Toric varieties with isolated singularity and smooth normalization

Maria Aparecida Soares Ruas; Maria Elenice Rodrigues Hernandes; Thais Maria Dalbelo

arxiv: 2412.05083 · v3 · submitted 2024-12-06 · 🧮 math.AG

Toric varieties with isolated singularity and smooth normalization

Thais Maria Dalbelo , Maria Elenice Rodrigues Hernandes , Maria Aparecida Soares Ruas This is my paper

Pith reviewed 2026-05-23 07:45 UTC · model grok-4.3

classification 🧮 math.AG

keywords toric varietiesisolated singularitysmooth normalizationsemigroup generatorsprenormal formtoric surfacesalgebraic geometry

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The pith

Toric varieties with isolated singularity at the origin and smooth normalization have a prenormal form for the generators of their semigroup.

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

The paper describes a prenormal form for the generators of the semigroup of a toric variety X subset C^p with isolated singularity at the origin and smooth normalization. This gives a structured way to present the algebraic generators of the variety. A complete description of the semigroup is given for varieties of dimension n in C^{2n}. For toric surfaces in C^4, generators of the defining ideal I are provided. These results matter for classifying and analyzing such singular toric varieties in algebraic geometry.

Core claim

We describe a prenormal form for the generators of the semigroup of a toric variety X subset C^p with isolated singularity at the origin and smooth normalization. A complete description of the semigroup is given when X is a variety of dimension n in C^{2n}. Moreover, for toric surfaces in C^4, we provide a set of generators of the ideal I defining X.

What carries the argument

The prenormal form for the generators of the semigroup of the toric variety.

Load-bearing premise

The semigroup is finitely generated and meets the toric conditions that ensure the normalization is smooth.

What would settle it

An example of a toric variety with isolated singularity at the origin, smooth normalization, but whose semigroup generators do not match the prenormal form described.

read the original abstract

In this work, we describe a prenormal form for the generators of the semigroup of a toric variety $X \subset \mathbb{C}^p$ with isolated singularity at the origin and smooth normalization. A complete description of the semigroup is given when $X$ is a variety of dimension $n$ in $\mathbb{C}^{2n}$. Moreover, for toric surfaces in $\mathbb{C}^4$, we provide a set of generators of the ideal $I$ defining $X$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies explicit prenormal forms and complete semigroup lists for toric varieties with isolated singularities at the origin and smooth normalization, focused on low-codimension cases.

read the letter

The main takeaway is that the authors give a prenormal form for the semigroup generators of a toric variety X in C^p that has an isolated singularity at 0 and smooth normalization. They also supply full semigroup descriptions when the variety has dimension n inside C^{2n}, plus generators for the defining ideal when it is a toric surface in C^4. These are presented as new explicit descriptions not found in the cited literature. If the constructions hold, they could shorten later calculations for people working inside toric singularity theory by handing over ready-made generators instead of deriving them each time. The paper stays within the standard toric setup of finitely generated semigroups and adds the isolated-singularity and smooth-normalization conditions to restrict the possibilities. That restriction is reasonable and leads directly to the listed forms. The contribution is therefore concrete rather than theoretical. The scope stays narrow: only isolated singularities with smooth normalization, and only the balanced codimension-n and surface-in-C^4 cases receive complete lists. Without the actual proofs and the explicit lists in hand it is hard to judge whether every case is covered or whether some generators were missed. The abstract gives no indication of verification steps or edge-case checks, so those details matter. This is a paper for specialists who already work with toric embeddings and need explicit normal forms for computations or examples. It will not shift the broader theory but can serve as a reference in that niche. I would send it to peer review. The claims are specific enough that referees can check the forms and lists directly, and the topic is a legitimate part of algebraic geometry even if the audience is small.

Referee Report

0 major / 2 minor

Summary. The paper describes a prenormal form for the generators of the semigroup of a toric variety X ⊂ ℂ^p with isolated singularity at the origin and smooth normalization. It gives a complete description of the semigroup when X has dimension n and is embedded in ℂ^{2n}. For toric surfaces in ℂ^4 it also supplies a set of generators for the defining ideal I.

Significance. If the stated forms are correct and generate the claimed semigroups and ideals, the work supplies explicit, usable descriptions in a setting where toric varieties are rarely given by concrete generators. Such explicit data can support classification efforts, computational checks, and further study of isolated singularities with smooth normalization.

minor comments (2)

The abstract states the main results but does not indicate the ambient lattice or the precise saturation conditions used to define the semigroup; a brief reminder of these standard toric hypotheses in §1 would improve readability.
Notation for the prenormal form (e.g., the precise meaning of “prenormal”) is introduced without an immediate reference to a numbered definition or example; adding a short displayed example in the introduction would clarify the form before the general statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper provides a direct description of a prenormal form for semigroup generators of toric varieties with isolated singularity and smooth normalization, plus explicit descriptions in low codimension cases. This is a classification result resting on the standard definition of affine toric varieties via finitely generated saturated semigroups. No equations or claims reduce to their own inputs by construction, no fitted quantities are relabeled as predictions, and no load-bearing steps invoke self-citations or imported uniqueness theorems. The derivation chain is self-contained within algebraic geometry definitions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of toric geometry (semigroups define affine toric varieties, normalization corresponds to saturation of the semigroup) and the assumption that the singularity is isolated with smooth normalization; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Toric varieties are defined by finitely generated semigroups in N^d; normalization is smooth precisely when the semigroup is saturated.
Invoked implicitly when the prenormal form and smoothness condition are stated.

pith-pipeline@v0.9.0 · 5608 in / 1216 out tokens · 23175 ms · 2026-05-23T07:45:32.654410+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

J. R. Borges-Zampiva, B. Or´ eﬁce-Okamoto, G. Pe ˜ nafort-Sanchis, J. N. Tomazella, Double points and image of reﬂection maps, arXiv: 2312.06792v2, ( 2024)

work page arXiv 2024
[2]

Ch´ avez Mart´ ınez and D

E. Ch´ avez Mart´ ınez and D. Duarte, On the zero locus of idea ls deﬁning the Nash blowup of toric surfaces. Comm. Algebra, 46 (3), 1048–1059 (2017)

work page 2017
[3]

D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, G raduate Studies in Mathematics 124, AMS, (2011)

work page 2011
[4]

Duarte, and A

D. Duarte, and A. E. Giles Flores, On the Lipschitz saturati on of toric singularities, arXiv: 2310.03216, ( 2024)

work page arXiv 2024
[5]

Eisenbud, and B

D. Eisenbud, and B. Sturmfels, Binomial ideals, Duke Math. J., 84 (1), 1–45 (1996)

work page 1996
[6]

Fulton, Introduction to toric varieties, Annals of Math ematics Studies, vol

W. Fulton, Introduction to toric varieties, Annals of Math ematics Studies, vol. 131, Princeton University Press, Princeton, ( 1993)

work page 1993
[7]

T. J. Gaffney , Properties of ﬁnitely determined germs, Ph. D. Thesis, Brandeis University , 107, (1975)

work page 1975
[8]

Mond, and J

D. Mond, and J. J. Nu ˜ no-Ballesteros, Singularities of map pings - the local behaviour of smooth and complex analytic mappings, Grundlehren der mathematis chen Wissenschaften 357. A se- ries of comprehensive studies in Mathematics, Springer, ( 2020). TORIC V ARIETIES WITH ISOLATED SINGULARITY AND SMOOTH NORMA LIZATION 23

work page 2020
[9]

T. Oda, Convex bodies and algebraic geometry - An introduct ion to the theory of toric varieties, A series of Modern Surveys in Mathematics, Springer-V erlag , Berlin Heidelberg, ( 2012)

work page 2012
[10]

Riemenschneider, Zweidimensionale Quotientensingul arit¨ aten: Gleichungen und Syzygien, Arch

O. Riemenschneider, Zweidimensionale Quotientensingul arit¨ aten: Gleichungen und Syzygien, Arch. Math. 37, 406–417 (1981)

work page 1981
[11]

M. E. Rodrigues Hernandes and M. A. S. Ruas, Parametrized mo nomial surfaces in 4-space, Quart. J. Math. 70, no. 2, 473–485 (2019)

work page 2019
[12]

M. E. Rodrigues Hernandes and M. A. S. Ruas, Join operation a nd A-ﬁnite map-germs, Quart. J. Math., 74 (4), 1415–1434 (2023)

work page 2023
[13]

C. T. C. Wall, Finite determinacy of smooth map-germs. Bull . London Math. Soc., 13 (6), 481– 539 (1981). Departamento de Matem´atica, Universidade Federal de S ˜ao Carlos - UFSCar, Brazil Email address: thaisdalbelo@ufscar.br Departamento de Matem´atica, Universidade Estadual de Maring ´a - UEM, B razil Email address: merhernandes@uem.br Instituto de Ciˆ...

work page 1981

[1] [1]

J. R. Borges-Zampiva, B. Or´ eﬁce-Okamoto, G. Pe ˜ nafort-Sanchis, J. N. Tomazella, Double points and image of reﬂection maps, arXiv: 2312.06792v2, ( 2024)

work page arXiv 2024

[2] [2]

Ch´ avez Mart´ ınez and D

E. Ch´ avez Mart´ ınez and D. Duarte, On the zero locus of idea ls deﬁning the Nash blowup of toric surfaces. Comm. Algebra, 46 (3), 1048–1059 (2017)

work page 2017

[3] [3]

D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, G raduate Studies in Mathematics 124, AMS, (2011)

work page 2011

[4] [4]

Duarte, and A

D. Duarte, and A. E. Giles Flores, On the Lipschitz saturati on of toric singularities, arXiv: 2310.03216, ( 2024)

work page arXiv 2024

[5] [5]

Eisenbud, and B

D. Eisenbud, and B. Sturmfels, Binomial ideals, Duke Math. J., 84 (1), 1–45 (1996)

work page 1996

[6] [6]

Fulton, Introduction to toric varieties, Annals of Math ematics Studies, vol

W. Fulton, Introduction to toric varieties, Annals of Math ematics Studies, vol. 131, Princeton University Press, Princeton, ( 1993)

work page 1993

[7] [7]

T. J. Gaffney , Properties of ﬁnitely determined germs, Ph. D. Thesis, Brandeis University , 107, (1975)

work page 1975

[8] [8]

Mond, and J

D. Mond, and J. J. Nu ˜ no-Ballesteros, Singularities of map pings - the local behaviour of smooth and complex analytic mappings, Grundlehren der mathematis chen Wissenschaften 357. A se- ries of comprehensive studies in Mathematics, Springer, ( 2020). TORIC V ARIETIES WITH ISOLATED SINGULARITY AND SMOOTH NORMA LIZATION 23

work page 2020

[9] [9]

T. Oda, Convex bodies and algebraic geometry - An introduct ion to the theory of toric varieties, A series of Modern Surveys in Mathematics, Springer-V erlag , Berlin Heidelberg, ( 2012)

work page 2012

[10] [10]

Riemenschneider, Zweidimensionale Quotientensingul arit¨ aten: Gleichungen und Syzygien, Arch

O. Riemenschneider, Zweidimensionale Quotientensingul arit¨ aten: Gleichungen und Syzygien, Arch. Math. 37, 406–417 (1981)

work page 1981

[11] [11]

M. E. Rodrigues Hernandes and M. A. S. Ruas, Parametrized mo nomial surfaces in 4-space, Quart. J. Math. 70, no. 2, 473–485 (2019)

work page 2019

[12] [12]

M. E. Rodrigues Hernandes and M. A. S. Ruas, Join operation a nd A-ﬁnite map-germs, Quart. J. Math., 74 (4), 1415–1434 (2023)

work page 2023

[13] [13]

C. T. C. Wall, Finite determinacy of smooth map-germs. Bull . London Math. Soc., 13 (6), 481– 539 (1981). Departamento de Matem´atica, Universidade Federal de S ˜ao Carlos - UFSCar, Brazil Email address: thaisdalbelo@ufscar.br Departamento de Matem´atica, Universidade Estadual de Maring ´a - UEM, B razil Email address: merhernandes@uem.br Instituto de Ciˆ...

work page 1981