Geometric Vertex Decomposition

• Geometric vertex decomposition is a generalization of vertex decomposability that recursively decomposes ideals using a blend of algebraic and combinatorial techniques.
• It unifies Gröbner degeneration with link/deletion phenomena, as seen in the Stanley–Reisner correspondence, yielding properties like radicality, Cohen–Macaulayness, and glicci.
• Applications range from toric ideals of graphs to Frobenius splitting and computational tests in Macaulay2, highlighting its versatility in algebraic geometry and combinatorics.

Geometric vertex decomposition is an ideal-theoretic generalization of vertex decomposability for simplicial complexes. In its recursive form, an ideal $I$ is required to be unmixed and either to lie in a base class, or to admit a decomposition with respect to a variable $y$ of the form

$$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$

with both contracted ideals again geometrically vertex decomposable. In the square-free monomial case, this recovers the usual combinatorial notion under the Stanley–Reisner correspondence, while in the general case it organizes Gröbner degeneration, link/deletion phenomena, and strong homological consequences such as radicality, Cohen–Macaulayness, and membership in the Gorenstein liaison class of a complete intersection (Cummings et al., 2022, Klein et al., 2020).

1. Algebraic formulation and recursive variants

Let

$R=k[x_1,\ldots,x_n]$

and let $y$ be one of the variables. A $y$-compatible monomial order $<$ is a monomial order such that for every $f\in R$,

$$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$

If

$$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$

then the initial $y$0-form is

$y$1

For an ideal $y$2,

$y$3

If a Gröbner basis $y$4 is written as

$y$5

with $y$6, then one defines

$y$7

A geometric vertex decomposition with respect to $y$8 is the identity

$y$9

This is the one-step decomposition introduced in the Knutson–Miller–Yong framework and used recursively by Klein–Rajchgot (Cummings et al., 2022, Klein et al., 2020).

An ideal $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$0 is geometrically vertex decomposable if $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$1 is unmixed and either $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$2, or $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$3 is generated by a subset of variables, or there is a variable $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$4 and a $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$5-compatible monomial order such that $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$6 has a geometric vertex decomposition with respect to $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$7 and the contractions of $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$8 and $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$9 to

$R=k[x_1,\ldots,x_n]$0

are again geometrically vertex decomposable. The package literature also records two refinements: $R=k[x_1,\ldots,x_n]$1-compatibly geometrically vertex decomposable, where a fixed lex order is used throughout the recursion, and weakly geometrically vertex decomposable, which distinguishes degenerate and nondegenerate decompositions and weakens the recursive requirements in the degenerate case (Cummings et al., 2022).

The liaison-oriented literature separates degenerate and nondegenerate decompositions. A decomposition is degenerate if either $R=k[x_1,\ldots,x_n]$2 or $R=k[x_1,\ldots,x_n]$3. If $R=k[x_1,\ldots,x_n]$4 has a geometric vertex decomposition with respect to $R=k[x_1,\ldots,x_n]$5, then $R=k[x_1,\ldots,x_n]$6 is squarefree in $R=k[x_1,\ldots,x_n]$7, and its reduced Gröbner basis has the form

$R=k[x_1,\ldots,x_n]$8

where no term of any $R=k[x_1,\ldots,x_n]$9 is divisible by $y$0 (Klein et al., 2020).

2. Simplicial-combinatorial model and related vertex decompositions

The combinatorial template comes from simplicial complexes. For a simplicial complex $y$1 and a vertex $y$2,

$y$3

where

$y$4

and

$y$5

For Stanley–Reisner ideals,

$y$6

This analogy is exact in the square-free monomial case: a simplicial complex $y$7 is vertex decomposable if and only if its Stanley–Reisner ideal $y$8 is geometrically vertex decomposable, and in that setting $y$9 corresponds to the star or link side while $y$0 corresponds to deletion (Klein et al., 2020, Cummings et al., 2022, Cummings et al., 2022).

This relationship clarifies the status of purely combinatorial vertex decomposability results. For extremal simplicial complexes, an extremal pure $y$1-dimensional simplicial complex $y$2 with $y$3-vector $y$4 is characterized by

$y$5

and the main result is

$y$6

Combined with

$y$7

this gives a combinatorial strengthening of the Herzog–Hibi theorem. However, that work does not discuss geometric vertex decomposition in the modern algebro-geometric sense; its connection to the geometric theory is indirect, through the combinatorial notion of vertex decomposability (Lasoń, 2013).

A second combinatorial criterion arises from facet counts. If $y$8 is a pure simplicial complex on $y$9 vertices of codimension

$<$0

and has at least

$<$1

facets, then $<$2 is vertex decomposable. The same source notes that if for some term order $<$3, the initial ideal $<$4 is the Stanley–Reisner ideal of a vertex decomposable complex, then $<$5 is geometrically vertex decomposable. Therefore, if $<$6 corresponds to a simplicial complex with at least $<$7 facets, then $<$8 is geometrically vertex decomposable (Dochtermann et al., 2024).

3. Liaison-theoretic interpretation and homological consequences

A central structural result is that geometric vertex decomposition and Gorenstein liaison encode the same recursive mechanism in different languages. If $<$9 is unmixed and has a nondegenerate geometric vertex decomposition with respect to $f\in R$0, and if $f\in R$1 is unmixed, then

$f\in R$2

as $f\in R$3-modules; in the homogeneous case this is graded of degree $f\in R$4,

$f\in R$5

Under the additional hypotheses that $f\in R$6 is homogeneous, saturated, and unmixed, that $f\in R$7 is Cohen–Macaulay and $f\in R$8, and that $f\in R$9 is unmixed, this yields an elementary $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$0-biliaison of height $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$1 from $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$2 to $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$3 (Klein et al., 2020).

The recursive consequence is that every homogeneous geometrically vertex decomposable proper ideal is linked by a finite chain of elementary $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$4-biliaisons of height $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$5 to a complete intersection, hence is glicci. The same paper records that geometrically vertex decomposable ideals are radical, and that homogeneous geometrically vertex decomposable proper ideals are Cohen–Macaulay. The graph-theoretic development adopts the same slogan in a compact form: $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$6 and states that a homogeneous geometrically vertex decomposable ideal is radical, Cohen–Macaulay, and glicci (Klein et al., 2020, Cummings et al., 2022).

The converse direction also exists. If there is an isomorphism

$$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$7

given by multiplication by $$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$8, with the Gröbner and non-zero-divisor hypotheses of the converse theorem and with

$$\operatorname{in}_<(f)=\operatorname{in}_<(\operatorname{in}_y(f)).$$9

then

$$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$0

is a geometric vertex decomposition of $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$1. This identifies a class of biliaison maps whose algebraic form forces geometric vertex decomposition (Klein et al., 2020).

The mixed case is treated separately. Using an alternative decomposition criterion tailored to the nonpure situation, one obtains that if $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$2 is homogeneous and $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$3 is Cohen–Macaulay, then

$$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$4

A plausible implication is that geometric vertex decomposition functions not only as a Cohen–Macaulay recursion for unmixed ideals, but also as a sequential Cohen–Macaulay recursion in the mixed setting (Klein et al., 2020).

4. Toric ideals of graphs and graph-theoretic recursion

For a finite simple graph $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$5 with $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$6 and $$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$7, the toric map is

$$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$8

and the toric ideal is

$$f=\sum_i \alpha_i y^i,\qquad \alpha_d\neq 0,\ \alpha_t=0 \text{ for } t>d,$$9

The graph-theoretic description of $y$00 is decisive: $y$01 is generated by binomials coming from closed even walks, and the primitive binomials corresponding to closed even walks form a universal Gröbner basis $y$02. This allows the $y$03- and $y$04-ideals of a geometric vertex decomposition to be read off from graph operations (Cummings et al., 2022).

Several recursive closure properties are established. Geometric vertex decomposability behaves under tensor products: $y$05 so for graphs one may reduce to connected components. If $y$06 is obtained from $y$07 by attaching a cycle of even length to $y$08 along one edge, then

$y$09

The first large family produced by these methods is bipartite graphs: $y$10 The same paper formulates the conjecture

$y$11

proves a general reduction to unmixedness of intermediate ideals $y$12, and proves the conjecture when $y$13 consists of quadratic binomials: $y$14 The same source records explicit families and examples: even cycles, odd cycles with $y$15, Ferrers graphs $y$16, complete bipartite graphs $y$17, the graphs $y$18 of Galetto et al., and gap-free graphs containing a $y$19-cycle, which are glicci (Cummings et al., 2022).

A complementary special-family result concerns graphs

$y$20

where $y$21 consists of $y$22 odd cycles, the $y$23-th cycle has length $y$24, and all cycles share one common central vertex. For this family, the toric ideal $y$25 is geometrically vertex decomposable. The proof uses an explicit universal Gröbner basis indexed by pairs of odd cycles, together with repeated initial-form steps in the odd edges of the last cycle, ending at a squarefree monomial complete intersection. The geometric vertex decomposition structure then yields the $y$26-polynomial formula

$y$27

and the regularity

$y$28

This is presented as a stronger structural statement than merely computing the Hilbert series or $y$29-polynomial (Bhaskara et al., 17 Apr 2025).

5. Frobenius splitting through geometric vertex decomposition

In characteristic $y$30, geometric vertex decomposition interacts with Frobenius splitting through link/deletion data. Over a perfect field $y$31 of prime characteristic $y$32, let

$y$33

and fix $y$34. Writing $y$35 for the $y$36-th standard basis weight vector, the relevant ideals are

$y$37

The ideal $y$38 admits a geometric vertex decomposition at $y$39 if

$y$40

and either

$y$41

or no minimal prime of $y$42 is a minimal prime of $y$43. When $y$44 is unmixed, this extra condition is automatic (Negri et al., 4 Sep 2025).

The descent theorem recalled there is due to Knutson. If $y$45 has degree $y$46, $y$47 is a lexicographic term order with $y$48 largest, and

$y$49

then $y$50 induces a splitting of $y$51. If this splitting compatibly splits $y$52, then $y$53 induces a splitting that compatibly splits

$y$54

This is the descent direction from an ideal to the link and deletion of a geometric vertex decomposition (Negri et al., 4 Sep 2025).

The main new result in that work is a partial converse. Assume that $y$55 has a nondegenerate geometric vertex decomposition at $y$56,

$y$57

and let $y$58, $y$59. If $y$60 has no term divisible by $y$61, if

$y$62

is a Frobenius splitting of $y$63 that compatibly splits both $y$64 and $y$65, and if there exists

$y$66

then there exists $y$67, with no term divisible by $y$68, such that

$y$69

compatibly splits $y$70. The proof uses the module isomorphism

$y$71

coming from the Klein–Rajchgot liaison framework (Negri et al., 4 Sep 2025).

The necessity of the extra hypothesis is demonstrated by the example

$y$72

At $y$73, one has

$y$74

and both are compatibly split by the standard splitting induced by

$y$75

Nevertheless the attempted lift

$y$76

does not compatibly split $y$77, and $y$78 is not Frobenius split. The obstruction is exactly the failure of the theorem’s nonzerodivisor-factor hypothesis (Negri et al., 4 Sep 2025).

The same paper applies the method to Li’s double determinantal varieties in the maximal-minor case. For $y$79 matrices $y$80 of size $y$81, with horizontal concatenation $y$82 and vertical concatenation $y$83, the maximal-minor ideal

$y$84

is shown to be a Knutson ideal of

$y$85

In particular,

$y$86

compatibly splits $y$87 (Negri et al., 4 Sep 2025).

6. Further families, neighborhood ideals, and computation

Geometric vertex decomposition also appears in square-free monomial ideals arising from graph neighborhoods. For a graph $y$88, the open neighborhood ideal is

$y$89

where $y$90. If $y$91 is a TD-unmixed tree, then the open neighborhood ideal of $y$92 is geometrically vertex decomposable; equivalently, after reduction to balanced trees and passage to the odd-open neighborhood ideal $y$93, one proves recursively that $y$94 is GVD. Since

$y$95

this implies that the corresponding Stanley–Reisner complex is vertex decomposable, and by the join decomposition of the stable complex the same conclusion extends to every TD-unmixed tree (Lim, 14 Dec 2025).

The same work places these ideals between two established classes. If $y$96 is TD-unmixed and balanced, then there exists an unmixed simplicial tree $y$97 on the even vertices such that

$y$98

where $y$99 is the facet ideal. Consequently,

$$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$00

It also shows that essentially any square-free monomial ideal can be realized from a chordal graph after adjoining a regular sequence of variables: $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$01 This suggests that open neighborhood ideals provide a large testing ground for geometric vertex decomposition beyond toric and determinantal settings (Lim, 14 Dec 2025).

The theory has also been implemented computationally in Macaulay2. The package GeometricDecomposability centers on the routine oneStepGVD, which checks whether

$$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$02

holds for a chosen variable $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$03, and returns a Boolean value together with $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$04 and $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$05. Auxiliary routines include CyI(I,y), NyI(I,y), and findOneStepGVD(I), while the full recursive testers are isGVD(I), isLexCompatiblyGVD(I, ...), isWeaklyGVD(I), and findLexCompatiblyGVDOrders(I) (Cummings et al., 2022).

The package note illustrates several distinctions that are already present in the theory. It exhibits an ideal that is GVD but not $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$06-compatibly GVD for any lex order, an ideal that is weakly GVD but not GVD, and the Stanley–Reisner ideal

$$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$07

of a vertex decomposable simplicial complex, for which oneStepGVD recovers the deletion/link structure algebraically. It also reports computational experiments on connected graphs with $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$08 to $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$09 edges: for $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$10 edges there is exactly one connected graph whose toric ideal is not GVD, and for $$\operatorname{in}_y(I)=C_{y,I}\cap (N_{y,I}+\langle y\rangle),$$11 edges there is exactly one connected graph whose toric ideal is weakly GVD but not GVD (Cummings et al., 2022).