question is explicit here: the paper proves upper-cluster realizations, conjectures local acyclicity as a mechanism for
\[
\mathcal A(v,\mathbf w)=\mathcal U(v,\mathbf w),
\]
and proves that equality in a product-length case [2506.10382].

These two directions are not themselves definitions of lower-upper varieties. They are, however, structurally adjacent. In the motivic setting, upper summands already determine numerical invariants relevant to incompressibility. In the cluster setting, varieties arise whose coordinate rings are naturally upper cluster algebras, while the lower-equals-upper problem remains a central issue. Together with the matrix, lattice, and nef-cone theories, they show that “lower-upper” has become a recurring organizing principle across several mathematically distant domains.

Source: https://www.emergentmind.com/topics/lower-upper-varieties