Pith. sign in

REVIEW 3 major objections 4 minor 42 references

The Universe of Deligne-Mostow Varieties

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that for all 85 Deligne–Mostow pairs the period map extends to an isomorphism between the Kirwan blow-up and the toroidal compactification exactly when condition (T) holds.

desk verdict The new (T)-criterion and the 85-pair census are genuinely useful, but the paper's reduction method is invalid: its own Table 4 gives a counterexample to Proposition 3.5(1). read the letter →

arxiv 2504.16235 v2 pith:Z4Z26WAA submitted 2025-04-22 math.AG math.NT

Klaus Hulek , Yota Maeda This is my paper · ORCID
classification math.AGmath.NT MSC 14L2414E3014G35
keywords Deligne-MostowvarietiesballquotientsperiodmapsKirwanblow-uptoroidalcompactificationconfigurationspaceslogminimalmodelprogramgeometricinvarianttheory
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 asks when Deligne–Mostow period maps—built from the monodromy of hypergeometric differential forms on configurations of $n$ ordered or partially unordered points on $\mathbb{P}^1$—lift from an open isomorphism to an isomorphism between two natural compactifications: the Kirwan blow-up of the GIT quotient and the toroidal compactification of the ball quotient. For the full universe of 85 Deligne–Mostow pairs, the paper proves a numerical criterion: the lift is an isomorphism exactly when condition (T) holds, meaning there is no block $T_1\subset S$ with at least three points and no set $T_2\subset S^{\mathrm{c}}$ such that the weights on $T_1\sqcup T_2$ sum to $1$. If (T) fails, neither the period map nor its inverse lifts, so the two compactifications are genuinely different. The result settles the compactification question uniformly, and a partial order on the universe reduces the verification to minimal and maximal entries.

What carries the argument

The load-bearing mechanism is the comparison of two divisors locally at polystable points. On the GIT side, the Kirwan blow-up $M^K_{w,S}$ resolves the polystable locus; on the ball-quotient side, the toroidal compactification $X^T_{w,S}$ adds the cusps. Around a polystable point whose support has two points, a local transverse slice to the group orbit (a Luna slice) gives local coordinates in which the discriminant divisor is a product of a monomial and binary discriminants $\operatorname{disc}(X^r+b_1X^{r-2}+\cdots+b_{r-1})$; blowing up the origin, the strict transform meets the exceptional divisor normally exactly when every such $r$ is at most $2$. Condition (T) is precisely the absence of a block with $r\ge 3$ and total weight $1$. Transversality up to finite quotients triggers the Borel extension theorem, producing the morphism, and a birational lemma using $Q$-factoriality (every divisor has a multiple that is Cartier) upgrades it to an isomorphism. A separate combinatorial device, the partial order $\prec$, shows that a check on minimal and maximal elements suffices.

What would settle it

Perform the omitted blow-up check for a block $T_1$ of size 3 or 5 whose weights sum to 1: at the corresponding two-point degeneration, write the discriminant as a product of the binary discriminants of a degree-$|T_1|$ polynomial and the remaining factor, blow up the origin, and determine whether the strict transform and the exceptional divisor meet transversally. The theorem predicts generic non-transversality; any transversal intersection would invalidate the only-if direction.

Watch

Extended reading notes

Core claim

In Theorem 2.6 the paper establishes the exact dichotomy for every Deligne–Mostow pair $(w,S)$. On one hand, when (T) holds, the period map $M_{w,S}\to X_{w,S}$ extends to an isomorphism $M^K_{w,S}\xrightarrow{\sim} X^T_{w,S}$; the proof uses transversality of the strict transform of the discriminant with the Kirwan exceptional divisor, the Borel extension theorem, $Q$-factoriality, and a birational criterion for isomorphisms. On the other hand, when (T) fails, a local computation shows that some component of the strict transform of the discriminant meets the Kirwan exceptional divisor generically non-transversally, while the toroidal boundary always meets the discriminant transversally; since both spaces are $Q$-factorial, no lift exists in either direction. The paper also records Corollary 1.6 on quotients by the symmetric group and Corollary 1.9: in the non-(T) case the Kirwan blow-up is not a semi-toroidal compactification, the pair $(M^K_{w,S}, \Delta^K_{w,S})$ is not a log canonical log minimal model, and it is not log $K$-equivalent to $(X^T_{w,S}, \Delta^T_{w,S})$.

Load-bearing premise

The proof that non-isomorphism really happens exactly when condition (T) fails relies on a local blow-up calculation at degenerations with three or five coincident points; for those two sizes the paper states the calculation is the same as the published four- and six-point cases but does not show it.

Editorial extensions

If this is right

  • The classification is complete: for all 85 Deligne–Mostow equivalence classes, one can decide the natural-isomorphism question by checking (T) on the weights, with no further geometric input.
  • Whenever $|S|\le 2$, condition (T) holds automatically, so the Kirwan blow-up and the toroidal compactification are always naturally isomorphic and the symmetric-group quotient commutes with the Kirwan blow-up.
  • Whenever (T) fails, the Kirwan blow-up is not a semi-toroidal compactification and is not log $K$-equivalent to the toroidal compactification, so it cannot serve as a log canonical log minimal model of the ball quotient.
  • By the reduction method, verifying the extremal elements (34 in the tables) is sufficient: failure at a minimal element propagates upward, and success at a maximal element propagates downward along $\prec$.

Reading between the lines

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

  • The same numerical criterion could be applied to other ball-quotient period maps, such as the generalized configurations mentioned in the paper's closing remarks; if the local normal forms are unchanged, the isomorphism question would again reduce to a weight-sum check.
  • The reduction to minimal and maximal elements makes the 85-case classification mechanically checkable: regenerate the weight tables, evaluate (T), and propagate; any mismatch with the paper's tables would indicate a hidden assumption in the enumeration.
  • The distinction between $|T_1|\le 2$ and $|T_1|\ge 3$ suggests that the local singularity type—an $A_1$ degeneration versus a worse one—is the actual geometric invariant behind the isomorphism statement; this could be tested by comparing exceptional-divisor geometry across pairs sharing the same $S$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Deligne-Mostow pairs (w,S), where w is a weight vector for n points on P^1 and S is a subset of indices acted on by a symmetric group. The main result, Theorem 2.6, gives a numerical criterion (T) for the Deligne-Mostow period map to extend to an isomorphism between the Kirwan blow-up M^K_{w,S} and the toroidal compactification X^T_{w,S}. The paper also introduces a partial order on the 85 Deligne-Mostow pairs and claims, in Reduction Method 3.8, that it suffices to verify Theorem 2.6 on minimal and maximal elements. Finally, the paper derives consequences for the log minimal model program, showing in Corollary 1.9 that when (T) fails the Kirwan blow-up is not a semi-toroidal compactification and not log K-equivalent to the toroidal compactification.

Significance. If Theorem 2.6 is correct, it completely and uniformly settles the natural-isomorphism question for all 85 Deligne-Mostow pairs, extending the earlier results of Gallardo-Kerr-Schaffler, Hulek-Maeda, and Hulek-Kondo-Maeda. The criterion (T) is genuinely easy to check, and the tables in Section 5 are a useful census of the Deligne-Mostow universe. The LMMP consequences in Section 4 are natural applications of the main criterion. However, the current manuscript contains a false statement in the reduction method and an incomplete verification of a load-bearing local computation, so the advertised significance is not yet fully established.

major comments (3)
  1. [§3.2, Proposition 3.5(1)] Proposition 3.5(1) is false as stated. Let A be the pair (5, (3/4,1/2,1/4,1/4,1/4), N{3,5}), written with sorted weights (1/4,1/4,1/4,1/2,3/4) and S={1,2,3}, and let B be (8, (1/4)^8, N3). Then A≺B by Definition 3.1: 5≤8, each weight of B is at most the corresponding weight of A, |S|=|S'|=3, and w(S)=w'(S')=3/4. For A, the only subset T1⊂S with |T1|≥3 is T1=S, whose weight sum is 3/4, and no subset of the complement {1/2,3/4} has weight 1/4, so (T) holds; indeed Table 4 records this row as T. For B, taking T1=S and T2={4} gives total weight 1, so (T) fails; Table 4 also records this row as NT. This directly contradicts Proposition 3.5(1). Consequently, the proof of Reduction Method 3.8 and the extremal counts in Table 2 are not justified as they stand. The assertion in the proof that the claim follows from |S|=|S'| and w(S)=w'(S') is insufficient, because condition (T) also involves weights outside S, and these may decrease under ≺.
  2. [§2, Proposition 2.2(2)] The proof of Proposition 2.2(2) is incomplete in a load-bearing way. The non-transversality direction requires a local blow-up computation in a Luna slice, but the cases |T1|=3 and |T1|=5 are only asserted to follow from "a similar computation," while the cases |T1|=4 and |T1|=6 are taken from earlier papers. Since Theorem 2.6's if-and-only-if statement depends on Proposition 2.2(2), the missing cases are not a cosmetic gap. The authors should either provide the full computation for |T1|=3 and |T1|=5, or supply a precise reference where these cases are proved. Since the computation is described as "computer-based," an appendix with explicit coordinates or reproducible code would also be appropriate.
  3. [§3, Reduction Method 3.8 and Theorem 3.7] Because Proposition 3.5(1) fails, the claimed reduction to minimal and maximal elements needs to be re-examined even if Theorem 2.6 is independently proved in Section 2. The reduction method relies on the monotonicity of (T) along ≺ to guarantee that a minimal element below a non-(T) pair is itself non-(T), and that a maximal element above a (T) pair is itself (T). The counterexample in Proposition 3.5(1) shows that such monotonicity does not hold in general. The authors should state precisely which weaker form of Reduction Method 3.8 remains valid, or prove the extremal reduction by a different argument that does not use Proposition 3.5(1).
minor comments (4)
  1. [Throughout] There are numerous typos and misspellings that should be corrected in a revision, including "rove" (Section 1.3), "Propoitipon" (Section 2), "thar" (Section 3.2), "unorderend" (Section 3.2), "a prior" (Section 1.1), and "poystable" (proof of Theorem 2.6).
  2. [Definition 3.1] The definition of ≺ is given on sorted representatives of equivalence classes. The authors should state explicitly that ≺ is well-defined on the equivalence classes in ADM, since the choice of representative is not completely fixed by the condition "we can do so without any restrictions."
  3. [Tables 4 and 5] The labels Max and Min in Tables 4 and 5 are not formally defined in the text. Clarify the intended definition: are these the maximal elements among pairs satisfying (T) and minimal elements among pairs not satisfying (T), or are they minimal and maximal with respect to ≺ in the whole universe ADM?
  4. [Remark 1.10(2)] The sentence beginning "Our approach to out to K-equivalene" contains a typo and should read "Our approach to K-equivalence".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the (T) criterion is a numerical condition verified by computation, and the self-citations to [HM25]/[HKM24] are independent published results; proof gaps and a flawed reduction lemma are correctness issues, not circularity.

full rationale

The central claim (Theorem 2.6) is not obtained by defining (T) as the answer: (T) is a purely combinatorial condition on weights, and the paper proves the non-transversality/transversality equivalence via Luna-slice computations (Proposition 2.2) before applying Borel extension and Lemma 2.7. No parameter is fitted to force the classification; the 85 cases are enumerated and checked against (T) in Tables 4–5. The cases |T1|=4,6 are delegated to [HM25, Theorem 3.4] and [HKM24, Theorem 2.5]; these are prior published theorems with their own parameter-free proofs, so citing them is real support rather than circularity. The statement that |T1|=3,5 'follow from a similar computation' is an omitted proof, not a reduction of the theorem to its own input. The reliance on [GKS21, Theorem 1.1] is external. The reduction method, however, has a genuine correctness problem: Proposition 3.5(1) claims (T) is preserved under ≺, with the proof 'This follows from the fact |S|=|S'| and w(S)=w'(S')', but this implication is not valid, and Table 4 itself appears to contain a counterexample, e.g. (5,32111,N{3,5}) ≺ (8,11111111,N3) with (T) true for the first pair and false for the second. This does not make Theorem 2.6 circular, because Theorem 2.6 is proved before and independently of the reduction method, but it does mean the advertised reduction via extremal elements is unsupported as stated.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central theorem uses standard compactification theory and prior published theorems. It introduces no fitted constants and no invented entities. The main unstated load is the delegated local blow-up computation for the remaining values of |T1|.

assumptions (5)
  • standard math Toroidal compactifications of ball quotients are unique and extend locally liftable holomorphic maps (Namikawa, AMRT, Faltings-Chai).
    Used in Theorem 2.5 to lift the period map from the Kirwan side to the unique toroidal compactification.
  • domain assumption The Gallardo-Kerr-Schaffler theorem identifies the Kirwan blow-up with the toroidal compactification in the INT case and is compatible with finite quotients.
    Basis for the claim that toroidal compactifications behave well under S[w]-quotients and for reducing to unordered cases.
  • domain assumption The moduli of weighted pointed stable curves equals the Kirwan blow-up (Kiem-Moon, Hassett).
    Used to identify M^K_{w,S} with the Kirwan blow-up of the GIT quotient.
  • domain assumption The list of arithmetic non-cocompact Deligne-Mostow pairs in GKS21 Tables 2 and 3 is complete.
    The 85-case enumeration and all extremal-case arguments depend on this list being the full universe ADM.
  • ad hoc to paper The omitted local blow-up computations for |T1|=3 and |T1|=5 give the same non-transversality as the published cases |T1|=4 and |T1|=6.
    Proposition 2.2(2) says these cases follow from a similar computation but does not show them; this is a load-bearing unproved assertion.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Universe of Deligne-Mostow Varieties." pith.science (2026). https://pith.science/paper/Z4Z26WAA

@misc{pith2026250416235,
  author       = {Pith},
  title        = {Pith review of: The Universe of Deligne-Mostow Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4Z26WAA}},
  note         = {Machine review of arXiv:2504.16235}
}
abstract

Deligne and Mostow investigated period maps on the configuration spaces $M_{0,n}$ of $n$ ordered points on $\mathbb{P}^1$. The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered $n$-tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log $K$-equivalent to the unique toroidal compactification.

Figures

Figures reproduced from arXiv: 2504.16235 by the authors.

Figure 1
Figure 1. Reduction method can reduce the proof to the study of extremal cases. Indeed, the theorem says the following: if (E) fails for (w ′ , S′ ) and (w ′S ′ ) ≺ (w, S), then (E) also fails for (w, S). Conversely, if (E) holds for (w, S) and (w ′S ′ ) ≺ (w, S), then (E) also holds for (w ′ , S′ ). This can be summarised by: Reduction Method 3.8. In order to prove Theorem 1.4, it is enough to prove this the￾orem for extrema… view at source ↗
Figure 2
Figure 2. Inclusions of XB wi in the Gaussian case 5.2. 85 elements in ADM. In Subsection 5.1, we reviewed the basic facts about ancestral Deligne-Mostow varieties. Gallardo, Kerr and Schaffler proved Theorem 1.3, reducing the problem to the ancestral cases by using Theorem 5.1. In this subsection, we give the specific forms of maximal and minimal elements that play a crucial role in Theorem 3.7 and the Reduction Method 3.8 … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 33 canonical work pages

  1. [1]

    Alexeev, A

    V. Alexeev, A. Brunyate, Extending the Torelli map to toroidal compactifications of Siegel space, Invent. Math. 188, No. 1, 175-196 (2012)

    work page 2012

  2. [2]

    Alexeev, A

    V. Alexeev, A. Brunyate, P. Engel, Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal, Geom. Topol. 26, No. 8, 3525-3588 (2022)

    work page 2022

  3. [3]

    Alexeev, P

    V. Alexeev, P. Engel, Compact moduli of K3 surfaces, Ann. Math. (2) 198, No. 2, 727-789 (2023)

    work page 2023

  4. [4]

    Alexeev, P

    V. Alexeev, P. Engel, C. Han, Compact moduli of K3 surfaces with a nonsymplectic automorphism, arXiv:2110.13834v2

    work page arXiv

  5. [5]

    Alexeev, R

    V. Alexeev, R. Livingston, J. Tenini, M. Arap, X. Hu, L. Huckaba, P. McFaddin, S. Musgrave, J. Shin, C. Ulrich, Extended Torelli map to the Igusa blow-up in genus 6, 7, and 8, Exp. Math. 21, No. 2, 193-203 (2012)

    work page 2012

  6. [6]

    A. Ash, D. Mumford, M. Rapoport, Y.-S. Tai, Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010

    work page 2010

  7. [7]

    Avritzer, H

    D. Avritzer, H. Lange, The moduli spaces of hyperelliptic curves and binary forms, Math. Z. 242, No. 4, 615-632 (2002)

    work page 2002

  8. [8]

    Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J

    A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differ. Geom. 6 (1972), 543--560

    work page 1972

Show all 42 references
  1. [9]

    Casalaina-Martin, S

    S. Casalaina-Martin, S. Grushevsky, K. Hulek, R. Laza, Non-isomorphic smooth compactifications of the moduli space of cubic surfaces, Nagoya Mathematical Journal, 1-51 (2023)

    2023

  2. [10]

    Casalaina-Martin, S

    S. Casalaina-Martin, S. Grushevsky, K. Hulek, R. Laza, The birational geometry of moduli of cubic surfaces and cubic surfaces with a line Moduli 2, Article ID e1, 24 p. (2025)

    2025

  3. [11]

    Deligne, G.D

    P. Deligne, G.D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IH\' E S 63 (1986) 5–89

    1986

  4. [12]

    H. Deng, P. Gallardo, Eigenperiods and the moduli of points in the line, Nagoya Mathematical Journal. Published online 2025:1-24

    2025

  5. [13]

    Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series

    I. Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series. 296 (2003)

    2003

  6. [14]

    Dolgachev, B

    I. Dolgachev, B. van Geemen, S. Kondō, A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces, J. Reine Angew. Math. 588, 99-148 (2005)

    2005

  7. [15]

    Doran, Hurwitz spaces and moduli spaces as ball quotients via pull-back, arXiv:0404363 (2004)

    B. Doran, Hurwitz spaces and moduli spaces as ball quotients via pull-back, arXiv:0404363 (2004)

    2004

  8. [16]

    Doran, C

    B. Doran, C. Doran, A. Harder, Picard–Fuchs uniformization of modular subvarieties, Uniformization, Riemann–Hilbert Correspondence, Calabi–Yau Manifolds & Picard–Fuchs Equations, Adv. Lect. Math. (ALM), vol. 42, Int. Press, Somerville, MA, 2018, pp. 21–54

    2018

  9. [17]

    Faltings, C.-L

    G. Faltings, C.-L. Chai, Degeneration of Abelian Varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin (1990)

    1990

  10. [18]

    Gallardo, M

    P. Gallardo, M. Kerr, L. Schaffler, Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces, Adv. Math. 381 (2021), Paper No. 107632

    2021

  11. [19]

    Hassett, Moduli spaces of weighted pointed stable curves, Adv

    B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316-352

    2003

  12. [20]

    Hulek, S

    K. Hulek, S. Kond\= o , Y. Maeda, Compactifications of the Eisenstein ancestral Deligne-Mostow variety, arXiv:2403.18345 (2024)

    2024 arXiv

  13. [21]

    Hulek, Y

    K. Hulek, Y. Maeda, Revisiting the moduli space of 8 points on ^1 , Adv. Math. 463 (2025)

    2025

  14. [22]

    Keel, Intersection theory of moduli space of stable n -pointed curves of genus zero, Trans

    S. Keel, Intersection theory of moduli space of stable n -pointed curves of genus zero, Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992)

    1992

  15. [23]

    Kiem, H-.B

    Y.-H. Kiem, H-.B. Moon, Moduli spaces of weighted pointed stable rational curves via GIT, Osaka J. Math. 48 (2011), no. 4, 1115–1140

    2011

  16. [24]

    Kiernan, S

    P. Kiernan, S. Kobayashi, Satake compactification and extension of holomorphic mappings, Invent. Math. 16, 237-248 (1972)

    1972

  17. [25]

    F. C. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. Math. (2) 122, 41-85 (1985)

    1985

  18. [26]

    Koll \'a r, S

    J. Koll \'a r, S. Mori, Birational geometry of algebraic varieties. With the collaboration of C . H . Clemens and A . Corti Cambridge Tracts in Mathematics, 134 (1998), Cambridge University Press

    1998

  19. [27]

    Laza, The KSBA compactification for the moduli space of degree two K3 pairs, J

    R. Laza, The KSBA compactification for the moduli space of degree two K3 pairs, J. Eur. Math. Soc. (JEMS) 18, No. 2, 225-279 (2016)

    2016

  20. [28]

    Looijenga, Semi-toroidal partial compactifications I, Report 8520 (1985), 72 pp., Catholic University Nijmegen

    E. Looijenga, Semi-toroidal partial compactifications I, Report 8520 (1985), 72 pp., Catholic University Nijmegen

    1985

  21. [29]

    Looijenga, New compactifications of locally symmetric varieties, Proceedings of the 1984 Vancouver conference in algebraic geometry, 341–364, CMS Conf

    E. Looijenga, New compactifications of locally symmetric varieties, Proceedings of the 1984 Vancouver conference in algebraic geometry, 341–364, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986

    1984

  22. [30]

    Looijenga, Compactifications defined by arrangements

    E. Looijenga, Compactifications defined by arrangements. I, - The ball quotient case, Duke Math. J. 118 (2003), no. 1, 151-187

    2003

  23. [31]

    Looijenga, Compactifications defined by arrangements

    E. Looijenga, Compactifications defined by arrangements. II, Locally symmetric varieties of Type IV, Duke Math. J. 119 (2003) no.3, 527-588

    2003

  24. [32]

    H. Moon, L. Schaffler, KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four, Proc. Edinb. Math. Soc., II. Ser. 64, No. 1, 99-127 (2021)

    2021

  25. [33]

    Mostow, Generalized Picard lattices arising from half-integral conditions, Publ

    G.D. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math. IH\'ES 63 (1986) 91-106

    1986

  26. [34]

    Mukai An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics 81

    S. Mukai An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics 81. 503 p. (2012)

    2012

  27. [35]

    Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math

    Y. Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52, 197-259 (1973)

    1973

  28. [36]

    Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties

    Y. Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties. I, Math. Ann. 221, 97-141 (1976)

    1976

  29. [37]

    Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties

    Y. Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties. II, Math. Ann. 221, 201-241 (1976)

    1976

  30. [38]

    Namikawa, Toroidal Compactification of Siegel Spaces, Lecture Notes in Math

    Y. Namikawa, Toroidal Compactification of Siegel Spaces, Lecture Notes in Math. 812, Springer, Berlin (1980)

    1980

  31. [39]

    Odaka, Semi-toroidal and toroidal compactifications as log minimal models, and applications to weak K-moduli, arXiv:2203.09120

    Y. Odaka, Semi-toroidal and toroidal compactifications as log minimal models, and applications to weak K-moduli, arXiv:2203.09120

    arXiv

  32. [40]

    Shah, A complete moduli space for K3 surfaces of degree 2, Ann

    J. Shah, A complete moduli space for K3 surfaces of degree 2, Ann. Math. (2) 112, 485-510 (1980)

    1980

  33. [41]

    W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, Geom. Topol. Monogr. 1, 511-549 (1998)

    1998

  34. [42]

    C. Yu, Z. Zheng, Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties, Algebra & Number Theory 14 (10) (2020), 2647-2683

    2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.