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RationalMaps, a package for Macaulay2

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper presents a Macaulay2 package, RationalMaps, that decides birationality of rational maps and computes their inverses from the rank of a weak Jacobian dual matrix.

desk verdict A genuinely useful Macaulay2 package for inverting rational maps between projective varieties, built on known Rees-algebra criteria; the main soft spot is the unquantified randomness in QuickRank, which should be documented or made off by default. read the letter →

arxiv 1908.04337 v2 pith:MU32G7NA submitted 2019-08-12 math.AG math.AC

C.J. Bott , S. Hamid Hassanzadeh , Karl Schwede , Daniel Smolkin This is my paper · ORCID
classification math.AGmath.AC MSC 14E0513A3014Q10
keywords rationalmapsbirationalmapinverseReesalgebraJacobiandualmatrixbaselocusclosedembeddingMacaulay2
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 introduces a Macaulay2 package that automates three core questions about rational maps between projective varieties: where a map is undefined, whether it is birational, and, when it is, what its inverse is. The package converts these geometry questions into algebra by studying the Rees algebra of the ideal generated by the coordinate forms of the map. Birationality is decided by the rank of a weak Jacobian dual matrix $\psi$: the map is birational exactly when that rank is one less than the embedding dimension of the source. When that holds, an inverse representative is obtained from a nonzero vector in the null space of $\psi$, or from signed maximal minors of a full-rank submatrix. A careful reader should care because these computations are routine in algebraic geometry and were previously much more labor-intensive.

What carries the argument

The central object is the weak Jacobian dual matrix $\psi$. One takes bihomogeneous relations $J_{1,*}$ of degree one in the source variables among the forms defining the map, lifts them to polynomials in $k[X,Y]$, differentiates with respect to the $X$-variables, and reduces the entries modulo the ideal of the target. The rank of the resulting matrix over the coordinate ring of the image is the birationality test. To form $\psi$, the package computes only the $(1,q)$ pieces of the Rees ideal until the rank reaches the required value, and the HybridStrategy saves already-found generators when it switches from this incremental search to a full Rees ideal computation. Once $\psi$ is available, inverse extraction is a null-space syzygy computation or a signed-minors computation.

What would settle it

Run inverseOfMap with CheckBirational=>true on a known birational map, for instance the degree-5 map on $\mathbb{P}^4$ used in the paper's timing tables, repeatedly with QuickRank=>true and once with QuickRank=>false; if any QuickRank run reports that the map is not birational, returns a different inverse representative, or fails to terminate, the package's central reliability claim is refuted.

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Extended reading notes

Core claim

For a rational map $F: X \dashrightarrow Y$ defined by forms $f_0,\ldots,f_m$ in the coordinate ring $R$ of an irreducible, non-degenerate projective variety, the paper's central claim is operational: $F$ is birational onto its image if and only if the weak Jacobian dual matrix $\psi$, attached to a minimal generating set of the bidegree $(1,*)$ part of the Rees presentation ideal, has rank $\operatorname{edim}(R)-1$. When this rank condition holds, the inverse map is represented either by the coordinates of any positive-degree homogeneous vector in the one-dimensional null space of $\psi$, or, when $R$ is a domain, by the ordered signed $(\operatorname{edim}(R)-1)$-minors of a rank $(\operatorname{edim}(R)-1)$ submatrix. The package computes enough of the Rees ideal, degree by degree, to form $\psi$ and check its rank, using a default hybrid strategy that switches from this incremental search to a full Rees ideal computation at a user-set limit. This makes a characteristic-free birationality criterion from the literature into a working implementation with several speed options.

Load-bearing premise

The package's answers depend on the computed rank of the weak Jacobian dual matrix being exactly right; with the default QuickRank option, ranks of selected submatrices come from randomized linear algebra routines that can, in principle, be too small.

Editorial extensions

If this is right

  • For any map satisfying the non-degeneracy and domain hypotheses, the default HybridStrategy either certifies birationality by reaching rank $\operatorname{edim}(R)-1$ during the incremental search, or, after HybridLimit, computes the full Rees ideal to decide non-birationality.
  • When the map is birational, the inverse is produced explicitly as a null-space vector of $\psi$ or as signed maximal minors, so the user receives a usable representative rather than an existence statement.
  • The same machinery gives automatic checks for closed embeddings: the base locus is empty, the inverse exists, and the inverse is regular.
  • The paper's timings show that the option choices (Strategy, HybridLimit, MinorsCount, QuickRank) materially change performance, so the package documents how to tune them for large examples.
  • The implementation handles maps whose inverse has very high degree, including examples from $\mathbb{P}^3$ to $\mathbb{P}^3$ that take hours but complete.

Reading between the lines

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

  • Inference: Because QuickRank is documented as containing randomness, a single negative birationality answer obtained with default settings is not a proof; rerunning with QuickRank=>false is a cheap way to make the answer deterministic.
  • Inference: The same Rees-algebra rank invariant could also compute other geometric data encoded in the Rees presentation ideal, such as the degree of a rational map, though the package does not currently do this.
  • Inference: The benchmark family with inverse degree $d^{n-1}$ indicates that the practical bottleneck is the cost of partial Groebner bases of the Rees ideal, so truncation or modular strategies could substantially extend the feasible range.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper describes RationalMaps, a Macaulay2 package for computing the base locus, birationality, inverse, and closed-embedding status of rational maps between projective varieties. The main algorithm is based on the theorem of Doria--Hassanzadeh--Simis: for a non-degenerate integral source, a map is birational onto its image if and only if the weak Jacobian dual matrix psi of a minimal generating set of the first-degree part of the Rees presentation ideal has rank edim(R)-1; the inverse is then read off from the null space of psi or from maximal minors. The package implements several strategies for computing the needed part of the Rees ideal (ReesStrategy, SimisStrategy, HybridStrategy, SaturationStrategy), options such as AssumeDominant, CheckBirational, MinorsCount, and QuickRank, and includes comparisons with the Cremona package and timings for maps attaining the Gabber degree bound.

Significance. If the package performs as documented, it provides a useful and generally applicable tool for a central operation in algebraic geometry. The theoretical backbone is an established published theorem (DHS12), and the paper demonstrates correctness of computed inverses in the worked examples by checking equality with isSameMap. The package also handles base loci, embeddings, and has already been used in a research application (He--Yang). The main weakness is that the paper does not specify the error semantics of the randomized linear-algebra routines used in QuickRank mode, leaving the user without a certified path to the rank condition on which Theorem 3.1 rests. A revision that addresses this documentation gap would make the correctness claims fully assessable.

major comments (1)
  1. [Section 3 (QuickRank paragraph) and Section 6 (timing examples)] The paragraph describing QuickRank states that FastLinAlg contains "a certain amount of randomness" and that rerunning a slow example can give a massive speedup, but it never states whether QuickRank is on by default, whether the randomized rank algorithms are Las Vegas (errors detected) or Monte Carlo (errors possible), or whether any post-hoc verification is performed. Since Theorem 3.1 is an iff criterion on the exact rank of the weak Jacobian dual matrix psi, and since both the birationality test and the inverse computation use this rank to decide birationality and to select the submatrix for the minor method, a rank underestimate can cause false negatives (failing to recognize a birational map or to find its inverse) and a rank overestimate can cause false positives (declaring a non-birational map birational and returning a spurious inverse). The paper should state the default value of QuickRank for every affected function, describe the error model of the FastLinAlg rank computations, and provide a deterministic setting that users can invoke to obtain certified output; ideally, when QuickRank is enabled, inverseOfMap should verify the result by composing with the candidate inverse.
minor comments (5)
  1. [Section 3] In the paragraph beginning "Note that a weak Jacobian matrix psi is not uniquely defined", there is a typo "on e set" for "one set"; also "non-degenerated" is used inconsistently for "non-degenerate".
  2. [Section 6 and References] The final section contains the typo "pacakge"; in addition, the text names "Zhuang He and Lei Yang" while the reference [HL19] lists "Z. He and Y. Lei" -- the author names should be harmonized.
  3. [Section 1 (Obtaining RationalMaps)] The package depends on a pre-release version of FastLinAlg, but no version number or commit hash is given; specifying one would make the examples and timings reproducible.
  4. [Section 6] The timing tables do not state the machine, the Macaulay2 version, or whether the default options (in particular QuickRank and HybridLimit) were used; this makes the comparisons with Cremona difficult to reproduce.
  5. [Section 4] The sentence "The following exam illustrates this" should read "The following example illustrates this".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the package applies published theorems and is benchmarked against external implementations.

full rationale

RationalMaps is a software package that implements known mathematical criteria rather than deriving new results from its own outputs. The central birationality and inverse-map criterion (Theorem 3.1) is explicitly quoted from [DHS12], and the base-locus characterization from [Sim04]. Although [DHS12] includes the second author, it is a published, peer-reviewed theorem whose content is independent of this package's computations; citing it is using external support, not making a prediction reduce to its own input. The paper also provides external validation by comparing outputs with the Cremona package via isSameMap and by testing against the sharp degree bounds from [HS17]. No parameter is fitted to a subset of data and then presented as a prediction, and no quantity is defined in terms of the result it is claimed to explain. The QuickRank option uses randomized rank computations and the paper explicitly discloses 'a certain amount of randomness' in FastLinAlg; this is a correctness and robustness caveat, not a circularity. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No new mathematical entities or fitted parameters are introduced. The central claim rests on published theorems (Sim04, DHS12, HS17), standard Groebner basis machinery, and the correctness of the Macaulay2 implementation. The observed performance gains come from algorithmic choices (SimisStrategy, HybridStrategy) rather than new mathematics.

assumptions (3)
  • domain assumption Theorem 3.1 (Doria-Hassanzadeh-Simis, 2012): birationality and inverse map computation via rank of weak Jacobian dual matrix
    This theorem is the theoretical core for isBirationalMap and inverseOfMap; the paper cites it without proof.
  • domain assumption The source variety X is irreducible, so its coordinate ring R is a domain, and X is non-degenerate
    Stated before Theorem 3.1: 'In the package we consider only the cases where X is irreducible i.e. R is a domain.' Violating this invalidates the inverse computation.
  • standard math Standard Groebner basis elimination theory is correct and implemented in Macaulay2
    The Rees algebra computation uses standard elimination; correctness depends on external computer algebra.

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Cite this review

Pith. "Pith review of RationalMaps, a package for Macaulay2." pith.science (2026). https://pith.science/paper/MU32G7NA

@misc{pith2026190804337,
  author       = {Pith},
  title        = {Pith review of: RationalMaps, a package for Macaulay2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MU32G7NA}},
  note         = {Machine review of arXiv:1908.04337}
}
read the original abstract

This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    H. Bass, E. H. Connell, and D. Wright : The J acobian conjecture: reduction of degree and formal expansion of the inverse , Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287--330. 663785

    work page 1982

  2. [2]

    Boehm : Parametrization, Package for Macaulay2, see \\ http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.14/share/doc/Macaulay2/Parametrization/html/

    J. Boehm : Parametrization, Package for Macaulay2, see \\ http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.14/share/doc/Macaulay2/Parametrization/html/

    work page

  3. [3]

    A. V. Doria, S. H. Hassanzadeh, and A. Simis : A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390--413. 2900548

    work page 2012

  4. [4]

    Eisenbud, A

    D. Eisenbud, A. Taylor, S. Popescu, and M. E. Stillman : ReesAlgebra: A Macaulay2 package. Version 2.2

    work page

  5. [5]

    Eisenbud, A

    D. Eisenbud, A. Taylor, S. Popescu, and M. E. Stillman : The ReesAlgebra package in Macaulay2 , The Journal of Software for Algebra and Geometry 8 (2018)

    work page 2018

  6. [6]

    S. H. Hassanzadeh and A. Simis : Bounds on degrees of birational maps with arithmetically C ohen- M acaulay graphs , J. Algebra 478 (2017), 220--236. 3621670

    work page 2017

  7. [7]

    He and Y

    Z. He and Y. Lei : The M ori dream space property of blow-ups of projective spaces at points and lines , Work in preparation

    work page

  8. [8]

    Hulek, S

    K. Hulek, S. Katz, and F.-O. Schreyer : Cremona transformations and syzygies, Math. Z. 209 (1992), no. 3, 419--443. 1152265

    work page 1992

Show all 16 references
  1. [9]

    Martinova, M

    B. Martinova, M. Robinson, K. Schwede, and Y. W. Yao : FastLinAlg.m2, a Macaulay2 package for sometimes faster function field arithmetic , A pre-release version of this package

  2. [10]

    Russo and A

    F. Russo and A. Simis : On birational maps and J acobian matrices , Compositio Math. 126 (2001), no. 3, 335--358. 1834742

    2001

  3. [11]

    J. G. Semple and J. A. Tyrrell : The C remona transformation of S_ 6 by quadrics through a normal elliptic septimic scroll ^ 1 R^ 7 , Mathematika 16 (1969), 89--97. 0249431

    1969

  4. [12]

    Simis : Two differential themes in characteristic zero, Topics in algebraic and noncommutative geometry ( L uminy/ A nnapolis, MD , 2001), Contemp

    A. Simis : Two differential themes in characteristic zero, Topics in algebraic and noncommutative geometry ( L uminy/ A nnapolis, MD , 2001), Contemp. Math., vol. 324, Amer. Math. Soc., Providence, RI, 2003, pp. 195--204. 1986124

    2001

  5. [13]

    Simis : Cremona transformations and some related algebras, J

    A. Simis : Cremona transformations and some related algebras, J. Algebra 280 (2004), no. 1, 162--179. 2081926

    2004

  6. [14]

    Staglian \`o : Cremona: Some computations for rational maps between projective varieties

    G. Staglian \`o : Cremona: Some computations for rational maps between projective varieties. Version 4.3

  7. [15]

    Staglian \`o : A Macaulay2 package for computations with rational maps , The Journal of Software for Algebra and Geometry 8 (2018)

    G. Staglian \`o : A Macaulay2 package for computations with rational maps , The Journal of Software for Algebra and Geometry 8 (2018)

    2018

  8. [16]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write mrnumber output.nonempty.mrnumber newline INTEGERS nameptr namesleft numnames F...

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