Lifting Frobenius splittings through geometric vertex decomposition

Elisa Gorla; Emanuela De Negri; Jenna Rajchgot; Lisa Seccia; Patricia Klein

Lifting Frobenius splittings through geometric vertex decomposition

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Pith's one-line read Frobenius splittings can be lifted through geometric vertex decomposition

desk verdict Solid partial converse to Knutson's Frobenius splitting descent, with a necessary hypothesis and a nice application to double determinantal ideals; referee it. read the letter →

arxiv 2509.04364 v2 pith:PG5M3RWO submitted 2025-09-04 math.AC math.AG

Emanuela De Negri , Elisa Gorla , Patricia Klein , Jenna Rajchgot , Lisa Seccia This is my paper · ORCID

classification math.ACmath.AG MSC 13A3513P1014M1213C40

keywords FrobeniussplittinggeometricvertexdecompositionGröbnerdegenerationKnutsonidealsGorensteinliaisondoubledeterminantalmaximalminorsF-split

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a lifting theorem for Frobenius splittings, a characteristic-p technique used to control singularities. The setting is a geometric vertex decomposition, which breaks an ideal I into two simpler ideals, the link C and the deletion N. If a splitting of the form Tr((x_n g)^{p-1}•) compatibly splits both C and N, and g has a factor u lying in C but outside every minimal prime of N, then the paper constructs r such that Tr((x_n g+r)^{p-1}•) compatibly splits I. This is a partial converse to the known descent theorem for such splittings through geometric vertex decomposition. The divisibility condition is essential: an example satisfying all other hypotheses but lacking u gives an ideal that is not Frobenius split at all. The construction also produces explicit splittings for Li's double determinantal varieties of maximal minors, showing they are Frobenius split.

What carries the argument

The key object is the module isomorphism ψ: C/N → I/N induced by a nondegenerate geometric vertex decomposition (Proposition 3.1). It is described as multiplication by (x_n q+r)/q; the element u lets the same isomorphism be described as multiplication by (x_n u+s)/u with x_n u+s∈I. This dual description converts the known splitting polynomial x_n g into a new polynomial x_n g+r inside I, so the trace map Tr((x_n g+r)^{p-1}•) compatibly splits I. For the double determinantal application, the matching concept is a Knutson ideal of f: the ideals obtained from (f) by repeatedly taking colons, sums, and intersections, all automatically compatibly split by Tr(f^{p-1}•).

What would settle it

A single counterexample satisfying all hypotheses of Theorem 3.2—including the factor u—but admitting no lift r would refute the central claim; the proof prescribes r as the preimage of the class of g under the isomorphism C/N ≅ I/N, so this is checkable case by case. Concretely, for small primes one can compute the explicit f of Theorem 5.3 and test in a computer algebra system whether Tr(f^{p-1}•) sends every generator of I=I_n(H)+I_n(V) into I; any violation would disprove the theorem.

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Extended reading notes

Core claim

The central claim, Theorem 3.2: for a nondegenerate geometric vertex decomposition in_{x_n}(I)=C∩(N+(x_n)), if Tr((x_n g)^{p-1}•) compatibly splits C and N and some u in C outside every minimal prime of N divides g, then some r with no x_n-divisible term makes Tr((x_n g+r)^{p-1}•) compatibly split I. The proof works through the isomorphism C/N ≅ I/N from Gorenstein liaison, described as multiplication by (x_n q+r)/q; u lets the same map be written as multiplication by (x_n u+s)/u with x_n u+s∈I, yielding f=(x_n u+s)g/u=x_n g+r. Example 3.3 shows the u-condition cannot be dropped. Theorem 5.3 adds that for s=t=m=n the double determinantal ideal I_n(H)+I_n(V) is a Knutson ideal of an explicit

Load-bearing premise

The load-bearing condition is that the polynomial g has a factor u lying in the link part C whose class in the quotient by the deletion part N is not annihilated by any nonzero element; without such a u the lifting construction has nothing to grip, and the paper's example shows the desired splitting need not exist.

Editorial extensions

If this is right

Every nondegenerate geometric vertex decomposition whose link and deletion are compatibly split by Tr((x_n g)^{p-1}•), with g having a suitable nonzerodivisor factor in C, is liftable, and the lifted splitting polynomial is explicitly constructed.
The counterexample of Example 3.3 shows that compatible splitting of the two pieces alone does not imply the original ideal is F-split; the divisibility hypothesis is a genuine obstruction, not a proof artifact.
In the degenerate case the lift holds without the extra factor: replacing x_n by any linear form containing it preserves the compatible splitting (Proposition 3.5).
The double determinantal ideals with s=t=m=n are Knutson ideals of an explicit f, so they are Frobenius split and carry the full family of compatibly split ideals attached to that splitting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The theorem suggests a broader principle: liftability is controlled by how the liaison isomorphism C/N ≅ I/N interacts with divisibility, so replacing the single factor u by the ideal of C-elements that are nonzerodivisors modulo N may yield a refined criterion covering the double determinantal example.
A natural testable weakening is to allow g to be a sum of multiples of elements of C outside the minimal primes of N rather than to require a single factor u|g; Example 5.4 already carries out this substitution in one case and is explicitly left open by the authors.
For Li's double determinantal ideals beyond maximal minors, the same vertex decomposition tree should produce explicit splitting polynomials; Theorem 5.3 is evidence that such a family exists.
The total failure in Example 3.3 can be read as a diagnostic: while building splittings inductively through a vertex decomposition tree, absence of a suitable factor u in the splitting polynomial is a cheap warning that the original ideal may not be F-split.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper gives a real partial converse to Knutson's descent theorem for Frobenius splittings through geometric vertex decomposition. The lift works when the splitting polynomial has a factor that is a nonzerodivisor modulo the deletion, and Example 3.3 shows that hypothesis is genuinely necessary — the ideal there satisfies everything else but is not F-split at all. The main trace computation in Theorem 3.2 is sound; I filled in the steps the stress test flags and they go through. The application to double determinantal varieties (Theorem 5.3) is the other solid piece: those ideals are Knutson ideals of an explicit polynomial, so an explicit splitting exists and the class of known F-split varieties grows.

Soft spots are mostly expositional, not mathematical. Proposition 3.1, which produces the module isomorphism bridging GVD and liaison, is deferred to a sketch of [KR21, Lemma 7.4]. The sketch covers the differences but a referee should insist on a full proof here. Example 3.3 delegates the non-F-splitting to Singh's argument; that is fine in an announcement but should be reproduced. Theorem 5.3 asserts f has the right leading term, but does not show that the leading term of f equals the product of all variables in the relevant term order. Given the determinants are diagonal, it is nearly immediate, but it is load-bearing for f to determine a splitting, so write it down. The citation to the authors' own [KR21] is not a problem: the lemma is about GVD-induced isomorphisms, independent of Frobenius splitting, so there is no circularity.

Bottom line: this is a competent paper with a new theorem, a meaningful counterexample to a full converse, and an honest open problem about why Section 5 does not follow from Theorem 3.2. It is for commutative algebraists working on F-splitting or Groebner degenerations. I would send it to a serious referee and expect publication after modest expansion.

Referee Report

0 major / 5 minor

Summary. This paper proves a lifting theorem for Frobenius splittings through geometric vertex decomposition. In Section 3, assuming in_{x_n}(I) = C ∩ (N + (x_n)) is a nondegenerate geometric vertex decomposition, and assuming Tr((x_n g)^{p-1} •) is a splitting that compatibly splits both C and N, the authors show that under an additional hypothesis—the existence of u ∈ C outside the minimal primes of N with u | g—there is r, with no term divisible by x_n, such that Tr((x_n g + r)^{p-1} •) compatibly splits I. The construction uses the isomorphism C/N → I/N from Proposition 3.1. Example 3.3 shows the divisibility hypothesis cannot be dropped, and Proposition 3.5 treats the degenerate case. Section 4 gives applications to determinantal ideals and to the toric ideal of a bipartite graph. Section 5 proves that Li's double determinantal ideals I = I_n(H) + I_n(V) for s = t = m = n are Knutson ideals of an explicitly defined polynomial f, hence Frobenius split.

Significance. If the results are correct—and I find them correct after close reading—the paper makes a substantial contribution. Theorem 3.2 is a genuine partial converse to Knutson's degeneration theorem, giving an explicit and parameter-free construction of the lifted splitting. The counterexample in Example 3.3 is valuable because it identifies a precise obstruction rather than a vague limitation. The double determinantal application in Section 5 is new and goes beyond what a direct iteration of Theorem 3.2 would provide. The main proof is a trace computation that is internally consistent; the paper also gives constructive examples that make the machinery easy to test. The principal caveats are expositional: a central supporting proposition is only sketched, and one key hypothesis in the Knutson-ideal application is not verified in the text.

minor comments (5)

[§3, Proposition 3.1] The statement is central to Theorem 3.2, but its proof is terse. In particular, Part (2) is not proved in the text; the reader is referred to specific paragraphs of [KR21, Theorem 4.1]. Since Proposition 3.1 removes the homogeneity and infinite-field assumptions from [KR21, Lemma 7.4], the authors should either expand Part (2) into a complete proof or restate the exact KR21 lemma and explain why the variant follows. This would make the paper substantially more self-contained.
[§3, Theorem 3.2 proof] In the last step of the proof, after deriving uTr(f^{p-1}i) = vγ + ν with γ ∈ C and ν ∈ N, the text says that Part (2) of Proposition 3.1 implies Tr(f^{p-1}i) ∈ I. This is correct, but the reasoning is compressed: one should explicitly note that the equation gives Tr(f^{p-1}i) ≡ ψ(γ) modulo N, and that ψ(γ) ∈ I/N. Adding this sentence would remove a real point of friction.
[§5, Theorem 5.3] The proof never explicitly verifies the defining hypothesis of a Knutson ideal: that the leading term of f with respect to the chosen term order is the squarefree product of all variables. This is a necessary condition for the cited Knutson splitting theorem to apply. The verification is short, since the factors det X_k and Δ_s, Δ_s are arranged along disjoint diagonals, but it should be written out.
[§5, Theorem 5.3] The proof silently uses the fact that a minimal prime over a Knutson ideal is again a Knutson ideal. This is true—for a minimal prime P of J, choose an element a in the intersection of the other minimal primes but not in P; then P = J : (a), and Knutson ideals are closed under colons—but the step should be stated or cited, since it is load-bearing in the argument that I_n(H_{i,i+1,i+2}) and I_n(H) are Knutson.
[§3, Example 3.3] The claim that S/I is not F-split at all is justified only by 'one can mimic Singh's argument in [Sin99, Proposition 3.1]'. Since this example is used to prove that the u-divisibility hypothesis cannot be dropped, a fuller argument or a precise pointer to the exact statement in Singh's paper should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lifting theorem is proved by a genuine trace computation, and the Knutson-ideal application is independent of any fitted or imported splitting.

full rationale

The main theorem (Theorem 3.2) assumes a splitting of C and N, then uses Proposition 3.1 (from [KR21, Lemma 7.4]) only to obtain the module isomorphism ψ and the element v = x_n u + s; that lemma concerns geometric vertex decomposition and does not assert any Frobenius splitting, and it is re-proved in sketch form here. The trace computation then proves Tr(f^{p-1}i) ∈ I directly from iu = vc + m and the compatible splitting of C and N. The hypothesis u | g is shown necessary by Example 3.3, so it is not a hidden restatement of the conclusion. Theorem 5.3 establishes I = I_n(H) + I_n(V) as a Knutson ideal of an explicit f by height comparisons and minimal-prime closure; it does not rename a fit or import the target splitting. The self-citations to [KR21], [FK20], and [Sec21] are to independent algebraic results or background, not to the Frobenius-splitting conclusion. The acknowledged limitations—the sketched proof of Proposition 3.1 and the omitted verification of in_<(f) in Theorem 5.3—are expositional/correctness caveats, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented entities appear. The paper rests on standard commutative algebra background, a published lemma from two of the authors, and known results on Frobenius splittings. The self-cited lemma is independent of the target result and is not being refitted here.

assumptions (6)

standard math Frobenius splitting and trace map properties from [BK05], including compatibly split ideals being radical and closed under sums, intersections, and quotients.
Used throughout the proof of Theorem 3.2 and in the definition of Knutson ideals.
standard math Geometric vertex decomposition facts from [KMY09] and [KR21], including the form of the Groebner basis and the equality in_{x_n}(I) = C ∩ (N + (x_n)).
Provides the decomposition structure that the theorem operates on; cited as known.
domain assumption [KR21, Lemma 7.4] as restated in Proposition 3.1: existence of q, r, and an isomorphism ψ : C/N → I/N.
This is the key structural lemma, authored by two of the present authors; the paper sketches a proof rather than including full details.
standard math Knutson's theorem [Knu09, Theorem 2] that if in_<(f) = x_1···x_n then f induces a splitting and the splitting descends through geometric vertex decomposition.
Used to guarantee Tr((x_n g)^{p-1} •) splits S and to set up the descent direction.
standard math The Fedder-Singh result that F-splitting does not deform, specifically that the ideal in Example 3.3 is not F-split.
Used to show the extra hypothesis in Theorem 3.2 is necessary.
domain assumption Fieldsteel-Klein [FK20] that double determinantal ideals have diagonal Groebner bases and are lex-compatibly geometrically vertex decomposable.
Used in Section 5 to define the polynomial f and compute heights of the relevant ideals.

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Pith. "Pith review of Lifting Frobenius splittings through geometric vertex decomposition." pith.science (2026). https://pith.science/paper/PG5M3RWO

@misc{pith2026250904364,
  author       = {Pith},
  title        = {Pith review of: Lifting Frobenius splittings through geometric vertex decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PG5M3RWO}},
  note         = {Machine review of arXiv:2509.04364}
}

abstract

Frobenius splitting, pioneered by Hochster and Roberts in the 1970s and Mehta and Ramanathan in the 1980s, is a technique in characteristic $p$ commutative algebra and algebraic geometry used to control singularities. In the aughts, Knutson showed that Frobenius splittings of a certain type descend through Gr\"obner degeneration of a certain type, called geometric vertex decomposition. In the present paper, we give a partial converse to Knutson's result. We show that a Frobenius splitting that compatibly splits both link and deletion of a geometric vertex decomposition can, under an additional hypothesis on the form of the splitting, be lifted to a splitting that compatibly splits the original ideal. We discuss an example showing that the additional hypothesis cannot be removed. Our argument uses the relationship between geometric vertex decomposition and Gorenstein liaison developed by Klein and Rajchgot. Additionally, we show that Li's double determinantal varieties defined by maximal minors are Frobenius split.

Figures

Figures reproduced from arXiv: 2509.04364 by the authors.

**Figure 1.** Tree recording geometric vertex decomposition from I G ∅ , where delei and lkei denote application of the operators N(ei , —) and C(ei , —), respectively. The leaves of this tree are all squarefree monomial ideals, hence all compatibly split by Tr(f p−1 3 •), for f3 = e1 · · · e11. Now e11 is an element of each of the leaves of this tree arising from a link taken at e3, i.e., I G\{e1,e2} {e3} = (e11), IG\{e1} {e2,e… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Geometrically vertex decomposable star configurations
math.AC 2026-07 accept novelty 6.5 of 10

Ideals of star configurations X(ℓ,c) are geometrically vertex decomposable precisely when their defining linear forms admit a triangular coefficient submatrix (for ℓ ≤ n+1), and this is equivalent to being Knutson.

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Works this paper leans on

22 extracted references · 21 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

231, Birkh\"auser Boston, Inc., Boston, MA, 2005

Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkh\"auser Boston, Inc., Boston, MA, 2005. 2107324

work page 2005

[2] [2]

Mike Cummings, Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl, Geometric vertex decomposition and liaison for toric ideals of graphs, Algebraic Combinatorics 6 (2023), 965--997

work page 2023

[3] [3]

Alexandru Constantinescu and Elisa Gorla, Gorenstein liaison for toric ideals of graphs, Journal of Algebra 502 (2018), 249--261

work page 2018

[4] [4]

3, 713--730

Aldo Conca and Matteo Varbaro, Square-free gr \"o bner degenerations , Inventiones mathematicae 221 (2020), no. 3, 713--730

work page 2020

[5] [5]

Emanuela De Negri, Elisa Gorla, Patricia Klein, and Lisa Seccia, Frobenius splitting and geometric vertex decomposition: Examples and obstructions, In preparation

work page

[6] [6]

Richard Fedder, F -purity and rational singularity , Trans. Amer. Math. Soc. 278 (1983), no. 2, 461--480. 701505

work page 1983

[7] [7]

Nathan Fieldsteel and Patricia Klein, Gr\"obner bases and the C ohen- M acaulay property of L i's double determinantal varieties , Proc. Amer. Math. Soc. Ser. B 7 (2020), 142--158. 4167593

work page 2020

[8] [8]

Justin Gebel, Frobenius splittings for toric ideals of bipartite graphs, Undergraduate thesis, McMaster University, 2025, Available at https://math.mcmaster.ca/undergraduate/useful-links/undergraduate-research-projects/

work page 2025