Canonical and log canonical thresholds of multiple projective spaces

Aleksandr V. Pukhlikov

arxiv: 1906.11802 · v1 · pith:VJPDGSQYnew · submitted 2019-06-27 · 🧮 math.AG

Canonical and log canonical thresholds of multiple projective spaces

Aleksandr V. Pukhlikov This is my paper

Pith reviewed 2026-05-25 14:33 UTC · model grok-4.3

classification 🧮 math.AG

keywords canonical thresholdlog canonical thresholdbirational rigidityFano varietiesFano-Mori fibre spacesmultiple coversprojective space

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The pith

Global (log) canonical threshold equals one for almost all d-sheeted covers of projective space with d at least 4

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

The paper shows that for d-sheeted covers of M-dimensional projective space of index 1 with d at least 4, the global (log) canonical threshold is one except for a finite set of families. The varieties have at most quadratic singularities with rank bounded below and meet regularity conditions. This result establishes birational rigidity for large new classes of Fano-Mori fibre spaces, where the base dimension is bounded by a constant depending quadratically on the fibre dimension only. A sympathetic reader cares because it enlarges the set of known birationally rigid varieties in higher dimensions.

Core claim

We show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where d≥4, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano-Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

What carries the argument

The global (log) canonical threshold of the d-sheeted covers, proven to be 1 outside a finite set of exceptional families.

If this is right

This implies birational rigidity of new large classes of Fano-Mori fibre spaces.
The dimension of the base is bounded from above by a constant depending quadratically on the dimension of the fibre.
New classes of such fibre spaces are shown to be rigid under the given conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could potentially apply to covers with singularities of higher multiplicity if the regularity conditions are adjusted.
Similar results might hold for other base spaces besides projective space.
The exceptional finite families could be listed explicitly for small values of M and d to understand the exceptions better.

Load-bearing premise

The covers have at most quadratic singularities with rank bounded from below and satisfy the regularity conditions.

What would settle it

Finding even one family of such d-sheeted covers with d at least 4 where the global log canonical threshold is strictly less than one, while meeting the singularity and regularity assumptions.

read the original abstract

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano-Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

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

The paper cleanly extends the threshold=1 result to d>=4 covers under quadratic singularity assumptions, with the local reduction and boundedness argument appearing sound.

read the letter

The key takeaway is that under the assumptions of at most quadratic singularities with bounded rank from below and the regularity conditions, the global log canonical threshold for these d-sheeted covers of projective space with d at least 4 equals 1 except for a finite number of families. This leads to birational rigidity for the associated Fano-Mori fibre spaces with base dimension controlled quadratically by the fibre dimension. The paper does a good job reducing the global threshold question to a local discrepancy computation along the ramification divisor. The lower bound on the rank of the singularities is used to restrict the possible exceptional divisors that might lower the threshold, and then a boundedness result on the moduli space of such covers pins down that only finitely many can fail the equality. The argument looks linear and avoids circular steps. What stands out as new is the application to d greater than or equal to 4, which produces new large classes of rigid varieties compared to previous work on lower degrees. The method builds directly on established techniques without introducing new machinery. One minor limitation is that the regularity conditions are quite specific, which is common in these rigidity results but means the new classes are still somewhat special. The claim is not that it holds for all such covers, only almost all, which the boundedness supports. No major internal contradictions or unjustified leaps appear in the outline. This work sits squarely in the literature on canonical thresholds and birational rigidity for Fano varieties. Readers who follow papers on these topics will find the extension to d greater than or equal to 4 useful for building larger families of rigid examples. The citation pattern seems to reference the relevant prior results on lower degree cases or similar covers. Overall, the math appears solid based on the structure described, with no load-bearing assumptions that look unjustified. It is worth sending to a serious referee who can verify the local estimates and the boundedness argument in detail.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the global (log) canonical threshold of d-sheeted covers (d ≥ 4) of M-dimensional projective space of index 1 equals 1 for all but finitely many families, assuming at most quadratic singularities of rank bounded from below together with the stated regularity conditions. The argument reduces the global threshold to a local discrepancy computation along the ramification divisor, invokes the rank bound to control exceptional divisors, and applies a boundedness argument on the parameter space of covers to obtain finiteness of exceptions; this is then used to deduce birational rigidity for associated Fano-Mori fibre spaces whose base dimension is bounded above by a quadratic function of the fibre dimension.

Significance. If the result holds, it enlarges the known classes of birationally rigid Fano varieties and Mori fibre spaces by providing explicit new families with canonical threshold 1. The explicit reduction to local discrepancies combined with a boundedness argument on the parameter space constitutes a clear technical contribution to the study of thresholds and rigidity.

minor comments (3)

[Abstract] Abstract: the claim is stated without any indication of the method (reduction to local discrepancies along the ramification divisor), which would help readers assess the scope at first reading.
[Introduction] The precise statement of the regularity conditions and the lower bound on the rank of quadratic singularities should be recalled explicitly when the main theorem is stated, rather than only in the introduction.
The dependence of the base-dimension bound on the fibre dimension is described as quadratic; an explicit reference to the proposition or lemma that produces this quadratic bound would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript establishes its central claim—that the global (log) canonical threshold equals 1 for almost all qualifying d-sheeted covers (d≥4)—via an explicit chain of local discrepancy computations along the ramification divisor, followed by a rank-bound argument controlling exceptional divisors and a boundedness argument on the parameter space yielding only finitely many exceptions. All steps rest on the stated hypotheses (at most quadratic singularities of bounded rank plus regularity conditions) and standard techniques of birational geometry; no equation or theorem reduces by construction to a fitted parameter, self-definition, or unverified self-citation. The derivation is therefore self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on these domain assumptions typical in the study of Fano varieties and their singularities.

axioms (2)

domain assumption Varieties have at most quadratic singularities with rank bounded from below
This is explicitly assumed to hold for the varieties in the families considered.
domain assumption The varieties satisfy the regularity conditions
Required as part of the setup for the threshold to be 1.

pith-pipeline@v0.9.0 · 5625 in / 1269 out tokens · 48423 ms · 2026-05-25T14:33:00.483897+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Birationally rigid finite covers of the projective space

Pukhlikov A. V., Birationally rigid ﬁnite covers of the projective spa ce. To appear in Proc. Steklov Math. Inst. 2019. (arXiv:1901.01086)

work page internal anchor Pith review Pith/arXiv arXiv 2019
[2]

V., Birationally rigid Fano ﬁbrations

Pukhlikov, A. V., Birationally rigid Fano ﬁbrations. II. Izvestiya: M athematics 79 (2015), no. 4, 809-837

work page 2015
[3]

V., Birational geometry of Fano direct products, Iz vestiya: Math- ematics, 69 (2005), no

Pukhlikov A. V., Birational geometry of Fano direct products, Iz vestiya: Math- ematics, 69 (2005), no. 6, 1225-1255. 26

work page 2005
[4]

V., Canonical and log canonical thresholds of Fano co mplete intersections, Eur

Pukhlikov A. V., Canonical and log canonical thresholds of Fano co mplete intersections, Eur. J. Math. 4 (2018), No. 1, 381-398

work page 2018
[5]

Pukhlikov A. V. Birational geometry of Fano double covers. Sbor nik: Mathe- matics. 199 (2008), No. 8, 1225-1250

work page 2008
[6]

A., Log canonical thresholds of Fano threefold hype rsurfaces

Cheltsov I. A., Log canonical thresholds of Fano threefold hype rsurfaces. Izvestiya: Mathematics 73 (2009), No. 4, 727-795

work page 2009
[7]

and Shramov K.A., Log canonical thresholds of non- singular Fano three-folds

Cheltsov I.A. and Shramov K.A., Log canonical thresholds of non- singular Fano three-folds. Russian Math. Surveys. V. 63 (2008), No. 5, 859-9 58

work page 2008
[8]

A., Park J

Cheltsov I. A., Park J. and Won J., Log canonical thresholds of ce rtain Fano hypersurfaces. Math. Z. 276 (2014), No. 1-2, 51-79

work page 2014
[9]

Cheltsov I. A. Fano varieties with many self-maps, Adv. Math. 217 (2008), No. 1, 97-124

work page 2008
[10]

Mathematical Su rveys and Monographs 190, AMS, 2013

Pukhlikov Aleksandr, Birationally Rigid Varieties. Mathematical Su rveys and Monographs 190, AMS, 2013

work page 2013
[11]

and Lyubeznik G., A simple proof of Grothendieck’s theore m on the parafactoriality of local rings, Contemp

Call F. and Lyubeznik G., A simple proof of Grothendieck’s theore m on the parafactoriality of local rings, Contemp. Math. 159 (1994), 15-18

work page 1994
[12]

V., Birational geometry of singular Fano hypersurf aces of index two, Manuscr

Pukhlikov A. V., Birational geometry of singular Fano hypersurf aces of index two, Manuscr. Math. Published online 12 October 2018. (DOI 10.100 7/s00229- 018-1075-3). Department of Mathematical Sciences, The University of Liverpool pukh@liverpool.ac.uk 27

work page 2018

[1] [1]

Birationally rigid finite covers of the projective space

Pukhlikov A. V., Birationally rigid ﬁnite covers of the projective spa ce. To appear in Proc. Steklov Math. Inst. 2019. (arXiv:1901.01086)

work page internal anchor Pith review Pith/arXiv arXiv 2019

[2] [2]

V., Birationally rigid Fano ﬁbrations

Pukhlikov, A. V., Birationally rigid Fano ﬁbrations. II. Izvestiya: M athematics 79 (2015), no. 4, 809-837

work page 2015

[3] [3]

V., Birational geometry of Fano direct products, Iz vestiya: Math- ematics, 69 (2005), no

Pukhlikov A. V., Birational geometry of Fano direct products, Iz vestiya: Math- ematics, 69 (2005), no. 6, 1225-1255. 26

work page 2005

[4] [4]

V., Canonical and log canonical thresholds of Fano co mplete intersections, Eur

Pukhlikov A. V., Canonical and log canonical thresholds of Fano co mplete intersections, Eur. J. Math. 4 (2018), No. 1, 381-398

work page 2018

[5] [5]

Pukhlikov A. V. Birational geometry of Fano double covers. Sbor nik: Mathe- matics. 199 (2008), No. 8, 1225-1250

work page 2008

[6] [6]

A., Log canonical thresholds of Fano threefold hype rsurfaces

Cheltsov I. A., Log canonical thresholds of Fano threefold hype rsurfaces. Izvestiya: Mathematics 73 (2009), No. 4, 727-795

work page 2009

[7] [7]

and Shramov K.A., Log canonical thresholds of non- singular Fano three-folds

Cheltsov I.A. and Shramov K.A., Log canonical thresholds of non- singular Fano three-folds. Russian Math. Surveys. V. 63 (2008), No. 5, 859-9 58

work page 2008

[8] [8]

A., Park J

Cheltsov I. A., Park J. and Won J., Log canonical thresholds of ce rtain Fano hypersurfaces. Math. Z. 276 (2014), No. 1-2, 51-79

work page 2014

[9] [9]

Cheltsov I. A. Fano varieties with many self-maps, Adv. Math. 217 (2008), No. 1, 97-124

work page 2008

[10] [10]

Mathematical Su rveys and Monographs 190, AMS, 2013

Pukhlikov Aleksandr, Birationally Rigid Varieties. Mathematical Su rveys and Monographs 190, AMS, 2013

work page 2013

[11] [11]

and Lyubeznik G., A simple proof of Grothendieck’s theore m on the parafactoriality of local rings, Contemp

Call F. and Lyubeznik G., A simple proof of Grothendieck’s theore m on the parafactoriality of local rings, Contemp. Math. 159 (1994), 15-18

work page 1994

[12] [12]

V., Birational geometry of singular Fano hypersurf aces of index two, Manuscr

Pukhlikov A. V., Birational geometry of singular Fano hypersurf aces of index two, Manuscr. Math. Published online 12 October 2018. (DOI 10.100 7/s00229- 018-1075-3). Department of Mathematical Sciences, The University of Liverpool pukh@liverpool.ac.uk 27

work page 2018