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Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

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Pith's one-line read For K3 moduli with order-3 and order-4 automorphisms, the compactifying boundary is explicitly identified as a semifan of ADE root lattices.

desk verdict Solid computation paper; order-3 part convincing, order-4 theorem needs the delegated smoothness/K-triviality check in Prop 4.8 actually done. read the letter →

arxiv 2412.11256 v2 pith:IFOTL57F submitted 2024-12-15 math.AG

Valery Alexeev , Anand Deopurkar , Changho Han This is my paper · ORCID
classification math.AG MSC 14J2814J5014D22
keywords K3surfacesnonsymplecticautomorphismKSBAcompactificationsemitoroidalKulikovdegenerationsADErootlatticestripleTschirnhausenconstructionquadrupleplanequartics
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 establishes what the boundary looks like when moduli spaces of K3 surfaces equipped with a nonsymplectic automorphism—one that multiplies the holomorphic 2-form by a nontrivial root of unity—are compactified by adding stable pairs. For the four maximal families with an order-3 automorphism whose fixed locus contains a curve of genus at least 2, and for cyclic degree-4 covers of the plane branched along a quartic, the KSBA compactification is identified as a semitoroidal compactification. In these settings the distinction among Baily–Borel, toroidal, and KSBA boundaries is controlled by a semifan: for each boundary cusp $J$, a primitive sublattice $F_J \subset J^\perp_{T_\rho}/J$ whose translates are contracted when passing from the fine toroidal boundary to the coarse stable-pair boundary. The main theorems list these sublattices explicitly, in terms of $A_1$, $A_2$, $E_6$, and $E_8$ root lattices. The upshot is that the geometric question of which K3 degenerations are identified at the boundary becomes a concrete lattice computation.

What carries the argument

The load-bearing mechanism is the triple Tschirnhausen construction: starting from an admissible triple cover $C\to\mathbb{P}^1$ with marked points, one blows up the threefold $\mathbb{P}E$ at the marked fibers, takes a normalized cyclic triple cover branched over the proper transforms, and blows down the resulting $(-1)$-curves; the output is a Kulikov degeneration of K3 surfaces with an order-3 automorphism. This construction is shown to dominate the boundary of the period domain, so every boundary period is realized by such a degeneration. The other half of the machinery is the lattice of numerically Cartier divisors of a Type II Kulikov surface, which is identified with $J^\perp/J$ and, in the $\rho$-equivariant setting, with $J^\perp_{T_\rho}/J$; the period of the degeneration lives in $J^\perp_{T_\rho}/J \otimes_{\mathbb{Z}[\zeta_3]} E$. Running the minimal model program on the divisor $R$, the fixed curve of the automorphism, produces the unique KSBA stable limit, and comparing which periods give the same stable limit yields exactly the sublattices $F_J$ in Table 1. For order 4 the construction is simpler: an explicit degeneration of $\mathbb{P}^2$ to two quadric cones, pulled back by the cyclic quadruple cover, gives the Kulikov surface, and the same contraction argument gives $F_J = A_1^{\oplus2}$.

What would settle it

Choose the cusp of the $(0,1)$ family with root lattice $E_8\oplus E_6\oplus A_2$ and compute the KSBA stable limit directly for two one-parameter families whose periods differ only in the $E_6$-summand; the paper's argument predicts these limits are distinct, by the Torelli theorem for anti-canonical pairs, so if direct calculation produced isomorphic stable pairs the claimed semifan would be a strict lower bound and the dominance step in Proposition 6.14 would fail.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.1, is that for the four maximal order-3 families the KSBA compactification $F^{\mathrm{KSBA}}_\rho$ is the semitoroidal compactification of the ball quotient $D_\rho/\Gamma_\rho$ with the semifan $F_J$ given in Table 1, and the paper proves this by splitting it into Theorems 7.1, 8.2, 9.3, and 10.5. The ambient lattice at a cusp is $J^\perp_{T_\rho}/J$, and the KSBA semifan is the saturation, or primitive closure, of the parenthesized $A_2$-summands, or zero where no summand is parenthesized. For $(n,k)=(0,2)$ this gives $F_J=0$, so the KSBA boundary is the toroidal boundary; for $(0,1)$, $(1,1)$, and $(2,1)$ the contracted directions are exactly the listed $A_2$ directions. Theorem 1.2 covers the order-4 case: the space of cyclic quadruple covers of $\mathbb{P}^2$ branched at a quartic has a unique cusp with $J^\perp_{T_\rho}/J=D_4^{\oplus2}\oplus A_1^{\oplus2}$, and the KSBA semifan is the $A_1^{\oplus2}$ summand, so the boundary contracts precisely the translates of $A_1^{\oplus2}\otimes_{\mathbb{Z}[i]}E$.

Load-bearing premise

The load-bearing premise is that every one-parameter degeneration near a cusp is produced by the triple Tschirnhausen construction for order 3, or by the stated degeneration model for order 4; if any degeneration is missed, the listed semifans would only be lower bounds on the true contractions.

Editorial extensions

If this is right

  • For each of the four maximal order-3 families, the KSBA compactification and its boundary strata are now explicitly computable from Table 1; the boundary points are stable pairs whose isomorphism type is determined by the period restricted to the non-contracted lattice directions.
  • In the $(n,k)=(0,2)$ case the KSBA and toroidal compactifications coincide, so the geometric boundary retains all of the period information of the toroidal boundary.
  • In the order-4 case, the KSBA boundary over the unique cusp is a finite quotient of an abelian variety with the translates of $A_1^{\oplus2}\otimes_{\mathbb{Z}[i]}E$ contracted; this gives a complete description of the KSBA compactification of the moduli space of cyclic quadruple covers of $\mathbb{P}^2$ branched at a quartic.
  • For the $(0,1)$ family, the quotient by the automorphism identifies the KSBA compactification with the compactification of stable log quadrics, so the boundary strata of the K3 moduli space match the geometrically defined boundary strata of that log-surface moduli space.

Reading between the lines

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

  • An extension of the paper's method would restrict Table 1 to the remaining, non-maximal order-3 families; since those families arise from the four maximal ones by specialization, one expects their KSBA semifans to be governed by the same $A_2$-coded contractions, though the paper does not carry out this restriction.
  • The pattern that the contracted directions are exactly the $A_2$-summands created by blowing up orbits of fixed points suggests a broader principle: for other cyclic automorphisms, the KSBA semifan may be determined by the root summands coming from such blow-ups, while the $E_6$ and $E_8$ summands inherited from del Pezzo surfaces remain uncontracted.
  • A testable extension is to run the same Tschirnhausen-based stable-model computation on an order-3 family with $g\ge 2$ that is not maximal, and check whether the predicted restriction of Table 1 reproduces the actual contractions; a mismatch would pinpoint where the dominance of the boundary period map breaks down.
  • The order-4 computation is explicitly ad hoc, so a natural next step is to adapt the triple-cover-style construction to cyclic quadruple covers branched along curves other than quartics, to see whether the $A_1^{\oplus2}$ contraction pattern persists or is special to this family.
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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

3 major / 5 minor

Summary. The paper studies moduli spaces of K3 surfaces carrying a purely non-symplectic automorphism of order 3 or 4 whose fixed locus contains a curve of genus at least 2. For the four maximal order-3 families with (n,k)=(0,2), (0,1), (1,1), (2,1) and for cyclic quadruple covers of P2 branched along a quartic, the authors identify the KSBA stable-pair compactification with an explicit semitoroidal compactification: the semifans are listed in Table 1 and Theorem 1.2 (FJ=A1^2 inside J-perp/J=D4^2+A1^2). The proof uses the general framework of [AEH24] to reduce KSBA compactifications to semitoroidal data, then computes the data via a new 'triple Tschirnhausen construction' relating degenerations of admissible triple covers to Type II Kulikov models, lattice computations, and the Torelli theorem for anti-canonical del Pezzo pairs.

Significance. If correct, this gives a complete and remarkably simple lattice-theoretic description of the boundary of these compactifications, and demonstrates that the KSBA boundary can be computed explicitly in nontrivial cases. The paper's main contribution is the identification of the exact semifan, not merely a lower bound, and the triple Tschirnhausen construction is an original and plausible computational tool. The lattice tables are detailed, and the logical architecture (reduction to boundary divisors, dominance via anti-canonical Torelli) is coherent. However, two load-bearing verifications are presented sketchily: the smoothness/K-triviality of the constructed order-4 Kulikov model in Proposition 4.8, and the generic finiteness/dominance statement for the order-3 case in Propositions 6.11 and 6.14. These gaps are localizable and appear fixable, but they should be completed before the theorems can be regarded as fully proved.

major comments (3)
  1. [§4.1, Proposition 4.8] Proposition 4.8, on which Theorem 4.10 and hence Theorem 1.2 rest, constructs a Kulikov model as a cyclic degree-4 cover of a singular threefold and concludes 'We leave it to the reader to verify that X is indeed smooth and K-trivial.' This is a load-bearing assertion: if the total space is singular or has nontrivial canonical class, the central fiber is not a genuine Type II Kulikov degeneration, and the period point psi used in Theorem 4.10 is not a period of a Kulikov degeneration. The earlier sentence 'It is easy to see that B subset ~Y can be smoothed' is part of the same gap. Smoothness of cyclic covers can fail when the branch locus is not transverse to the singular locus of the base, and K-triviality is not automatic for such covers. Please provide a complete verification, for example by local coordinate analysis near the double locus and the blown-up fibers, or by a precise reference covering this specific degeneration.
  2. [§6.3–6.4, Propositions 6.11 and 6.14] The exactness of all four order-3 semifans in Table 1 depends on the dominance statement of Proposition 6.14: without it, the computations in Sections 7–10 can only show that the listed sublattice is contained in the KSBA semifan. The proof of Proposition 6.14 reduces to Proposition 6.11, whose generic-finiteness argument is compressed. In particular, the case d=9 is dismissed as 'automatic', and the finiteness for blow-ups in Z/3Z-orbits is justified by 'only finitely many embeddings A2 to Lambda'. Since a failure of generic finiteness would leave the equality FJ = ... unproved, I ask that this argument be expanded to a complete proof, or that the missing finiteness statement be stated and proved as a separate lemma.
  3. [§10.3, Theorem 10.5] Theorem 10.5 lists the fourth cusp as (J-perp/J)_root = E8 + A2 with FJ = <A2>^sat, but Section 10.1, the table in Theorem 5.5, and Table 1 in the introduction all give this cusp as E8 + A2^2, and the proof in §10.3.4 concludes that FJ is the saturation of the complementary A2^2 summand. Please correct the statement of Theorem 10.5 (and any parallel statement in Theorem 1.1 or Table 1 if needed) so that the theorem, the table, and the proof are mutually consistent.
minor comments (5)
  1. [Table 1] The first data row of Table 1 prints (n,k) as (0,0); it should be (0,2).
  2. [Proof of Theorem 4.10] There is a typo in the expression 'A1^2 tensor Z[i]]E'; the stray bracket should be removed.
  3. [Section 5.3, Theorem 5.5] The notation R* for an index-3 overlattice is defined only after the theorem statement; it would be clearer to define it before or inside the statement.
  4. [Section 4.1, before Proposition 4.5] The sentence 'The pairs that give a type II surface correspond to those with reducible Y' is imprecise because only generic points of the locus Z are discussed; please clarify the genericity assumption.
  5. [Section 8.4] In Theorem 8.4 the map Y_rho to the Baily-Borel compactification is asserted to be regular; the surrounding text would benefit from a sentence explaining why this map is regular rather than merely rational.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the explicit KSBA semifans are derived from independent period, Torelli, and dimension arguments; the only flagged gap is an unproved smoothness/K-triviality check in Proposition 4.8, which is a correctness issue, not a circular one.

full rationale

The derivation of the KSBA semifans is not circular. The paper takes from [AEH24, Theorem 3.26] the structural fact that the KSBA compactification is a semitoroidal compactification with some semifan F, and then independently computes F by geometric analysis of Kulikov degenerations: Proposition 6.6 constructs Type II models from admissible triple covers; Proposition 6.9 computes root sublattices; Proposition 6.11 and Theorem 6.12 supply a Torelli/dominance statement for anti-canonical pairs; Proposition 6.14 transfers this dominance to the toroidal boundary. For each cusp, the authors exhibit which period translates are contracted by showing that the stable model depends only on certain del Pezzo period summands (e.g., the E6 summands in Sections 8.3.1 and 9.3.1) and by a dimension count in Theorem 4.10. The semifan values in Table 1 and Theorem 4.10 are outputs of these computations, not assumed inputs. The self-citations to [AEH24] are load-bearing but not circular: they are published structural theorems whose assumptions do not include the explicit semifan values being derived. The only flagged deficiency is in Proposition 4.8, where the proof of the order-4 Kulikov model says 'We leave it to the reader to verify that X is indeed smooth and K-trivial' and 'It is easy to see that B ⊂ ~Y can be smoothed to a divisor B ⊂ ~Y'. That is a correctness gap in the proof of Theorem 4.10, but it is not circular reasoning, so it does not raise the circularity score.

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

Pure mathematics preprint relying on established theorems of K3 period maps, Kulikov models, and Torelli results. No fitted parameters or invented entities.

assumptions (7)
  • domain assumption The period map identifies the separated moduli space of rho-markable K3 surfaces with (D_rho minus Delta_rho)/Gamma_rho (Theorem 2.1, citing [AEH24]).
    Load-bearing for transferring Hodge-theoretic compactifications to geometric moduli.
  • domain assumption For (X,sigma) satisfying the genus-bound condition, the KSBA compactification F_KSBA is a semitoroidal compactification determined by a semifan ([AEH24, Theorem 3.26]).
    This is the bridge that lets the paper compute a semifan instead of building the KSBA compactification from scratch.
  • standard math Every one-parameter degeneration of K3 surfaces admits a Kulikov model, and type II models have the stated normal-crossing and lattice structure ([Kul77], [PP81], [FS86], [Kon85]).
    The boundary period computations in Section 3.3 rely on this classification.
  • standard math Torelli theorem for rational surfaces with an anti-canonical cycle: an anti-canonical pair is determined up to isomorphism by its period from K-perp to Jac(D) (Theorem 6.12, based on [Loo81]).
    Used to convert computations on individual Kulikov surfaces into statements about all periods.
  • standard math Nikulin's classification of 2-elementary lattices and the uniqueness of the relevant order-3 and order-4 automorphisms of the K3 lattice ([Nik79], [AS08], [BH23]).
    Used in Section 5 and Proposition 4.2 to determine the lattices and cusp quotients.
  • standard math Casnati-Ekedahl structure theorem for triple covers and Pardini's abelian cover construction ([CE96], [Par91]).
    Underlies the triple Tschirnhausen construction in Section 6.1.
  • standard math The Hurwitz space of admissible triple covers is irreducible of the stated dimension and its boundary divisors have the stated dual graphs ([HM82], [RW06], [Deo14]).
    The order-3 cases are parameterized by Hurwitz spaces.

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Pith. "Pith review of Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism." pith.science (2026). https://pith.science/paper/IFOTL57F

@misc{pith2026241211256,
  author       = {Pith},
  title        = {Pith review of: Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IFOTL57F}},
  note         = {Machine review of arXiv:2412.11256}
}
abstract

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism contains a curve of genus $\ge2$. For order $3$, we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain $ADE$ root lattices.

Figures

Figures reproduced from arXiv: 2412.11256 by the authors.

Figure 1
Figure 1. We obtain a degeneration of P 2 to Blp P(1, 1, 2) ∪ P(1, 1, 2) by blowing up succesively in two lines and blowing down a P 2 . The figure shows the central fibers in this process. The numbers next to the edges represent self-intersections. By taking a cyclic degree 4 cover, we obtain a type II Kulikov degeneration [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. The pinched triple Tschirnhausen construction curves are pairwise disjoint. Set Fb = F i Fbi . Let ǫ ⊂ PbE be the exceptional divisor of the blow-up. Then ǫ is the disjoint union of 3n smooth rational curves of self-intersection −1. Consider the cyclic triple cover PbE branched over Cb+ 2Fb. This is constructed as follows. Let H ∈ Pic PbE be the class of the pull-back of OPE(1). Observe that in Pic PbE, we have Cb +… view at source ↗
Figure 3
Figure 3. In the case m = 0 and C = (1, 3), the surface X (left) is the blow-up of a dP3 X (right). The double curve is blue, the ramification curve is red, and the numbers are self-intersections. We have the triple cover Xb → PbE branched over Cb and F i Fbi and the blow-down Xb → X along the pre-images of Fbi . We denote by αe the image in X of the pre-image of αb in Xb. Note that αe may be disconnected. Let D ⊂ X be the pr… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: In the case m = 0 and C = (0, 1) ⊔ (2, 2), the surface X (left) is the blow-up of a dP1 X (right). The double curve is blue, the ramification curve is red, and the numbers indicate self-intersections. 9 1 [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: In the case m = 1 and C = (0, 3), the surface X (left) is the blow-up of P 2 (right). The double curve is blue, the ramification curve is red, and the numbers indicate self-intersections. 6.2.3. Case m = 1 and Γ = {(0, 3)}. In this case, rk H2 (X,Z) ρ = 4. We have PE =…
Figure 6
Figure 6. Figure 6: In the case m = 1 and C = (0, 1) ⊔ (1, 2), the surface X (left) is the blow-up of a dP3 X (right). The double curve is blue, the ramification curve is red, and the numbers indicate self-intersections. Note that σe is not nef. But Qe is nef, and it is zero only on σe an…
Figure 7
Figure 7. Figure 7: The surface X (left) as the blow-up of P 2 (right), together with the double curve (blue) and the ramification curve (red). The numbers indicate self-intersection. rational curves of self-intersections −1 and −2. These three chains are mutually disjoint. Furthermore, Q…
Figure 8
Figure 8. Figure 8: For the case n = 0 and k = 2, we consider degenerations of triple covers whose central fibers have the dual graph shown above. 7.2. Kulikov models. We use the notation in Section 6.4. In particular, we let H be the moduli space of triple covers φ: Q ⊔ P 1 → P 1 and H i…
Figure 9
Figure 9. Figure 9: The stable model is obtained by contracting (Z/3Z)-fixed (−1)- curves on each component (dashed blue) disjoint from the divisor R (red). The image of contracted curves is a single Z/3Z-fixed point (blue dot) of the double curve of X, which is the unique isolated Z/3Z-f…
Figure 10
Figure 10. Figure 10: The dual graphs of the admissible covers that give Kulikov degenerations in the case n = 0 and k = 1. (3, 3). Therefore, X is simply the triple cover of P 1×P 1 branched over a curve of class (3, 3). Artebani–Sarti give the same description using explicit equations [A…
Figure 11
Figure 11. Figure 11: The stable model in the case i = 1 is obtained by contracting a (Z/3Z)-orbit of (−1)-curves on each component (dashed) disjoint from the divisor (red). For j = 0, 1, let ψj be the restriction of the period point ψ of X to the first and the second E6-summand. Then ψj i…
Figure 12
Figure 12. Figure 12: The stable model in the case i = 2 is obtained by an M1 mod￾ification along a (−1)-curve on the left (dashed red) followed by contracting a (Z/3Z)-orbit of (−1)-curves on the right (dashed). The divisor is shown in red (dashed and solid) [PITH_FULL_IMAGE:figures/full…
Figure 13
Figure 13. Figure 13: The stable model in the case i = 3 is obtained by M1 modifi￾cations along the (−1)-curves on the left and the right (dashed red). The divisor is shown in red (dashed and solid). Conclusion of the proof of Theorem 8.2. Having settled the cases of the three cusps in Sec…
Figure 14
Figure 14. Figure 14: The dual graphs of the admissible covers that give Kulikov degenerations in the case n = 1 and k = 1. Proposition 9.2. The extended period map H 99K Dρ/Γρ BB maps ∆i to the i-th cusp in Section 9.1. Proof. The proof is analogous to the proof of Proposition 8.1, based …
Figure 15
Figure 15. Figure 15: The stable model in the case i = 1 is obtained by contracting a (Z/3Z)-orbit of (−1)-curves on the left (dashed) and three such (Z/3Z)- orbits on the right (dashed). All are disjoint from the divisor (red). 9.3.2. The case i = 2. In this case, C0 = P0 ⊔ Q0, where Q0 i…
Figure 16
Figure 16. Figure 16: The stable model in the case i = 2 is obtained by an M1 modi￾fication along a (−1)-curve on the left (dashed red) followed by a morphism that contrats the entire right-hand component, and folds the double curve into a P 1 , resulting in a non-normal surface [PITH_FUL…
Figure 17
Figure 17. Figure 17: The stable model in the case i = 3 is obtained by an M1 modifi￾cation along a (−1)-curve on the right (dashed red) followed by contracting a (−1)-curve on the right (dashed) and three (−1)-curves on the left (dashed). The divisor is shown in red (dashed and solid). By…
Figure 18
Figure 18. Figure 18: The stable model in the case i = 4 is obtained by M1 modifica￾tions along the (−1)-curves on the left and the right (dashed red) followed by the contraction of a (−1)-curve on the right (dashed). The divisor is shown in red (dashed and solid). anti-canonical pairs (Th…
Figure 19
Figure 19. Figure 19: ). For i = 1, 2, let σi be the images of σei . Observe that the contracted surface is a P 1 -bundle over P 1 with sections σi satisfying σ 2 i = 0. Therefore, it is P 1 × P 1 . The image of C of Ce in P 1 × P 1 is a curve of class 3σi + 2f. Let C → P 1 be the degree 3…
Figure 20
Figure 20. Figure 20: The dual graphs of the admissible covers that give Kulikov degenerations in the case n = 2 and k = 1. 10.3. Stable models and the KSBA semifan. We now identify the KSBA semifan F that gives the KSBA compactification F KSBA ρ . We follow the notation of Theorem 8.2. Th…
Figure 21
Figure 21. Figure 21: The stable model in the case i = 1 is obtained by contracting three (Z/3Z)-orbits of (−1)-curves on each side (dashed). All are disjoint from the divisor (red) [PITH_FULL_IMAGE:figures/full_fig_p050_21.png]
Figure 22
Figure 22. Figure 22: The stable model in the case i = 2 is obtained by an M1 modi￾fication along a (−1)-curve on the left (dashed red) followed by a morphism that contracts the entire right-hand component, contracts a (−1)-curve on the left (dashed), and folds the double curve into a P 1 …
Figure 23
Figure 23. Figure 23: The stable model in the case i = 3 is obtained by M1 modi￾fications along the (−1)-curves (dashed red) followed by the contraction of the (−1)-curves (dashed). The divisor is shown in red (dashed and solid). 10.3.4. The case i = 4. Write C0 = P0 ∪ Q0, where Q0 is a sm…
Figure 24
Figure 24. Figure 24: ). Now R′ 0 ⊂ X′ 0 is ample but R′ 1 ⊂ X′ 1 is still negative on Q′ 1 (which is now a (−1)-curve). We make another M1 modification along Q′ 1 , resulting in (X′′, R′′). Now R′′ 0 ⊂ X′′ 0 is nef and trivial only on the double curve. On the other hand R′′ 1 = 0. It foll…

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