Irrationality and monodromy for cubic threefolds

Ivan Smith

REVIEW 3 major objections 5 minor 36 references

Irrationality and monodromy for cubic threefolds

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The monodromy of cubic threefolds does not lift to a genus-five surface.

desk verdict A new monodromy non-factorization theorem for cubic threefolds, mostly solid, with one informal curve-configuration check in Section 4 that should be tightened. read the letter →

arxiv 1908.06667 v3 pith:2YGGOXY6 submitted 2019-08-19 math.AG math.GRmath.SG

Ivan Smith This is my paper · ORCID

classification math.AGmath.GRmath.SG MSC 14J3014H1032G2020F36

keywords cubicthreefoldsintermediateJacobianmonodromymappingclassgroupirrationalityArtinbraidrigidityDehntwists

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

The paper proves that the cohomological monodromy $\pi_1(M_{3,3}) \to \mathrm{Sp}(10;\mathbb{Z})$ of the universal family of smooth cubic threefolds does not factor through the genus-five mapping class group $\Gamma_5$. This matters because a rational threefold has intermediate Jacobian equal to a product of Jacobians of curves; for the family as a whole, rationality would force the monodromy on $H^3$ to factor through a mapping class group. The theorem therefore gives a group-theoretic obstruction to rationality that rational cohomology cannot see. It also rules out a particular mechanism by which all cubics could be rational with a fixed singularity-codimension type of $\theta$ divisor.

What carries the argument

The key object is the Artin group $G(\Gamma)$ associated to the graph on the sixteen vertices $\{0,1\}^4$: its generators satisfy braid relations along edges and commute off them, and a theorem of the paper's background gives $\pi_1(M_{3,3})$ as a quotient of $G(\Gamma)$ by extra relations. The load-bearing mechanism is the combination of a rigidity theorem for homomorphisms $Br_n \to \Gamma(\Sigma)$ with $n \ge 6$ and the change-of-coordinates principle for simple closed curves. The rigidity theorem forces each generator to be a Dehn twist in a nonseparating curve; the change-of-coordinates principle turns the Artin relations into a finite graph of geometric intersection numbers. The final contradiction is that this graph cannot be embedded in the curve graph of a genus-five surface while containing the affine $A_8$ chain supplied by the braid-group extensions.

What would settle it

Draw seven simple closed curves on a closed genus-five surface whose pairwise intersections realise the $A_7$ chain from Lemma 2.9 together with the curves $u$, $v$, and $w_+$ (or $w_-$) of Section 4; the paper's final contradiction says this configuration cannot exist. Alternatively, exhibit any homomorphism $G(\Gamma) \to \Gamma_5$ lifting the homological monodromy $G(\Gamma) \to \mathrm{Sp}(10;\mathbb{Z})$, which would overturn Theorem 1.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1.1: the homomorphism $\pi_1(M_{3,3}) \to \mathrm{Sp}(10;\mathbb{Z})$ given by parallel transport in the local system $R^3\pi_*\mathbb{Z}$ does not factor through the genus-five mapping class group $\Gamma_5$. The proof assumes such a factorization and uses a presentation of $\pi_1(M_{3,3})$ as a quotient of an Artin group $G(\Gamma)$ attached to the sixteen-vertex hypercube graph, together with rigidity constraints on braid-group homomorphisms into mapping class groups. It shows that every Artin generator would have to map to a positive Dehn twist in a nonseparating simple closed curve, and then that the required intersection pattern of these curves cannot be drawn on a closed surface of genus five. Corollary 4.3 extends the non-factorization to the mapping class group of any surface of total genus five.

Load-bearing premise

The argument rests on a rigidity theorem for homomorphisms from braid groups $Br_n$ into mapping class groups of small-genus surfaces; if that theorem's hypotheses fail in any of the applications at Lemmas 3.5, 3.7, 3.10 or 3.13, or if its centralising factor acted nontrivially on homology, the reduction of each generator to a positive Dehn twist would collapse.

Editorial extensions

If this is right

The monodromy map $\pi_1(M_{3,3}) \to \mathrm{Sp}(10;\mathbb{Z})$ does not factor through $\pi_1(\mathcal{M}_5)$ or $\pi_1(\mathcal{M}_5^{ct})$, because either would produce a factorization through $\Gamma_5$.
For every surface $\Sigma$ of total genus five, connected or disconnected, the monodromy does not factor through $\Gamma(\Sigma)$; the disconnected case is Corollary 4.3.
A rationality scenario in which every smooth cubic threefold has intermediate Jacobian whose theta-divisor singular locus has fixed codimension at least 2 is ruled out, since it would yield the forbidden mapping-class lift.
The obstruction is not visible in rational cohomology: a known result cited in the paper says there is no rational cohomological obstruction to such a factorization.

Reading between the lines

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

A testable extension is to see how few vertices of the Artin graph already force an impossible curve configuration; the paper notes four vertices and the triangle relations are left unused, so a smaller obstruction may exist.
The paper's suggested computation of the pullback of the separating-twist extension class to $H^2(\pi_1(M_{3,3}), \Lambda^3 H/H)$ would give an independent obstruction to lifting through the compact-type Torelli space, if the class is nonzero.
In the symplectic picture sketched in Section 1.3, the same non-factorization would obstruct realising the universal cubic family by symplectic blow-ups and blow-downs from a surface, once the heuristic relation between mapping class groups and autoequivalences of Fukaya categories is made precise.

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.

Desk Editor's Note

read the letter

You should know this paper proves something genuinely new: the cohomological monodromy of the universal cubic threefold family does not factor through the genus five mapping class group. Irrationality itself is old (Clemens-Griffiths et al.), but the factorization obstruction is not in the prior literature, and it gives a fresh geometric group theory perspective rather than a repackaged Hodge argument. The proof is honest: it explicitly separates the known irrationality from the new rigidity statement, and it relies on published theorems (Lonne's presentation, Castel's rigidity, Beauville's monodromy analysis) rather than on fitting parameters.

What the paper does well: the reduction in Section 3 is quite careful. Castel's theorem is applied only in the range n ≥ 6, g ≤ n/2, and the argument that the centralizer element w acts trivially on homology is justified by the transvection condition; the stress-test worry about w having nontrivial homology action does not survive contact with Lemma 2.6. The homological irreducibility arguments (Lemmas 2.7 and 2.8) are solid, and Corollary 4.3's reduction to the connected case is sensible.

The main soft spot is the final step in Section 4. After reducing to positive Dehn twists about nonseparating curves with a specific intersection graph, the contradiction is a planar curve-configuration statement about the genus-five surface. The proof invokes Figures 5–6 and a short paragraph: u and v must lie on opposite sides of h, and the two ends of w+ cannot be joined. This is a finite case check, but it is not formalized, and the statement about the additional handle's position is exactly the kind of thing that can hide an overlooked configuration. I don't see a concrete counterexample, and the informal argument is plausible, but this is where I would want the author to write out the cases or provide a small computer verification. The stress-test note identifies this correctly; it is not a fatal flaw, but it is the least secure step.

The citation pattern looks fair. The paper cites Lonne, Castel, Beauville, Hefez-Lazzeri, and the contextual remarks correct earlier errors in the discussion (per acknowledgements). The "aspirational context" on symplectic nonrationality is appropriately speculative and labeled as such.

Who is this for? Geometric group theorists and people working on monodromy, rigidity, and the symplectic side of rationality. It deserves a serious referee, despite the Section 4 informality, because the theorem is new and the main reduction is sound. My recommendation: send it to peer review. The referee should push for a more explicit enumeration in Section 4, but the paper should not be desk-rejected.

Referee Report

3 major / 5 minor

Summary. The paper proves that the cohomological monodromy representation π1(M_{3,3}) → Sp(10;Z) for the universal family of smooth cubic threefolds does not factor through the mapping class group of any closed surface of total genus 5. The proof uses Lönne's presentation of π1(M_{3,3}) as a quotient of the Artin group G(Γ) associated with a 4-dimensional hypercube graph, combined with Castel's rigidity theorem for braid-group representations into mapping class groups. The argument first shows that, under a hypothetical factorization, each generator of G(Γ) would have to be a positive Dehn twist about a nonseparating curve, and then rules out the required curve configuration on a genus-5 surface. Corollary 4.3 extends the non-factorization from the connected case to disconnected surfaces of total genus 5. The paper is explicit that this gives a geometric group theory perspective on irrationality rather than a new proof of irrationality of cubic threefolds.

Significance. If the proof is correct, the result is a clean and interesting obstruction at the level of orbifold fundamental groups: the universal family of cubic threefolds cannot be lifted through M5, even though there is no rational cohomological obstruction. The paper makes honest use of external theorems (Lönne, Castel, Beauville, Hefez–Lazzeri) and keeps the main algebraic reductions transparent. Its strength is that the conclusion is a precise, falsifiable statement about a concrete representation, and the proof reduces it to a finite curve-configuration problem. The main weakness is that the final configuration contradiction is not fully formalized, and the rank computation in Lemma 2.8 is asserted rather than documented; both points are load-bearing but appear fixable within the manuscript's scope.

major comments (3)

[Section 4, Figures 5–6] The final contradiction, which rules out a hypothetical factorization, rests on an informal planar case analysis. After establishing that the sixteen generators must act by positive Dehn twists with intersection graph Γ, the proof asserts that u and v must lie on opposite sides of h, and that the two dotted ends of w+ cannot be rejoined. These assertions are not fully justified: the additional handle may lie on either side of h, and u or v may wind around it, so the configuration space is larger than the figure suggests. Since this is the load-bearing step for Theorem 1.1, I ask for a rigorous argument, for example via Euler-characteristic bounds and the change-of-coordinates principle, or a finite enumeration of the possible embeddings of the subgraph in a genus-5 surface.
[Lemma 2.8] The claim that the sublattice spanned by the 14 non-extremal vertices has rank 10 is asserted by 'Direct computation' and is used in Corollaries 3.9 and 3.15 to contradict the irreducibility of the monodromy. Because this rank statement is load-bearing, the paper should include the actual intersection submatrix, or an explicit reproducible computation, so that the reader can verify it without reconstructing the Hefez–Lazzeri formula and checking sixteen-by-sixteen signs.
[Lemmas 3.5–3.13] The applications of Castel's Theorem 2.18 exclude the hyperelliptic involution in the centralizer of the chain only implicitly. In Lemma 3.7, for example, the conclusion that w is a power of the boundary twist requires ruling out the hyperelliptic involution of the chain neighbourhood; this follows from the fact that such an involution would have eigenvalue −1 on homology, contradicting the transvection action, but the paper does not say so. I recommend adding this one-sentence justification, as the same point recurs in Lemmas 3.10 and 3.13.

minor comments (5)

[Section 1.1] There is a typo in the sentence beginning 'This gives a geometric group theory perspective on the well-known irrationality'; the surrounding text uses 'allbeit' instead of 'albeit'.
[Lemma 2.9] The statement says that for each v there is a homomorphism Br6 → Z(σv), but the proof only exhibits this for v = (0001). The conclusion for arbitrary v follows from conjugacy of the generators; this should be stated explicitly.
[Figure 4 and the paragraph after it] The labels A, B, C for the regions in Figure 5 are used in the formal paragraph but are not marked in the figure; please add them or describe the regions verbally.
[Section 3.5, Corollary 3.17] The sentence treating the case r = 2, where γ1 ∪ γ2 separates the surface, is very compressed; a sentence explaining why the previous bounding-pair arguments apply verbatim would improve readability.
[Remark 2.16] The reference to 'the proof of [Sal, Lemma 5.10]' is appropriate, but the terminology 'Σ_b^1' in the remark is not defined in the main text; please clarify the notation for the surface with one boundary component.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained against external published theorems and never assumes the factorization it seeks to disprove.

full rationale

The paper's central claim is that the cohomological monodromy G(Γ) → Sp(10; Z) does not factor through the genus-five mapping class group. The proof assumes such a factorization for contradiction and derives constraints on the images of the Artin generators: each must be a positive Dehn twist in a nonseparating curve, with intersection pattern governed by Γ. These constraints are obtained from Lönne's external presentation of π1(M3,3), Castel's rigidity theorem applied within its stated hypotheses (n ≥ 6, g ≤ n/2), Picard–Lefschetz theory for transvections, and standard mapping class group facts about essential reduction systems. No parameter is fitted to the target conclusion, and the non-factorization claim is not used as an input anywhere. The contextual discussion of irrationality is explicitly presented as motivation rather than as a proof ingredient, and the self-citations [AS], [Smi12], and [Sei14] appear only in the aspirational symplectic discussion, not in the load-bearing argument. The final curve-configuration contradiction in Section 4 is informal and could merit formalization, but that is a rigor or correctness concern, not a circularity: the contradiction is derived from the assumed factorization and standard intersection-number relations, and it does not reduce to the theorem being proved. Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities appear. The central claim rests on published background theorems, a finite intersection-matrix computation, and the geometric rigidity theorem of Castel. The proof does not introduce new objects with independent evidence requirements.

assumptions (7)

standard math Lonne's presentation: π1(M_{3,3}) is a quotient of the Artin group G(Γ) attached to the four-dimensional cube by triangle and nonlocal relations.
Imported from Lonne 2009 and stated as Proposition 2.2. It is used to translate a hypothetical factorization through Γ_5 into a homomorphism G(Γ) → Γ_5.
standard math Castel's rigidity theorem: for n ≥ 6, any homomorphism Br_n → Γ(Σ_g^b) with g ≤ n/2 is either cyclic or given by twists along a chain of curves.
Stated as Theorem 2.18 and used throughout Sections 3.3 and 3.4 to constrain the induced braid group representations.
standard math Beauville's monodromy theorem: the homological monodromy of the universal cubic threefold has irreducible image in Sp(10;Z) preserving a quadratic refinement.
Stated as Lemma 2.7 and used to contradict reducible curve configurations.
standard math Hefez-Lazzeri intersection matrix: the 14 non-extremal vanishing cycles span a rank 10 sublattice of H3 of the cubic threefold.
Stated as Lemma 2.8 with a direct computation from the Hefez-Lazzeri formula, and used in Corollaries 3.9, 3.15, and 3.17.
standard math Birman-Lubotzky-McCarthy: centralizers of pseudo-Anosov mapping classes are virtually cyclic, and commuting mapping classes preserve each other's essential reduction systems.
Stated as Lemmas 2.12 and 2.13 and used to constrain braid group actions in the centralizer of a candidate reduction.
standard math Thurston's classification of mapping classes into periodic, reducible, and pseudo-Anosov types.
Stated in Section 2.2 and used throughout the reduction-system analysis in Section 3.
standard math Artin's theorem: for k < n, any homomorphism Br_n → Sym_k has cyclic image.
Cited as Art47 and used in Corollary 4.3 to handle disconnected surfaces and permutations of components.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Irrationality and monodromy for cubic threefolds." pith.science (2026). https://pith.science/paper/2YGGOXY6

@misc{pith2026190806667,
  author       = {Pith},
  title        = {Pith review of: Irrationality and monodromy for cubic threefolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YGGOXY6}},
  note         = {Machine review of arXiv:1908.06667}
}

read the original abstract

We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

Figures

Figures reproduced from arXiv: 1908.06667 by the authors.

**Figure 1.** The Artin graph for cubic surfaces Consider the Artin group G(Γ) generated by elements σv , for v ∈ Vert(Γ), and with relations (3) [σv, σw] = 1 if (v,w) 6∈ Edge(Γ); σvσwσv = σwσvσw if (v,w) ∈ Edge(Γ). We will call the generators σv the ‘standard generators’ of G(Γ). Remark 2.1 The vertices (0000) and (1111) are connected by edges to all others; we will call these two vertices extremal, and denote either by vext . F… view at source ↗

**Figure 2.** A copy of Br6 Proof An ordered sequence of vertices v1, . . . , vk defines a representation Brk+1 → G(Γ) exactly when the subgraph spanned by the vi inside Γ is a linear chain, so vi is joined by an edge to vi+1 for 1 ≤ i ≤ k − 1 and there are no other edges between these vertices. One checks the following ordered sequence of vertices has this property: (0001) → (0101) → (0100) → (0110) → (0010) → (1010) → (1000). T… view at source ↗

**Figure 3.** An A7 -chain of curves After cutting along the curves labelled a, c, e, g one obtains a torus with 8 boundary components. The remaining curves of the configuration Γ can now be drawn on this bordered surface Σ 8 1 . In particular, if we label by h the generator which comes from the vertex (1001) and which extends the A7 -chain to a cycle of 8 curves, as defining the Braff 9 -representation of Lemma 2.11, then this l… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The affine braid group generators after Σ has been cut open along the A7 -chain (the handle may be on the other side of h) h h u ′ u ′′ g g e f f e d d c c b b a a [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Two possible positions u ′ and u ′′ for the curve u, which meets only {a, e} Since u meets a, it meets it in precisely one of the two intervals into which a has already been divided by the intersections of a with b and h. These two options are shown in [PITH_FULL_IMAG…

**Figure 6.** Figure 6: Impossible curve configurations with {u,w +} respectively {v,w −} manifest: v w − a ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ b ❏ ❏ ❏ ❏ ❏ ❏ c ✐✐ ✐✐ ✐✐ ✐✐ ✐✐ ✐✐ ✐✐ sssssssssssssssss d tt tt tt t t t t t t t e ✐✐✐✐✐✐✐✐✐✐✐✐✐✐ s s s s s s s s s s s s s s s s s f …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[1]

Tarik Aougab, Ian Biringer, and Jonah Gaster, Determining the finite subgraphs of curve graphs, Groups Geom. Dyn. 13 (2019), no. 1, 31--45. 3900763

work page 2019
[2]

Allan Adler, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1978), no. 6, 1275--1280. 522700

work page 1978
[3]

Artin, Braids and permutations, Ann

E. Artin, Braids and permutations, Ann. of Math. (2) 48 (1947), 643--649. 0020989

work page 1947
[4]

Denis Auroux and Ivan Smith, Spherical objects in F ukaya categories of surfaces , Manuscript in preparation, 2019

work page 2019
[5]

1194, Springer, Berlin, 1986, pp

Arnaud Beauville, Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections compl\`etes, Complex analysis and algebraic geometry ( G \" o ttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 8--18. 855873

work page 1985
[6]

, Non-rationality of the symmetric sextic F ano threefold , Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Z\" u rich, 2012, pp. 57--60. 2987652

work page 2012
[7]

Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math

Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107--1120. 726319

work page 1983
[8]

378, vi+175

Fabrice Castel, Geometric representations of the braid groups, Ast\' e risque (2016), no. 378, vi+175. 3499033

work page 2016

Show all 36 references

[9]

Fabrizio M. E. Catanese, Infinitesimal T orelli theorems and counterexamples to T orelli problems , Topics in transcendental algebraic geometry ( P rinceton, N . J ., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 143--156. 756850

1981
[10]

Herbert Clemens and Phillip A

C. Herbert Clemens and Phillip A. Griffiths, The intermediate J acobian of the cubic threefold , Ann. of Math. (2) 95 (1972), 281--356. 0302652

1972
[11]

Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek, and Radu Laza, Extending the P rym map to toroidal compactifications of the moduli space of abelian varieties , J. Eur. Math. Soc. (JEMS) 19 (2017), no. 3, 659--723, With an appendix by Mathieu Dutour Sikiri\' c . 3612866

2017
[12]

Alberto Collino, A cheap proof of the irrationality of most cubic threefolds, Boll. Un. Mat. Ital. B (5) 16 (1979), no. 2, 451--465. 546468

1979
[13]

Georgios Daskalopoulos, Ludmil Katzarkov, and Richard Wentworth, Harmonic maps to T eichm\" u ller space , Math. Res. Lett. 7 (2000), no. 1, 133--146. 1748294

2000
[14]

48, Princeton University Press, Princeton, NJ, 2012, Translated from the 1979 French original by Djun M

Albert Fathi, Fran c ois Laudenbach, and Valentin Po\' e naru, Thurston's work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012, Translated from the 1979 French original by Djun M. Kim and Dan Margalit. 3053012

2012
[15]

49, Princeton University Press, Princeton, NJ, 2012

Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. 2850125

2012
[16]

(N.S.) 17 (2011), no

Masahiro Futaki and Kazushi Ueda, Homological mirror symmetry for B rieskorn- P ham singularities , Selecta Math. (N.S.) 17 (2011), no. 2, 435--452. 2803848

2011
[17]

Tawanda Gwena, Degenerations of cubic threefolds and matroids, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1317--1323. 2111937

2005
[18]

Vieweg, Braunschweig, 1999, pp

Richard Hain, Locally symmetric families of curves and J acobians , Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 91--108. 1722540

1999
[19]

, Finiteness and T orelli spaces , Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 57--70. 2264131

2006
[20]

Hefez and F

A. Hefez and F. Lazzeri, The intersection matrix of B rieskorn singularities , Invent. Math. 25 (1974), 143--157. 0396555

1974
[21]

Algebraic Geom

Richard Hain and David Reed, Geometric proofs of some results of M orita , J. Algebraic Geom. 10 (2001), no. 2, 199--217. 1811554

2001
[22]

Dennis Johnson, The structure of the T orelli group. II . A characterization of the group generated by twists on bounding curves , Topology 24 (1985), no. 2, 113--126. 793178

1985
[23]

Katzarkov, Homological mirror symmetry and algebraic cycles, Homological mirror symmetry, Lecture Notes in Phys., vol

L. Katzarkov, Homological mirror symmetry and algebraic cycles, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 125--152. 2596637

2009
[24]

Maxim Kontsevich and Yuri Tschinkel, Specialization of birational types, Invent. Math. 217 (2019), no. 2, 415--432. 3987175

2019
[25]

Michael L\" o nne, Fundamental group of discriminant complements of B rieskorn- P ham polynomials , C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 93--96. 2343559

2007
[26]

, Fundamental groups of projective discriminant complements, Duke Math. J. 150 (2009), no. 2, 357--405. 2569617

2009
[27]

John McCarthy, Normalizers and centralizers of pseudo- A nosov mapping classes , Unpublished manuscript, 1982, available at https://users.math.msu.edu/users/mccarthy/publications/normcent.pdf

1982
[28]

Shigeyuki Morita, The extension of J ohnson's homomorphism from the T orelli group to the mapping class group , Invent. Math. 111 (1993), no. 1, 197--224. 1193604

1993
[29]

Dimitri Markushevich and Xavier Roulleau, Irrationality of generic cubic threefold via W eil's conjectures , Commun. Contemp. Math. 20 (2018), no. 7, 1750078, 12. 3864287

2018
[30]

J. P. Murre, Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of M umford , Compositio Math. 27 (1973), 63--82. 0352089

1973
[31]

C. A. M. Peters and J. H. M. Steenbrink, Degeneration of the L eray spectral sequence for certain geometric quotients , vol. 3, 2003, \ Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday\ , pp. 1085--1095, 1201. 2078574

2003
[32]

Nick Salter, On the monodromy group of the family of smooth plane curves, Preprint, available at arXiv:1610.04920

arXiv
[33]

Paul Seidel, Abstract analogues of flux as symplectic invariants, M\' e m. Soc. Math. Fr. (N.S.) (2014), no. 137, 135. 3289398

2014
[34]

Ivan Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012), no. 1, 149--250. 2929086

2012
[35]

122 (2000), no

Paul Schmutz Schaller, Mapping class groups of hyperbolic surfaces and automorphism groups of graphs, Compositio Math. 122 (2000), no. 3, 243--260. 1781329

2000
[36]

Chengbo Yue, Mostow rigidity of rank 1 discrete groups with ergodic B owen- M argulis measure , Invent. Math. 125 (1996), no. 1, 75--102. 1389961

1996

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Tarik Aougab, Ian Biringer, and Jonah Gaster, Determining the finite subgraphs of curve graphs, Groups Geom. Dyn. 13 (2019), no. 1, 31--45. 3900763

work page 2019

[2] [2]

Allan Adler, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1978), no. 6, 1275--1280. 522700

work page 1978

[3] [3]

Artin, Braids and permutations, Ann

E. Artin, Braids and permutations, Ann. of Math. (2) 48 (1947), 643--649. 0020989

work page 1947

[4] [4]

Denis Auroux and Ivan Smith, Spherical objects in F ukaya categories of surfaces , Manuscript in preparation, 2019

work page 2019

[5] [5]

1194, Springer, Berlin, 1986, pp

Arnaud Beauville, Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections compl\`etes, Complex analysis and algebraic geometry ( G \" o ttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 8--18. 855873

work page 1985

[6] [6]

, Non-rationality of the symmetric sextic F ano threefold , Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Z\" u rich, 2012, pp. 57--60. 2987652

work page 2012

[7] [7]

Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math

Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107--1120. 726319

work page 1983

[8] [8]

378, vi+175

Fabrice Castel, Geometric representations of the braid groups, Ast\' e risque (2016), no. 378, vi+175. 3499033

work page 2016