Pith. sign in

REVIEW 2 major objections 4 minor 151 references

Normal stable degenerations of Noether-Horikawa surfaces

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper classifies all normal stable Horikawa surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities, proves a local-to-global criterion for their $\mathbb{Q}$-Gorenstein smoothability, and describes the…

desk verdict A large, mostly credible classification of stable Horikawa degenerations, but the proof of the key good-involution reduction has a real gap for non-standard surfaces, so the headline theorem is not yet fully established. read the letter →

arxiv 2507.17633 v1 pith:AWAXN4SO submitted 2025-07-23 math.AG math.DGmath.GT

Hiroto Akaike , Makoto Enokizono , Masafumi Hattori , Yuki Koto This is my paper · ORCID
classification math.AGmath.DGmath.GT MSC 14J2914J1714B07
keywords HorikawasurfacesstablelogcanonicalsingularitiesQ-GorensteinsmoothabilitymodulispaceKSBAcompactificationT-singularitiesellipticfibrations
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 aims to complete the classification of normal stable Horikawa surfaces, the surfaces of general type on the Noether line $K_X^2=2p_g(X)-4$. It asserts that among such surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities, exactly seven families occur: Gorenstein double covers of minimal degree surfaces, $p_g=4$ surfaces whose bicanonical map double-covers an elliptic cone, Lee-Park type surfaces with two T-singularities of type $\frac{1}{(p_g-1)^2}(1,p_g-2)$, and four $p_g=3$ configurations built from $\frac{1}{50}(1,29)$, the strictly lc singularity $(2,2,2,2)$, and two $\frac{1}{4}(1,1)$ T-singularities. It then reduces global $\mathbb{Q}$-Gorenstein smoothability to a local condition: every elliptic double cone singularity must be equivariantly smoothable with respect to the unique good involution. If correct, the boundary of the KSBA moduli space of normal stable Horikawa surfaces is explicitly stratified, and the moduli space is connected exactly when $p_g=6$, $p_g=10$, or $p_g-2$ is not divisible by $4$.

What carries the argument

The argument is carried by four objects. The log Noether inequality (Theorem 4.2) is a logarithmic version of Noether's inequality for normal stable surfaces: it replaces the surface by its minimal resolution together with the reduced exceptional divisor over its elliptic singularities, and it forces every non-standard Horikawa surface's minimal model to be an elliptic fibration over a rational or elliptic curve, organized by the trichotomy $(g,n_X,l_{Eh})=(0,1,0),(0,2,0),(1,1,1)$. Extended T-chains are combinatorial records of exceptional curves that arise by contracting T-chains separated by $(-1)$-curves; their P-admissibility test decides which singularity configurations can actually occur. A good involution is the unique involution that a smoothable normal stable Horikawa surface inherits as the limit of the covering involution of the canonical map on smooth fibers, and it lets the paper replace the surface by the quotient pair $(W,\frac{1}{2}B)$. An anti-P-resolution is a birational modification $W^-\to W$ with $\mathbb{Q}$-Gorenstein smoothable slc singularities and ample relative anticanonical divisor; it is the device that makes the relevant cohomology vanish and turns local equivariant smoothability into a global smoothing.

What would settle it

Compute the invariant triple $(g,n_X,l_{Eh})$ for the minimal resolution of any purported counterexample: Proposition 5.3(2) says a Q-Gorenstein smoothable normal stable Horikawa surface must fall into exactly three cases, so a surface with an elliptic fibration over $\mathbb{P}^1$ with one horizontal component and one elliptic singularity would immediately refute the classification. A second check is the singularity list: any Q-Gorenstein smoothable normal stable Horikawa surface whose non-Gorenstein singularities are not among the seven configurations in Theorem 1.3 would also refute it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: every $\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surface belongs to exactly one of seven families. They are the Gorenstein standard surfaces whose canonical map is a double cover of a minimal-degree surface; the $p_g=4$ Gorenstein surfaces whose bicanonical map is a double cover of an elliptic cone; Lee-Park type surfaces with two T-singularities of type $\frac{1}{(p_g-1)^2}(1,p_g-2)$; and four $p_g=3$ configurations assembled from the T-singularity $\frac{1}{50}(1,29)$, the strictly lc singularity $(2,2,2,2)$, and two $\frac{1}{4}(1,1)$ T-singularities. The companion Theorem 1.4 says a surface in this list is globally $\mathbb{Q}$-Gorenstein smoothable exactly when all its elliptic double cone singularities are equivariantly smoothable with respect to the good involution; in particular, surfaces with no elliptic double cone singularity are automatically smoothable. From this the paper derives an explicit stratification of the boundary of the moduli space of $\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surfaces, and proves that the moduli space is connected precisely when $p_g=6$, $p_g=10$, or $p_g-2$ is not divisible by $4$.

Load-bearing premise

The load-bearing premise is the log Noether inequality for normal stable surfaces: if some log canonical surface with elliptic singularities escapes the trichotomy it imposes, the seven-family list would be incomplete.

Editorial extensions

If this is right

  • Because Theorem 1.4 reduces global smoothability to a local condition, a Horikawa surface with no elliptic double cone singularity is automatically Q-Gorenstein smoothable, and one with such a singularity is smoothable exactly when that singularity is equivariantly smoothable.
  • The seven-family classification means that the boundary of the moduli space of Q-Gorenstein smoothable normal stable Horikawa surfaces is stratified in the explicit way the paper diagrams, with the degeneration behavior of Lee-Park and standard type surfaces depending on the geometric genus.
  • The moduli space is connected exactly for $p_g=6$, $p_g=10$, or $p_g-2$ not divisible by $4$; in particular, the two Gieseker components for $p_g=10$ are joined through supersingular standard Horikawa surfaces.
  • The list includes the first known stable Horikawa surface with a strictly lc singularity of type $(2,4,4)[3]$, constructed through the paper's recipe from an elliptic fibration with prescribed singular fibers.
  • For $p_g\geq 4$, among non-Gorenstein klt Horikawa surfaces only those of Lee-Park type can be Q-Gorenstein smoothable, sharpening earlier classifications of the klt case.

Reading between the lines

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

  • The same log Noether inequality and extended T-chain analysis should classify normal stable surfaces on the next line above the Horikawa line, namely surfaces with $K^2=2p_g-3$; the paper sketches this as its planned continuation.
  • Anti-P-resolutions give a general technique for equivariant cusp smoothing when a finite group acts non-freely outside the singularity, a setting broader than the free-action cases treated in earlier deformation theory.
  • The $p_g=10$ connectedness result reopens the Horikawa problem: the two Gieseker components can now be joined through lc degenerations, so the remaining question is whether such lc degenerations preserve diffeomorphism type in the same way that degenerations through T-singularities do.
  • Because the good involution is unique on every smoothable normal stable Horikawa surface, the quotient-pair viewpoint is intrinsic rather than a choice: any smoothable surface must carry this symmetry, so no surface is missed by studying double covers of quotients.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper classifies normal stable Horikawa surfaces (surfaces of general type with K^2 = 2p_g - 4) whose singularities are log canonical and Q-Gorenstein smoothable. The main theorem (Theorem 1.3) asserts that every such Q-Gorenstein smoothable surface falls into one of seven families: standard double covers of minimal degree surfaces; pg=4 Gorenstein surfaces with bicanonical double cover of an elliptic cone; Lee-Park type surfaces with two T-singularities; and four pg=3 configurations involving the T-singularity 1/50(1,29), the strictly lc singularity (2,2,2,2), and two 1/4(1,1) T-singularities. Theorem 1.4 reduces global Q-Gorenstein smoothability to equivariant smoothability of elliptic double cone singularities with respect to the good involution. The paper also describes the KSBA boundary strata (Theorem 1.14) and proves connectedness of the moduli space for pg=6, pg=10, and pg-2 not divisible by 4 (Theorem 1.17). The technical machinery includes a log Noether inequality, extended T-chains, good involutions, and anti-P-resolutions. The paper is very long (212 pages), with the classification and deformation proofs concentrated in Section 7, Section 9, and Appendices A-C.

Significance. If the classification is correct, it is a substantial advance: explicit descriptions of KSBA boundary strata for surface moduli are rare, and the paper gives a complete normal-locus boundary description for Horikawa surfaces except for one explicitly flagged supersingular pg=10 case. The reduction in Theorem 1.4 to local equivariant smoothability is clean and useful, and the anti-P-resolution machinery appears to be a new and potentially transferable tool. The paper is also honest about its limitations: Remark 1.2(2) states that existence is not verified for most candidates in the broader non-standard classification, and the pg=10 supersingular case is explicitly left partially open. The extensive case analysis is not mechanically verified, which sets a ceiling on confidence, but the parts I could check are careful in their attributions and caveats. The main weakness is a load-bearing gap in the proof of Proposition 5.7, on which the reduction to good involutions depends.

major comments (2)
  1. [Section 5.2, Proposition 5.7] The proof of Proposition 5.7 states: 'Let W° denote the image of X° under φ_{X°/C}. Then φ_{X°/C}: X° → W° is a double covering.' This is valid only for standard Horikawa surfaces. For a non-standard Horikawa surface, Definition 5.1 says the canonical linear system is composed with a pencil, so the relative canonical morphism over C\{0} has positive-dimensional generic fibers and one-dimensional image; it is not a finite double cover and does not define a covering involution σ°. The later portion of the proof treats the non-standard case, but it appears to assume the existence of the involution (for example, in forming W := X/σX and the fiberwise involution eσ) rather than constructing it from the smoothing family. Since Proposition 5.7 is the basis for the assertions in Section 1.4 and Theorem 1.3 that every Q-Gorenstein smoothable normal stable Horikawa surface admits a unique good involution, the reduction of the classification to surfaces with a good involution is incomplete as written.
  2. [Section 1.4 and Theorem 1.3] Section 1.4 asserts that every Q-Gorenstein smoothable normal stable Horikawa surface carries the good involution 'as the limit of the good involutions of smooth Horikawa surfaces.' For smooth non-standard Horikawa surfaces the relevant involution is the hyperelliptic involution of the canonical pencil, not the covering involution of the canonical map; the text does not prove that this involution extends to the limit in the singular family. This is not merely a presentation issue: if the extended involution fails to exist or is not the one induced by the smoothing, then the list in Theorem 1.3, which is obtained by interspersing the good-involution classification with the smoothability analysis, could miss a family. The authors should either supply a direct construction of σ for the non-standard case or prove separately that every Q-Gorenstein smoothing of a non-standard Horikawa surface is equivariant with respect to the hyperelliptic involution of the general fibers.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'Quenstion' in Section 1.6, 'toransform' in the proof of Proposition 6.2, 'suffces' in Lemma 2.7, and the running title contains 'NOETHER-HORIKA W A' with a broken spacing. These should be corrected in the final version.
  2. [Notation 3.1(2)] The operation 'a, 2^{-1}, b' in a string is defined as a+b-2, but the example '[3,2^{-1},3] denotes [4]' is correct; however, the discussion would be clearer if the operation were described as a legal substring replacement rather than as a separate number inserted into the chain.
  3. [Figures 1-7] Several figures, especially Figures 2, 6, and 7, are difficult to read in the arXiv version because of small labels and dense decorations. Since the stratification results in Section 9 refer to these figures repeatedly, higher-resolution or vector versions would materially help the reader.
  4. [Remark 1.2(2)] The statement 'Except for a few cases, we do not verify the existence of a non-standard Horikawa surface whose set of non-Du Val singularities coincides with a given candidate' could be misunderstood as applying to Theorem 1.3. It applies to the broader classification Theorem 7.49; the text would be clearer if this distinction were repeated at the point of Theorem 7.49.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation chain is anchored to external classification and smoothing results; the good-involution reduction is a substantive proof step, not a circular definition, and the flagged non-standard case is a correctness concern rather than circularity.

full rationale

The paper's central claim, Theorem 1.3, is derived from Theorem 6.8 (quoted from Chen [32]) and Theorem 7.52, which in turn rests on the log Noether inequality (Theorem 4.2, adapted from [32]) and on the independent classification of non-standard surfaces (Theorem 7.49). The external anchors—Horikawa [76], Kollár–Shepherd-Barron [97], Wahl [125], Pinkham [124], and the KSBA framework [2, 72, 94]—do not depend on the conclusions of this paper. The notion of a good involution is introduced ad hoc (Definition 5.4), but the paper proves, rather than assumes, that every Q-Gorenstein smoothable Horikawa surface admits one (Proposition 5.7). Even if the proof of Proposition 5.7 is incomplete for non-standard surfaces, as the skeptic's objection suggests, that is a potential correctness gap, not a circular reduction: the statement is not equivalent to its inputs by construction. The internal forward reference to Theorem 7.42/7.44 is explicitly acknowledged in Remark 5.6 and is stated to be independent of the involution classification. The vanishing theorem of the second author [48] is used as a parameter-free external theorem whose assumptions do not include the target classification, so it does not constitute load-bearing self-citation. No fitted parameter is renamed as a prediction, no result is defined in terms of another result in the same paper, and no uniqueness claim is imported solely from the authors' prior work. I therefore find no significant circularity.

Assumptions & free parameters 4 free parameters · 7 assumptions · 4 invented entities

No numerical constants are fitted in this paper; the classification is indexed by genuinely free discrete invariants, listed as free_parameters with descriptive values to mark that they are degrees of freedom of the theorem, not tuned numbers. The load-bearing axioms are external published inputs: the classification of Q-Gorenstein smoothable strictly lc rational singularities ([125, Theorem 1.2], Lemma-Definition 2.15), the smoothability of simple elliptic singularities for b <= 9 ([124], Lemma 2.14), properness of the KSBA stack and finiteness of automorphism group schemes ([2, 72, 63, 101, 94], Theorem 2.13), the vanishing theorems of Fujino [60] and of the second author [48], the classification of minimal degree surfaces [118], and the Kodaira canonical bundle formula [25]. The paper's own new objects, good involutions, anti-P-resolutions, and extended T-chains, all carry independent checkable content (uniqueness and existence theorems, a complete classification, and combinatorial tables), so none of them are postulated without a handle.

free parameters (4)
  • pg (geometric genus) = integer, pg >= 3
    The Horikawa relation sets K^2 = 2pg - 4, so pg indexes the entire classification; it is a free discrete invariant of the theorem, not a fitted constant.
  • Type index d for standard Horikawa surfaces = 0 <= d <= min(pg-4, pg/2+1), pg-d even; type (d)' has d = pg-2
    Indexes the Hirzebruch surface quotient in Theorem 6.8; enumerates genuinely different surfaces.
  • Double cone singularity indices (k1, k2) = k1, k2 >= 0 integers
    Local branch data in Proposition 6.6(6) classifying Gorenstein double cone singularities of type (d; k1, k2).
  • Kodaira fiber index n and blow-up count beta = n >= 0, 0 <= beta <= n+8 (varies by case)
    Enumerate the decorated graphs of Proposition 7.5 and the Appendix C tables; these are classification data, not fitted values.
assumptions (7)
  • standard math Log resolutions, discrepancy formalism, and the surface MMP for log pairs
    Foundational background used throughout Sections 2 to 4 (Definition 2.1, Section 4.1), assumed without proof.
  • domain assumption Q-Gorenstein smoothable strictly lc rational surface singularities are exactly the four families (2,2,2,2)[b1..bn] with sum(bi-3) <= 3, (3,3,3)[b] with b=2,3,4, (2,4,4)[b] with b=2,3, and (2,3,6)[2]
    Cited to [125, Theorem 1.2] as Lemma-Definition 2.15; this completeness is used throughout Section 7 (Lemmas 7.6 to 7.15) whenever a strictly lc rational singularity appears. If the list were incomplete, the classification would miss configurations.
  • domain assumption A simple elliptic singularity of type Ell_b is smoothable if and only if b <= 9
    Cited to [124, Section 5] as Lemma 2.14; used to decide smoothability of elliptic singularities, including double cone singularities of simple elliptic type.
  • domain assumption The KSBA moduli functor is a proper Deligne-Mumford stack, and automorphism group schemes of stable surfaces are finite
    Theorems 2.13 and the argument in Proposition 5.7 use this to extend the canonical involution from smooth fibers to a central stable fiber.
  • domain assumption Fujino's vanishing theorem for slc schemes [60] and the Z-positivity vanishing theorem of the second author [48]
    Used in Sections 8 and 9 to prove the crucial vanishing H^1(OW-(B-)) = 0 in the smoothing construction. [48] is prior published work by an author of this paper, with stated hypotheses, so it is external support under the review rules.
  • standard math Classification of minimal degree surfaces in projective space: P2, Hirzebruch surfaces, and cones over rational normal curves
    Cited to [118] and used in Theorem 6.8 to list possible canonical images of standard Horikawa surfaces.
  • standard math Kodaira's classification of singular fibers and the canonical bundle formula for elliptic surfaces
    Cited to [25, V.12]; used in Sections 4, 7, and Appendix A for the structure of elliptic fibrations underlying non-standard surfaces.
invented entities (4)
  • Good involution (Definition 5.4) independent evidence
    purpose: A canonical involution on normal stable Horikawa surfaces that reduces the classification to double covers X -> W over minimal degree surfaces; uniqueness (Proposition 5.5) and necessary existence for smoothable surfaces (Proposition 5.7) organize Theorems 1.3 and 1.5.
    Tied to an external benchmark: on standard surfaces it must equal the covering involution of the canonical double cover, and on smoothable surfaces it is forced as the limit of the canonical involutions of smooth fibers (Proposition 5.7).
  • Anti-P-resolution (Section 8.4, Definition 8.17, Theorem 8.21) independent evidence
    purpose: A small birational morphism W- -> W with Q-Gorenstein smoothable slc singularities and -KW- ample over W, used to prove the local-to-global smoothability criterion (Theorem 1.4) and to construct explicit smoothings (Example 1.23).
    Admissible anti-P-resolutions are completely classified into three types (Theorem 8.21) and concrete instances are exhibited with explicit blow-up sequences (Example 1.23, Figures 6 and 7).
  • Extended T-chains and T-trains with P-admissibility (Definition 3.6) independent evidence
    purpose: Combinatorial encoding of the possible configurations of non-Du Val singularities under the minimal resolution flow; the backbone of the Section 7 classification (Propositions 7.5, 7.16, Theorems 7.17 to 7.49).
    The notion reduces to the classical T-chain classification of [97, Proposition 3.11], is checked by explicit combinatorial computations in Appendix C, and overlaps with the independent works [53] and [114] as the authors document in Remark 1.19.
  • Mild singularities and double cone singularities (Definitions 6.1 and 6.4) independent evidence
    purpose: Split the singularities of standard Horikawa surfaces according to whether their quotient image is smooth or a cone; elliptic double cone singularities are exactly the potential obstruction to global smoothability (Theorem 1.4).
    Both are local analytic singularity classes with explicit classifications (Propositions 6.2 and 6.6); double cone singularities of type (d; k1, k2) are concretely cusps or simple elliptic singularities with given dual graphs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Normal stable degenerations of Noether-Horikawa surfaces." pith.science (2026). https://pith.science/paper/AWAXN4SO

@misc{pith2026250717633,
  author       = {Pith},
  title        = {Pith review of: Normal stable degenerations of Noether-Horikawa surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWAXN4SO}},
  note         = {Machine review of arXiv:2507.17633}
}
abstract

We classify all normal stable Horikawa surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities. Furthermore, we provide a criterion for their global $\mathbb{Q}$-Gorenstein smoothability and describe the boundary strata of the moduli space of $\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surfaces.

Figures

Figures reproduced from arXiv: 2507.17633 by the authors.

Figure 1
Figure 1. Horikawa surfaces of general/infinite Lee-Park type 5 5 2 2 2 2 blow-ups ←−−−−− 6 2 2 5 2 1 6 2 2 5 2 1 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Horikawa surfaces of special Lee-Park type I with pg = 5 In Section 9, it turns out that the KSBA boundary is stratified as follows: • All Horikawa surfaces of special Lee-Park type with pg = 5 and an elliptic double cone singularity are partially Q-Gorenstein smoothable to Horikawa surfaces of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Standard Horikawa surfaces of type (pg − 2)′ Taking into account the subdivisions introduced above, we now describe the strati￾fications of the boundary of the moduli space and an answer to Question 1.1 (3). For [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Anti-P-resolution over Σ3 As an application of anti-P-resolutions, we give the following explicit examples of normal Horikawa surfaces of geometric genus 10 which are degenerations of smooth Horikawa surfaces of both types (0) and (6): Example 1.23. Let us fix two dist…
Figure 6
Figure 6. Figure 6: Smoothing to Type (0). ∆0 Γ Γ ′ ψ1 ←− ψ2 ←− ψ3 ←− ψ4 ←− ψ5 ←− ψ6 ←− 9 2 2 2 2 2 6 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Smoothing to Type (6). 1.5. Relation to other works and further questions. Describing the KSBA moduli of slc Horikawa surfaces is not only interesting in itself but also related to various problems. We will explain these relationships in turn. Horikawa problem From the…
Figure 10
Figure 10. Figure 10: pg = 3 Case G 4,2 LPS G 4,1 LPS G 4,0 LPI G 4,2 std G 4 LPI G 4 G cone 4,1 std G 4,0 std G 4 dV,(0) [PITH_FULL_IMAGE:figures/full_fig_p192_10.png]
Figure 11
Figure 11. Figure 11: pg = 4 Case [PITH_FULL_IMAGE:figures/full_fig_p192_11.png]
Figure 12
Figure 12. Figure 12: pg = 5 Case G 6,2 LPS G 6,1 LPS G 6 LP∞ G 6,2 std G 6,1 std G 6 G dV,(∞) 6 dV,(0) G 6 LPI G 6 LPI∗ [PITH_FULL_IMAGE:figures/full_fig_p193_12.png]
Figure 13
Figure 13. Figure 13: pg = 6 Case [PITH_FULL_IMAGE:figures/full_fig_p193_13.png]
Figure 14
Figure 14. Figure 14: pg ≥ 7 Case where pg − 2 ̸∈ 4Z G pg G LPS pg LPI∗ G pg LPI G pg stdnF G pg stdF G pg dV,(0) G pg dV,( pg 2 +1) [PITH_FULL_IMAGE:figures/full_fig_p194_14.png]
Figure 15
Figure 15. Figure 15: pg ≥ 7 Case where pg − 2 ∈ 4Z Proof. First, the case where pg ≥ 7 follows from Theorem 1.5, Propositions 9.24, 9.9, 9.45, 9.46, 9.64, 9.65, Corollaries 9.35, and 9.43. We now treat the case where pg = 3. The most difficult part of this case has already been handled in…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

151 extracted references · 72 canonical work pages

  1. [3]

    Alexeev, Complete moduli in the presence of semiabelian group action, Ann

    V. Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math., (2), 155, no. 3 (2002), 611–708

    2002

  2. [1]

    On the Gonality type invariants and the slope of a fibered $3$-fold

    H. Akaike, On the Gonality type invariants and the slope of a fibered 3-fold, arXiv:2404.05216

    work page Pith review arXiv

  3. [2]

    Alexeev, Boundedness and K 2 for log surfaces, Internat

    V. Alexeev, Boundedness and K 2 for log surfaces, Internat. J. Math., 5 (6), (1994), 779–810

    1994

  4. [4]

    Alexeev, Kappa classes on KSBA spaces, Moduli, 2 (2025), e4

    V. Alexeev, Kappa classes on KSBA spaces, Moduli, 2 (2025), e4

    2025

  5. [5]

    Alexeev, A

    V. Alexeev, A. Brunyate, P. Engel, Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal, Geom. Topol. 26 (2022) 3525–3588

    2022

  6. [6]

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

    V. Alexeev, A. Deopurkar, C. Han. Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism, arXiv:2412.11256

    work page Pith review arXiv

  7. [7]

    Alexeev, P

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

    2023

  8. [8]

    Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution

    V. Alexeev, P. Engel, Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution, arXiv:2208.10383

    work page Pith review arXiv

Show all 151 references
  1. [9]

    Alexeev, P

    V. Alexeev, P. Engel, D. Z. Garza, L. Schaffler, Compact moduli of Enriques surfaces with a numerical polarization of degree 2, to appear in Nagoya Math. J

  2. [10]

    Alexeev, P

    V. Alexeev, P. Engel, A. Thompson, Stable pair compactification of moduli of K3 surfaces of degree 2, J. Reine. Angew. Math. 799 (2023), 1–56

    2023

  3. [11]

    Alexeev, X

    V. Alexeev, X. Hu, Explicit compactifications of moduli spaces of secondary Burniat surfaces, arXiv:2309.11397

    arXiv

  4. [12]

    Alexeev, R

    V. Alexeev, R. Pardini, Non-normal abelian covers, Compositio Math. 148 (2012), 1051–1084

    2012

  5. [13]

    Alexeev, R

    V. Alexeev, R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, accepted by Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, it is available online. DOI: 10.2422/2036-2145.202312 008

  6. [14]

    Anthes, Gorenstein stable surfaces with K 2 X = 2 and χ(OX ) = 4, Annali SNS di Pisa Vol

    B. Anthes, Gorenstein stable surfaces with K 2 X = 2 and χ(OX ) = 4, Annali SNS di Pisa Vol. XXI, issue special (pp. 1137-1186) 2020

    2020

  7. [15]

    Artin, Some numerical criteria for contractability of curves on algebraic surfaces

    M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84 (1962), 485–496

    1962

  8. [16]

    Ascher, D

    K. Ascher, D. Bejleri, Moduli of weighted stable elliptic surfaces and invariance of log plurigenera, Proc. Lond. Math. Soc. (3) 122 (2021), no. 5, 617–677. With an appendix by G. Inchiostro

    2021

  9. [17]

    Ascher and D

    K. Ascher and D. Bejleri, Compact moduli of elliptic K3 surfaces. Geom. Topol., 27, (5) (2023), 1891–1946. NORMAL STABLE DEGENERATIONS OF NOETHER-HORIKA W A SURF ACES 213

    2023

  10. [18]

    Ascher, D

    K. Ascher, D. Bejleri, G. Inchiostro, Zs. Patakfalvi, Wall crossing for moduli of stable log pairs, Ann. of Math. (2) 198 no. 2 (2023), 825–866

    2023

  11. [19]

    Ascher, K

    K. Ascher, K. DeVleming, and Y. Liu, Wall crossing for K-moduli spaces of plane curves. Proc. Lond. Math. Soc. (3), 128(6):Paper No. e12615, 113, 2024

    2024

  12. [20]

    Ascher, K

    K. Ascher, K. DeVleming, Y. Liu, and X. Wang. Hassett–Keel Program in genus four. arXiv:2504.20312

    arXiv

  13. [21]

    Ashikaga and K

    T. Ashikaga and K. Konno, Global and local properties of pencils of algebraic curves, in: Algebraic Geometry 2000 Azumino, S. Usui et al. eds, 1–49 Adv. Stud. Pure Math. 36, math. Soc. Japan, Tokyo, 2002

    2000

  14. [22]

    Auroux, The canonical pencils on Horikawa surfaces, Geom

    D. Auroux, The canonical pencils on Horikawa surfaces, Geom. Topol.(4), 10 (2006), 2173–2217

    2006

  15. [23]

    M. A. Barja, Lower bounds of the slope of fibred threefolds, Internat. J. Math. 11 (2000), no 4, 461–491

    2000

  16. [24]

    M. A. Barja, Generalized Clifford-Severi inequality and the volume of irregular varieties, Duke. Math. J 164 (2015), no 3, 541–568

    2015

  17. [25]

    Barth, K

    W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact complex surfaces, second edition, Springer-Verlag, Berlin, 2004

    2004

  18. [26]

    Behnke, J

    K. Behnke, J. A. Christophersen, M-resolutions and deformations of quotient singularities, Amer. J. Math. 116 (1994), no.4, 881–903

    1994

  19. [27]

    Bejleri, J

    D. Bejleri, J. Foster, A. Fernandez Herrero, G. Inchiostro, S. Makarova, J. Zhao, Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves, arXiv:2407.05539

  20. [28]

    Catanese, Surfaces with K 2 = pg = 1, and their period mapping, in Algebraic Geometry, Lect

    F. Catanese, Surfaces with K 2 = pg = 1, and their period mapping, in Algebraic Geometry, Lect. Notes in Math., 732 (1979), 1–29

    1979

  21. [29]

    Catanese, The moduli and the global period mapping of surfaces with K 2 = pg = 1: A counter example to the global Torelli problem, Compositio Math., 41, (1980), no.3, 401–414

    F. Catanese, The moduli and the global period mapping of surfaces with K 2 = pg = 1: A counter example to the global Torelli problem, Compositio Math., 41, (1980), no.3, 401–414

    1980

  22. [30]

    Catanese, Generalized Kummer surfaces and differentiable invariants of Noether-Horikawa surfaces

    F. Catanese, Generalized Kummer surfaces and differentiable invariants of Noether-Horikawa surfaces. I, Manifolds and geometry (Pisa, 1993), 132–177. Sympos. Math., XXXVI Cambridge University Press, Cambridge, 1996

    1993

  23. [31]

    Catanese, S

    F. Catanese, S. Manfredini, The orbifold fundamental group of Persson-Noether-Horikawa sur- faces, Geometric theory of singular phenomena in partial differential equations (Cortona, 1995), 71–90. Sympos. Math., XXXVIII Cambridge University Press, Cambridge, 1998

    1995

  24. [32]

    Chen, The minimal and next minimal volumes of normal KSBA stable surfaces with pg ≥ 2, arXiv:2308.01473

    J. Chen, The minimal and next minimal volumes of normal KSBA stable surfaces with pg ≥ 2, arXiv:2308.01473

    arXiv

  25. [33]

    J. A. Chen, M. Chen, and C. Jiang, The Noether inequality for algebraic 3-folds, with an appendix by J. Koll´ ar, Duke Math. J.169 (9) (2020), 1603–1645

    2020

  26. [34]

    J. A. Chen, M. Chen, and C. Jiang, Addendum to ”The Noether inequality for algebraic 3-folds”, Duke Math. J. 169 (11) (2020), 2199–2204

    2020

  27. [35]

    M. Chen, Y. Hu, and C. Jiang, On the moduli spaces of canonical threefolds with small genera and minimal volumes, arXiv:2407.01276

    arXiv

  28. [36]

    Ciliberto, R

    C. Ciliberto, R. Pardini, On some boundary divisors in the moduli spaces of stable Horikawa surfaces with K 2 = 2pg − 3, arXiv:2502.16322

    arXiv

  29. [37]

    Codogni, L

    G. Codogni, L. Tasin and F. Viviani, Slope inequalities for KSB-stable and K-stable families, Proc. London Math Soc. (3) 126 (2023), 1394–1465

    2023

  30. [38]

    Cornalba and J

    M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applica- tions to the moduli space of curves, Annales Sci. ´Ecole Norm. Sup. (4) 21 (1988), 455–475

    1988

  31. [39]

    Coughlan, M

    S. Coughlan, M. Franciosi, R. Pardini, J. Rana, and S. Rollenske, On T-divisors and intersections in the moduli space of stable surfaces M1,3. J. Lond. Math. Soc. (2), 107 (2023), 750–776

    2023

  32. [40]

    Coughlan, M

    S. Coughlan, M. Franciosi, R. Pardini, and S. Rollenske, Degeneration of Hodge structures on I- surfaces, to appear in the volume ”Current developments in Hodge theory. Proceedings of Hodge theory at IMSA”, Simons Symposia series, Springer

  33. [41]

    Coughlan, Y

    S. Coughlan, Y. Hu, R. Pignatelli, and T. Zhang, Threefolds on the Noether line and their moduli spaces, 2024. arXiv:2409.17847. 214 HIROTO AKAIKE, MAKOTO ENOKIZONO, MASAFUMI HATTORI, AND YUKI KOTO

    2024

  34. [42]

    Cox, J.B

    D.A. Cox, J.B. Little, H.K. Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, 2011

    2011

  35. [43]

    Deligne, D

    P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes ´Etudes Sci. 36 (1969), 75–109

    1969

  36. [44]

    Demazure, Surfaces de del Pezzo II–V, in S´ eminaire sur les Singularit´ es des Surfaces, Lecture Notes in Math

    M. Demazure, Surfaces de del Pezzo II–V, in S´ eminaire sur les Singularit´ es des Surfaces, Lecture Notes in Math. 777, Springer, 1980, 23–69

    1980

  37. [45]

    DeVleming, L

    K. DeVleming, L. Ji, P. Kennedy-Hunt, and M. H. Quek, The K-moduli space of a family of conic bundle threefolds, arXiv:2403.09557

    arXiv

  38. [46]

    DeVleming, D

    K. DeVleming, D. Stapleton, Smooth limits of plane curves of prime degree and Markov numbers, J. ´Ec. polytech. Math. 11 (2024), 683–731

    2024

  39. [47]

    Engel, A proof of Looijenga’s conjecture via integral-affine geometry, J

    P. Engel, A proof of Looijenga’s conjecture via integral-affine geometry, J. Differential Geom. 109 (3), (2018) 467-495

    2018

  40. [48]

    Enokizono, An integral version of Zariski decompositions on normal surfaces, Eur

    M. Enokizono, An integral version of Zariski decompositions on normal surfaces, Eur. J. Math. 10 (2024), no. 38

    2024

  41. [49]

    Enokizono, Slope inequality of fibered surfaces, Morsification conjecture and moduli of curves, arXiv:2307.04311

    M. Enokizono, Slope inequality of fibered surfaces, Morsification conjecture and moduli of curves, arXiv:2307.04311

    arXiv

  42. [50]

    Enriques, Le Superficie Algebriche

    F. Enriques, Le Superficie Algebriche. Zanichelli, Bologna (1949)

    1949

  43. [51]

    Evans, A

    J. Evans, A. Simonetti, G. Urz´ ua, Tropical methods for stable octic double planes, arXiv:2405.02735

  44. [52]

    Fantechi, et

    B. Fantechi, et. al. Fundamental Algebraic Geometry: Grothendieck’s FGA Explained. Amer. Math. Soc. 2005

    2005

  45. [53]

    Figueroa, J

    F. Figueroa, J. Rana and G. Urz´ ua, Optimal bounds for many T-singularities in stable surfaces, arXiv:2308.05624

    arXiv

  46. [54]

    Fintushel, R

    R. Fintushel, R. Stern, Rational blowdowns of smooth 4-manifolds, J. Differential Geom. 46 (1997), no.2, 181–235

    1997

  47. [55]

    Franciosi, R

    M. Franciosi, R. Pardini, and S. Rollenske, Gorenstein stable surfaces with K 2 X = 1 and pg > 0. Math. Nachr., 290 (2017), 794–814

    2017

  48. [56]

    Franciosi, R

    M. Franciosi, R. Pardini, J. Rana and S. Rollenske. I-surfaces with one T-singularity. Bollettino dell’Unione Matematica Italiana, 15 (2022), 173–190

    2022

  49. [57]

    Friedman, A new proof of the global Torelli theorem for K3 surfaces, Ann

    R. Friedman, A new proof of the global Torelli theorem for K3 surfaces, Ann. of Math. 120 (1984), 237-269

    1984

  50. [58]

    Friedman, P

    R. Friedman, P. Griffiths, Deformations of I-surfaces with elliptic singularities, arXiv:2402.17677

    arXiv

  51. [59]

    Friedman, P

    R. Friedman, P. Griffiths, Limiting mixed Hodge structures associated to I-surfaces with simple elliptic singularities, arXiv:2408.02062

    arXiv

  52. [60]

    Fujino, Fundamental theorems for semi log canonical pairs, Algebraic Geometry 1 (2) (2014) 194–228

    O. Fujino, Fundamental theorems for semi log canonical pairs, Algebraic Geometry 1 (2) (2014) 194–228

    2014

  53. [61]

    Fujino, Some remarks on the minimal model program for log canonical pairs, J

    O. Fujino, Some remarks on the minimal model program for log canonical pairs, J. Math. Sci. Univ. Tokyo. 22 (2015), 149–192

    2015

  54. [62]

    Fujino, Foundations of the minimal model program, MSJ Mem

    O. Fujino, Foundations of the minimal model program, MSJ Mem. 35, (Mathematical Society of Japan, 2017)

    2017

  55. [63]

    Fujino, Semipositivity theorems for moduli problems, Ann

    O. Fujino, Semipositivity theorems for moduli problems, Ann. of Math., 187 (2018), no. 3, 639– 665

    2018

  56. [64]

    Gallardo, G

    P. Gallardo, G. Pearlstein, L. Schaffler, Z. Zhang, Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces, Math. Nachr. 297, No. 2, (2024) 595–628

    2024

  57. [65]

    Gieseker, Global Moduli for Surfaces of General Type, Invent

    D. Gieseker, Global Moduli for Surfaces of General Type, Invent. Math. 47 (1977), 233-282

    1977

  58. [66]

    D. Greb, S. Kebekus, S. Kov´ acs, Extension theorems for differential forms and Bogomolov- Sommese vanishing on log canonical varieties, Compositio Math. 146 (1) (2010), 193–219

    2010

  59. [67]

    Greco, C

    S. Greco, C. Traverso, On seminormal schemes, Compositio Math. 40, no. 3 (1980), 325–365

    1980

  60. [68]

    Griffiths, Topics in Transcendental Algebraic Geometry, Ann

    P. Griffiths, Topics in Transcendental Algebraic Geometry, Ann. Math. Studies, Princeton Univ. Press, Princeton, NJ, (1984). NORMAL STABLE DEGENERATIONS OF NOETHER-HORIKA W A SURF ACES 215

    1984

  61. [69]

    Gross, P

    M. Gross, P. Hacking and S. Keel, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes ´Etudes Sci., 122 (2015), 65-168

    2015

  62. [70]

    Hacking, Compact moduli of plane curves, Duke Math

    P. Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), 213–257

    2004

  63. [71]

    Hacking, Y

    P. Hacking, Y. Prokhorov, Smoothable del Pezzo surfaces with quotient singularities, Compositio Math. 146 (2010), 169–192

    2010

  64. [72]

    C. D. Hacon, J. M cKernan, C. Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20 (2018), no. 4, 865–901

    2018

  65. [73]

    Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No

    R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977

    1977

  66. [74]

    Hashizume, A note on lc-trivial fibrations, Bull

    K. Hashizume, A note on lc-trivial fibrations, Bull. London Math. Soc. 56 (2024), no. 2, 551-565

    2024

  67. [75]

    Horikawa, On deformations of quintic surfaces, Invent

    E. Horikawa, On deformations of quintic surfaces, Invent. Math. 31 (1975), 43–85

    1975

  68. [76]

    Horikawa, Algebraic surfaces of general type with small c2 1, I

    E. Horikawa, Algebraic surfaces of general type with small c2 1, I. Ann. of Math. 104 (1976), no. 2, 357–387

    1976

  69. [77]

    Horikawa, Algebraic surfaces of general type with small c2 1, II, Invent

    E. Horikawa, Algebraic surfaces of general type with small c2 1, II, Invent. Math. 37 (1976), no. 2, 121–155

    1976

  70. [79]

    III. Invent. Math. 47 (1978), no. 3, 209–248

    1978

  71. [81]

    IV. Invent. Math. 50 (1978), no. 2, 103–128

    1978

  72. [82]

    Horikawa, Algebraic surfaces of general type with small c2

    E. Horikawa, Algebraic surfaces of general type with small c2

  73. [83]

    V. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, (1981), 745–755

    1981

  74. [84]

    Horikawa, On algebraic surfaces with pencils of curves of genus 2, In: Complex Analysis and Algebraic Geometry, a volume dedicated to K

    E. Horikawa, On algebraic surfaces with pencils of curves of genus 2, In: Complex Analysis and Algebraic Geometry, a volume dedicated to K. Kodaira, Tokyo and Cambridge, Iwanami Shoten Publishers and Cambridge University Press, (1977), 79–90

    1977

  75. [85]

    Hu and T

    Y. Hu and T. Zhang, Fibered varieties over curves with low slope and sharp bounds in dimension three, J. Algebraic Geom. 30 (2021), 57–95

    2021

  76. [86]

    Hu and T

    Y. Hu and T. Zhang, Relative Severi inequality for fibrations of maximal Albanese dimension over curves, Forum of Math. Sigma 10 e45 (2022), 1–31

    2022

  77. [87]

    Illusie, Complexe Cotangent et Deformations, I,Lecture Notes in Math

    L. Illusie, Complexe Cotangent et Deformations, I,Lecture Notes in Math. 239, Springer, Berlin, 1971

    1971

  78. [88]

    Illusie, Complexe Cotangent et Deformations, II,Lecture Notes in Math

    L. Illusie, Complexe Cotangent et Deformations, II,Lecture Notes in Math. 283, Springer, Berlin, 1972

    1972

  79. [89]

    Ishii, Introduction to singularities, Springer, 2018

    S. Ishii, Introduction to singularities, Springer, 2018

    2018

  80. [90]

    Jiang, Equivariant smoothing of cusp singularities, arXiv:2302.00637

    Y. Jiang, Equivariant smoothing of cusp singularities, arXiv:2302.00637

    arXiv

  81. [91]

    Kawakita, Inversion of adjunction on log canonicity

    M. Kawakita, Inversion of adjunction on log canonicity. Invent. Math. 167 (2007), 129-133

    2007

  82. [92]

    Kawakita, Complex Algebraic Threefolds, Cambridge Studies in Advanced Mathematics 209, Cambridge University Press, 2023

    M. Kawakita, Complex Algebraic Threefolds, Cambridge Studies in Advanced Mathematics 209, Cambridge University Press, 2023

    2023

  83. [93]

    Kawamata, Crepant blowings-up of three-dimensional canonical singularities and its application to degenerations of surfaces

    Y. Kawamata, Crepant blowings-up of three-dimensional canonical singularities and its application to degenerations of surfaces. Ann. of Math. 127, no. 1 (1988), 93–163

    1988

  84. [94]

    Kobayashi, On the Noether inequality for 3-folds, J

    M. Kobayashi, On the Noether inequality for 3-folds, J. Math. Soc. Japan 44 (1992), 145–156

    1992

  85. [95]

    S. Keel, K. Matsuki, and J. M cKernan, Log abundance theorem for threefolds, Duke Math. J. 75 (1994), 99–119

    1994

  86. [96]

    Koll´ ar,Singularities of the Minimal Model Program,Cambridge Tracts in Mathematics 200

    J. Koll´ ar,Singularities of the Minimal Model Program,Cambridge Tracts in Mathematics 200. Cambridge University Press, Cambridge, 2015

    2015

  87. [97]

    Koll´ ar,Families of varieties of general type, Cambridge Tracts in Mathematics 231

    J. Koll´ ar,Families of varieties of general type, Cambridge Tracts in Mathematics 231. Cambridge University Press, Cambridge, 2024

    2024

  88. [98]

    Koll´ ar and S

    J. Koll´ ar and S. Mori, Classification of Three-Dimensional Flips, J. Amer. Math. Soc., 5 (1992), 533–703

    1992

  89. [99]

    Koll´ ar and S

    J. Koll´ ar and S. Mori,Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathe- matics, 134, Cambridge University Press, Cambridge, 1998. 216 HIROTO AKAIKE, MAKOTO ...

    1998

  90. [100]

    Koll´ ar and N

    J. Koll´ ar and N. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338

    1988

  91. [101]

    J. Koll´ ar et al., Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August

  92. [102]

    V. I. Kynev, An example of a simply connected surface of general type for which the local Torelli theorem does not hold, C. R. Acad. Bulgare Sci. 30 (1977), no. 3, 323–325

    1977

  93. [103]

    Konno, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann

    K. Konno, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Sc. Norm. Sup. Pisa ser. IV, 20 (1993), 575–595

    1993

  94. [104]

    Konno, Chain-connected component decomposition of curves on surfaces, J

    K. Konno, Chain-connected component decomposition of curves on surfaces, J. Math. Soc. Japan 62 (2010), 467–486

    2010

  95. [105]

    Kov´ acs, Zs

    S. Kov´ acs, Zs. Patakfalvi, Projectivity of the moduli space of stable log-varieties and subaddi- tivity of log-Kodaira dimension. J. Amer. Math. Soc. 30 (2017), 959–1021

    2017

  96. [106]

    Liu, Stable degenerations of surfaces isogenous to a product II, Trans

    W. Liu, Stable degenerations of surfaces isogenous to a product II, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2411–2427

    2012

  97. [107]

    Lee, Numerical bounds for degenerations of surfaces of general type, Internat

    Y. Lee, Numerical bounds for degenerations of surfaces of general type, Internat. J. Math. 10 (1999), 79–92

    1999

  98. [108]

    Y. Lee, J. Park, A construction of Horikawa surface via Q-Gorenstein smoothings, Math. Z. 267 (2011), no.1–2, 15–25

    2011

  99. [109]

    Liu, The minimal volume of surfaces of log general type with positive geometric genus, arXiv:1706.03716

    W. Liu, The minimal volume of surfaces of log general type with positive geometric genus, arXiv:1706.03716

    arXiv

  100. [110]

    Manetti, Normal degenerations of the complex projective plane, J

    M. Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89–118

    1991

  101. [111]

    W. Liu, S. Rollenske, Geography of Gorenstein stable log surfaces, Trans. Amer. Math. Soc. 368 (2016), 2563-2588

    2016

  102. [112]

    Y. Liu, C. Zhou. Non-proportional wall crossing for K-stability, arXiv:2412.15725

    arXiv

  103. [113]

    Looijenga, Rational surfaces with an anticanonical cycle

    E. Looijenga, Rational surfaces with an anticanonical cycle. Ann. of Math. (2) 114 (1981), no. 2, 267–322

    1981

  104. [114]

    Monreal, J

    V. Monreal, J. Negrete, G. Urz´ ua, Classification of Horikawa surfaces with T-singularities, arXiv:2410.02943

    arXiv

  105. [115]

    Manetti, On the moduli space of diffeomorphic algebraic surfaces

    M. Manetti, On the moduli space of diffeomorphic algebraic surfaces. Invent. Math. 143, (2001), 29–76

    2001

  106. [116]

    Miranda, The moduli of Weierstrass fibrations overP1, Math

    R. Miranda, The moduli of Weierstrass fibrations overP1, Math. Ann. 255 (1981), no. 3, 379–394

    1981

  107. [117]

    F. Meng, Z. Zhuang, MMP for locally stable families and wall crossing for moduli of stable pairs, arXiv:2311.01319

    arXiv

  108. [118]

    Nagata, On rational surfaces I, Mem

    M. Nagata, On rational surfaces I, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 32 (1960), 351-370

    1960

  109. [119]

    Moriwaki, Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves, J

    A. Moriwaki, Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 no.3 (1998), 569–600

    1998

  110. [120]

    Mumford, Abelian varieties, Oxford university press, (1974)

    D. Mumford, Abelian varieties, Oxford university press, (1974)

    1974

  111. [121]

    Murakami, Infinitesimal Torelli theorem for surfaces of general type with certain invariants, Manuscripta Math

    M. Murakami, Infinitesimal Torelli theorem for surfaces of general type with certain invariants, Manuscripta Math. 118 (2005), 151–160

    2005

  112. [122]

    Pardini, The Severi inequality K 2 ≥ 4χ for surfaces of maximal Albanese dimension, Invent

    R. Pardini, The Severi inequality K 2 ≥ 4χ for surfaces of maximal Albanese dimension, Invent. Math. 159 (2005), 669–672

    2005

  113. [123]

    Ohno, Some inequalities for minimal fibrations of surfaces of general type over curves, J

    K. Ohno, Some inequalities for minimal fibrations of surfaces of general type over curves, J. Math. Soc. Japan, 44 (1992), no 4, 643–666

    1992

  114. [124]

    Olsson, Algebraic spaces and stacks

    M. Olsson, Algebraic spaces and stacks. Colloquium Publications 62, American Mathematical Society (2016)

    2016

  115. [125]

    Pardini, On the period map for abelian covers of projective varieties, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 26 (4) (1998), 719–735

    R. Pardini, On the period map for abelian covers of projective varieties, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 26 (4) (1998), 719–735

    1998

  116. [126]

    Rana, A boundary divisor in the moduli space of stable quintic surfaces

    J. Rana, A boundary divisor in the moduli space of stable quintic surfaces. Int. J. Math, 28, No. 4 (2017)

    2017

  117. [127]

    Pearlstein, Z

    G. Pearlstein, Z. Zhang, A generic global Torelli theorem for certain Horikawa surfaces, Algebraic Geometry 6 (2), (2019) 132–147

    2019

  118. [128]

    Pinkham, Deformations of algebraic varieties with Gm-actions, Ast´ erisque,20, (1974)

    H. Pinkham, Deformations of algebraic varieties with Gm-actions, Ast´ erisque,20, (1974). NORMAL STABLE DEGENERATIONS OF NOETHER-HORIKA W A SURF ACES 217

    1974

  119. [129]

    Prokhorov, Log-canonical degenerations of del Pezzo surfaces in Q-Gorenstein families, Kyoto J

    Y. Prokhorov, Log-canonical degenerations of del Pezzo surfaces in Q-Gorenstein families, Kyoto J. Math. 59, no.4 (2019), 1041–1073

    2019

  120. [130]

    Reid, Problems on pencils of small genus, Preprint (1990) (see his home page)

    M. Reid, Problems on pencils of small genus, Preprint (1990) (see his home page)

    1990

  121. [131]

    J. Rana, S. Rollenske, Standard stable Horikawa surfaces, Algebraic Geometry 11 (4) (2024), 569–592

    2024

  122. [132]

    J. Rana, G. Urz´ ua, Optimal bounds for T-singularities in stable surfaces, Adv. Math.345 (2019), 814–844

    2019

  123. [133]

    Reyes and G

    J. Reyes and G. Urz´ ua, Exotic surfaces, Math. Ann. 391 (2025), no. 1, 227–254

    2025

  124. [134]

    van Opstall, Stable degenerations of surfaces isogenous to a product of curves, Proc

    M. van Opstall, Stable degenerations of surfaces isogenous to a product of curves, Proc. Amer. Math. Soc. 134 (2006), no. 10, 2801–2806

    2006

  125. [135]

    Reider, On the Infinitesimal Torelli Theorem for certain irregular surfaces of general type, Math

    I. Reider, On the Infinitesimal Torelli Theorem for certain irregular surfaces of general type, Math. Ann. 280 (1988), 285–302

    1988

  126. [136]

    Reider, Infinitesimal Torelli theorem for double coverings of surfaces of general type

    I. Reider, Infinitesimal Torelli theorem for double coverings of surfaces of general type. (English summary), J. Algebraic Geom. 14, no. 4 (2005), 691–704

    2005

  127. [137]

    Rollenske, Compact moduli for certain Kodaira fibrations, Ann

    S. Rollenske, Compact moduli for certain Kodaira fibrations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 4, 851–874

    2010

  128. [138]

    Tevelev, G

    J. Tevelev, G. Urz´ ua, Categorical aspects of the Koll´ ar–Shepherd-Barron correspondence, arXiv: 2204.13225

    arXiv

  129. [139]

    Simonetti, Z/2Z-Equivariant smoothings of cusp singularities, arXiv:2201.02871

    A. Simonetti, Z/2Z-Equivariant smoothings of cusp singularities, arXiv:2201.02871

    arXiv

  130. [140]

    Steenbrink, Mixed Hodge structures associated with isolated singularities, in Singularities, Arcata 1981, Proc

    J. Steenbrink, Mixed Hodge structures associated with isolated singularities, in Singularities, Arcata 1981, Proc. Sympos. Pure Math. 40, A.M.S. (1983), vol. 2, 513–536

    1983

  131. [141]

    Stevens, Degenerations of elliptic curves and equations for cusp singularities, Math

    J. Stevens, Degenerations of elliptic curves and equations for cusp singularities, Math. Ann. 311 (1998), 199–222

    1998

  132. [142]

    Usui, Local Torelli theorem for some nonsingular weighted complete intersections, Proc

    S. Usui, Local Torelli theorem for some nonsingular weighted complete intersections, Proc. of the Int. Symp. on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Kinokuniya Book Store, Tokyo, 1978), 723–734

    1977

  133. [143]

    The authors of stacks project, Stacks project

  134. [144]

    Tsunoda and D

    S. Tsunoda and D. Q. Zhang. Noether’s inequality for noncomplete algebraic surfaces of general type. Publ. Res. Inst. Math. Sci., 28(1):21–38, 1992

    1992

  135. [145]

    Urz´ ua, N

    G. Urz´ ua, N. Vilches, On wormholes in the moduli space of surfaces. Algebr. Geom.9 (1), (2022), 39–68

    2022

  136. [146]

    Xu, K-stability of Fano varieties, New Mathematical Monographs 50 Cambridge University Press, Cambridge, 2025

    C. Xu, K-stability of Fano varieties, New Mathematical Monographs 50 Cambridge University Press, Cambridge, 2025

    2025

  137. [147]

    Usui, The problem of Torelli type, Sugaku (3), 49 (1997), 235–252

    S. Usui, The problem of Torelli type, Sugaku (3), 49 (1997), 235–252

    1997

  138. [148]

    Wahl, Smoothings of normal surface singularities, Topology 20 (1981), 219–246

    J. Wahl, Smoothings of normal surface singularities, Topology 20 (1981), 219–246

    1981

  139. [149]

    Xiao, Fibred algebraic surfaces with low slope, Math

    G. Xiao, Fibred algebraic surfaces with low slope, Math. Ann. 276 (1987), 449–466

    1987

  140. [151]

    Zhao, Log canonical models for moduli of genus six curves and K-stability, arXiv:2304.13259

    J. Zhao, Log canonical models for moduli of genus six curves and K-stability, arXiv:2304.13259

    arXiv

  141. [152]

    Zhou, On wall-crossing for K-stability, Adv

    C. Zhou, On wall-crossing for K-stability, Adv. Math. 413 (2023), Paper No. 108857

    2023

  142. [153]

    Zhou, On the Shape of the K-semistable Domain and Wall Crossing for K-stability

    C. Zhou, On the Shape of the K-semistable Domain and Wall Crossing for K-stability. Peking Math. J. (2024). 218 HIROTO AKAIKE, MAKOTO ENOKIZONO, MASAFUMI HATTORI, AND YUKI KOTO Institute of Mathematics, Tohoku University,6-3 Aramaki Aza, Aoba, Aoba W ard, Sendai, Miyagi 980-85...

    2024

  143. [1991]

    Soci´ et´ e Math´ ematique de France, Paris, 1992

    Ast´ erisque No.211 (1992). Soci´ et´ e Math´ ematique de France, Paris, 1992

    1992

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.