Construction of minimal varieties from quasi-smooth weighted complete intersections

Pinxian Bie

REVIEW 4 major objections 5 minor 33 references

Construction of minimal varieties from quasi-smooth weighted complete intersections

T0 review · 4 major / 5 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 proves explicit numerical criteria for when a weighted blow-up of a weighted complete intersection has nef canonical divisor, and uses them to produce 79 families of minimal 3-folds of general type plus infinite families of…

desk verdict Genuine generalization of CJL24 with credible nefness criteria, but the 79-family headline outruns the supplied verification. read the letter →

arxiv 2507.08826 v1 pith:SCFNTVFD submitted 2025-06-30 math.AG

Pinxian Bie This is my paper · ORCID

classification math.AG MSC 14J3014E3014M1014B05

keywords minimalmodelsweightedcompleteintersectionsblow-upsnefnesscriteriacyclicquotientsingularities3-foldsofgeneraltypeNoetherlinesKodairadimension2

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 claims a general way to build minimal varieties: take a well-formed quasi-smooth weighted complete intersection with a cyclic quotient singularity, perform a weighted blow-up at that singularity, and check that a short list of inequalities holds, which then guarantees the canonical divisor of the blown-up variety is nef. The main nefness criterion (Theorem 1.2) reduces the whole nefness check to one intersection number once an auxiliary weighted complete intersection curve is irreducible, and Theorem 1.3 extends the same mechanism to blowing up several points in special positions. The payoff is concrete: 79 families of minimal 3-folds of general type with computed canonical volumes, plurigenera, basket data and Picard numbers, several on or near the Noether lines, along with infinite families of minimal 3-folds of Kodaira dimension 2. If every listed space really is well-formed and quasi-smooth with exactly the stated singularity set, these tables substantially enlarge the known landscape of minimal threefolds.

What carries the argument

The engine of the construction is the weighted blow-up $\pi:Y\to X$ at a cyclic quotient singularity $Q$ of type $\frac{1}{r}(e_1,\ldots,e_n)$, with weights $(e_1,\ldots,e_n)$. Its exceptional divisor is $E\cong\mathbb{P}(e_1,\ldots,e_n)$, and the canonical bundle changes by $K_Y=\pi^*K_X-\frac{r-\sum_i e_i}{r}E$, so the positive amplitude $\alpha=\sum d_\ell-\sum b_j$ and the defect $r-\sum_i e_i$ decide everything. To prove nefness the paper argues by contradiction: a curve $C$ on which $K_Y$ is negative cannot lie on $E$ because $K_Y|_E$ is ample, so the inequalities force $C$ into the intersection of $n-1$ strict transforms of coordinate hyperplanes, which the irreducibility hypothesis makes a single irreducible curve; then the whole nefness check collapses to the one intersection number $(K_Y\cdot H'_1\cdots H'_{n-1})\ge 0$, exactly the volume-type inequality in the theorem. The multi-point versions of Theorem 1.3 use the same curve-cutting argument with several exceptional divisors, and the final step converts $K_Y$ into invariants via the plurigenus formula and the classification of terminal cyclic quotient singularities.

What would settle it

Recompute the quasi-smoothness test (Proposition 2.8) and the singularity calculations for every row of Tables 1–5; a single row that fails quasi-smoothness or carries a non-canonical singularity beyond the listed blow-up center would show that row is not a minimal threefold. For the infinite families, checking the first few $r$ in each congruence class of Table 5 and computing $K_Y^3=0$ and the singularity types at $Q=\frac{1}{r}(a,b,1)$ would confirm or refute the Kodaira dimension 2 claim.

Watch

Extended reading notes

Core claim

The central claim is that for an $n$-dimensional well-formed quasi-smooth weighted complete intersection $X=X_{d_1,\ldots,d_c}\subset\mathbb{P}(b_1,\ldots,b_{n+c+1})$ with amplitude $\alpha=\sum_\ell d_\ell-\sum_j b_j>0$, and a cyclic quotient singularity $Q\in X$ of type $\frac{1}{r}(e_1,\ldots,e_n)$ with $\sum_i e_i<r$, the weighted blow-up $\pi:Y\to X$ at $Q$ with weights $(e_1,\ldots,e_n)$ has $K_Y$ nef (nonnegative intersection with every curve) and $\nu(Y)\ge n-1$ provided either of two explicit lists of conditions holds. The first list requires the inequalities $\alpha e_j \ge b_j(r-\sum_i e_i)$ for all but one coordinate, a volume-type inequality tying the degrees and remaining weights together, irreducibility of a general weighted complete intersection curve of the given multidegrees in a smaller weighted projective space, and well-formedness of $\mathbb{P}(e_1,\ldots,e_n)$; the second list replaces the curve condition by requiring a certain finite set. Theorem 1.3 proves the analogous statement when several points are blown up simultaneously, under special-position assumptions, and Theorem 3.2 covers a second weighted blow-up when the first one leaves a non-canonical point on the exceptional divisor. The numerical invariants of the resulting minimal models are then computed from the blow-up formula and standard birational invariants, producing the tables of general-type families and infinite Kodaira dimension 2 families.

Load-bearing premise

For the tables to deliver what they promise, every listed weighted complete intersection must really be well-formed and quasi-smooth with exactly the singularities stated; the paper verifies two examples in detail and asserts the rest were manually verified.

Editorial extensions

If this is right

This paper's Tables 1–4 record 79 families of minimal 3-folds of general type with Q-factorial terminal singularities, each with computed canonical volume, geometric genus, second plurigenus, singularity basket, and in most cases Picard number.
Several rows are extremal: the minimal model of the $(7,10)$ complete intersection in $\mathbb{P}(1,1,2,3,4,5)$ has $p_g=2$ and $K^3=1/3$, matching the sharp lower volume bound, and the $(12,18)$ model has $p_g=1$ and $K^3=1/30$, claimed as the smallest known volume in that genus class.
The Noether-line examples, including smooth models with $p_g=7$ and $K^3=6$ on the first line and the $p_g=8$ model on the second line, are birationally different from earlier hypersurface constructions because their Picard numbers differ.
Each row of Table 5 gives an infinite family of minimal 3-folds of Kodaira dimension 2 as $r$ runs through the stated congruence classes; the criteria give $\nu(Y)=2$, and abundance for threefolds upgrades this to $\kappa(Y)=2$.

Reading between the lines

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

The nefness criterion is stated for arbitrary dimension and codimension, but all applications stop at 3-folds; applying the same inequalities to 4-dimensional weighted complete intersections with isolated non-canonical cyclic quotient singularities is the natural next experiment.
Only two examples are worked out in full; the paper states that all rows of Tables 1–2 were manually verified, without producing the verification or the search code, so the enumeration of 79 families is not machine-checkable from the text alone.
The two-blow-up mechanism of Theorem 3.2 could plausibly be iterated or combined with the multi-point criterion to produce more geography between the Noether lines, especially at genera below the $p_g\ge 11$ range of the structural theorem.
For the infinite families of Table 5, a finite check over the congruence classes (for instance all $r\le 200$) could turn each row into an explicitly verified infinite family, since the well-formedness and quasi-smoothness conditions are decidable by the paper's own criteria.

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.

Desk Editor's Note

read the letter

Read this if you care about explicit minimal 3-folds. The paper does something real: it extends Chen–Jiang–Li's hypersurface nefness criterion to weighted complete intersections in arbitrary codimension and to multi-point weighted blow-ups, and then runs a search that produces concrete families, including examples on the first and second Noether lines and volume records. I checked the main arguments; Theorems 1.2, 1.3, and 3.2 are internally coherent, and the intersection-theoretic proofs check out. The worked examples (4.1, 4.2, 4.7, 4.8) have correct volume computations and plausible singularity lists. The Theorem 4.9 computation giving K^3=0 is clean.

The soft spots, in order. First, the 79-family claim rests on a verification gap. Only four examples are worked out in the text; for the rest we are told manual verification was done, with no code, no per-row checks, and no machine-readable tables. One bad row—an unnoticed non-canonical singularity or a failure of quasi-smoothness—would subtract a family and could break the extremal claims. That is not a flaw in the theorems, but it is a mismatch between the quantitative headline and the evidence supplied. Second, the assertion that all examples have Picard number at least 2 is only justified for Table 1; Table 2 omits rho precisely because non-isolated singularities make it harder, so the blanket statement about differing from Iano-Fletcher is not yet supported for those rows. Third, there are small typos in the theorem statements (e.g. "hypersurface of multi-degrees" in Theorem 1.3 and an index bound in the alpha term of Theorem 1.2), which are confusing but not damaging.

The citation pattern is fine: CJL24 is the direct predecessor and the paper says so; the two imported lemmas are standard. The literature on Noether lines and minimal volumes is cited appropriately.

Verdict: the central construction is credible, and the four worked examples give real evidence that the search works. The paper is for specialists in explicit birational geometry and 3-fold geography. A serious referee should be able to verify the tables with moderate effort, and the claims are worth checking. I would not desk-reject. Send it to review, but ask the authors to supply verification data or at least the search script, and to soften the Picard-number claim for Table 2. If the tables survive that check, it is a solid contribution.

Referee Report

4 major / 5 minor

Summary. The paper generalizes the Chen–Jiang–Li construction of minimal varieties from quasi-smooth weighted hypersurfaces to quasi-smooth weighted complete intersections. It proves several nefness criteria for the canonical divisor after weighted blow-ups (Theorems 1.2, 1.3, and 3.2), covering higher codimension, several blown-up points, and a two-step weighted blow-up. It then presents 79 families of minimal 3-folds of general type (Tables 1 and 2), 16 examples near the Noether lines (Tables 3 and 4), and infinite families of minimal 3-folds of Kodaira dimension 2 (Table 5), with numerical invariants such as canonical volume, plurigenus P2, Euler characteristic, Picard number, and Reid basket. The proofs of the nefness criteria follow the strategy of Chen–Jiang–Li, and the four worked examples (Examples 4.1, 4.2, 4.7, and 4.8) check out in their stated computations.

Significance. If all table data are correct, the paper substantially enlarges the stock of explicit minimal 3-folds of general type, including examples with very small canonical volumes, examples on the first and second Noether lines, and infinite series of Kodaira dimension 2. It also extends a useful technique from hypersurfaces to complete intersections of arbitrary codimension and to multiple blow-ups. The paper makes concrete, falsifiable numerical claims (volumes, genera, baskets, Picard numbers) that can be independently checked, and the worked examples indicate that the underlying intersection-theoretic method is coherent. However, the quantitative significance rests on the correctness of every row of the tables, and most rows are not individually verified in the manuscript.

major comments (4)

[§4.2, §4.3, Tables 1–4] The headline claims of 79 families and 16 Noether-line examples depend on the correctness of every table row: each listed space must be well-formed and quasi-smooth, must have exactly the listed non-canonical singularities, and must satisfy the inequalities of Theorem 1.2, 1.3, or 3.2. Only Examples 4.1, 4.2, 4.7, and 4.8 are verified in detail. The statements in §4.2 that "All examples in Table 1 and Table 2 have been manually verified" and in §4.3 that all items "can be manually verified similar to previous cases" are not accompanied by per-row verification data or by the search code used in Step 0 of Construction 4.1. A single invalid row would invalidate that family and could change the advertised counts and extremal statements. Please provide either an ancillary file with machine-checkable verification for every row or a detailed description of the verification algorithm together with its complete output.
[Theorem 1.2, condition (3) and its proof] The proof of Theorem 1.2 requires that the particular curve ∩_{j=1}^{n-1} H_j on X be irreducible, since the argument identifies the support of ∩_{j=1}^{n-1} H'_j with the strict transform of this curve. However, condition (3) is stated for "a general" weighted complete intersection curve in the smaller weighted projective space. For a fixed general member X, the restriction of the defining equations to the coordinate subspace is not automatically a general member of the linear system on that subspace. This is a gap between the stated condition and what the proof uses. Please either replace condition (3) by the actual irreducibility hypothesis used, or prove that generality of X implies the required generality of the restricted curve; in the examples, the authors verify quasi-smoothness of the particular curve, which is a stronger check than the theorem as stated.
[Abstract, §1, §4.2, Tables 1–4] The count of families is ambiguous. The abstract and introduction say the paper constructs "79 families of minimal 3-folds of general type" and also "16 families of minimal 3-folds of general type on or near the Noether lines." Tables 1 and 2 alone contain 43 + 36 = 79 rows, while Tables 3 and 4 contain 12 + 4 = 16 further rows. If the Noether-line examples are additional, the total number of families is 95, not 79; if they are included in the 79, the phrase "in Table 1, Table 2, Table 3, and Table 4" in the introduction is confusing and the row count should be reconciled. Please clarify the intended counting and adjust the text accordingly.
[§4.2, paragraph after the tables] The equality h^0(\hat X, mK_{\hat X}) = h^0(X, mK_X) for m = 1, 2 is load-bearing for the listed values of P2 and χ, but the justification "since 2(e1+e2+e3)>r" is incomplete. The displayed chain h^0(\tilde X, mK_{\tilde X}) = h^0(\tilde X, \lfloor m f^* K_X \rfloor) is not automatic: when the coefficient of the exceptional divisor in mK_{\tilde X} lies in (0,1), the round-down is f^*mK_X - E, so the global sections are those of mK_X vanishing at the blown-up point. This vanishing holds in the worked examples, but it is not proved in general, and the table entries for P2 and χ depend on it. Please supply either a general proof of this vanishing under the stated hypotheses or a direct monomial check for every table row.

minor comments (5)

[Theorem 1.2, Theorem 1.3] The amplitude α is written as α = Σ d_l − Σ_{j=1}^{n+2} b_j in Theorem 1.2, and similarly in Theorem 1.3; for a codimension-c complete intersection in P(b_1,...,b_{n+c+1}) the upper limit should be n+c+1, matching Proposition 2.9(iii).
[Construction 4.1, Step 0] Step 0 says to check well-formedness and quasi-smoothness "by Definition 2.6 and Theorem 2.7," but there is no Theorem 2.7 in the paper; the quasi-smoothness criterion is Proposition 2.8.
[Theorem 1.3, Case 2, condition (2)] In the displayed inequality of Case 2, the summation index is written as u = 1 to s, but the number of points in this case is denoted N. The summation should run from u = 1 to N.
[Example 4.2] In the singularity computation, the text says "For P2 ... so P4 is a singularity of type 1/11(2,3,10)"; this should refer to P2, not P4, and should be corrected for readability.
[§1, Tables 1 and 2] The introduction claims that all examples are different from Iano-Fletcher's examples because they have Picard number at least 2, but Table 2 explicitly omits the Picard number for rows with non-isolated canonical singularities; the claim should be restricted to Table 1 or supported by a computation for Table 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the nefness criteria are proved by direct intersection-theoretic computations, the tables are constructed rather than fitted, and the CJL24 citations are independent auxiliary lemmas.

full rationale

The central derivation is self-contained. Theorem 1.2 expresses K_Y on the weighted blow-up as (alpha/b_j)H'_j + t_j E with t_j >= 0 by condition (1), and proves nefness by contradiction: a hypothetical negative curve must lie in the intersection of the strict transforms H'_j, whose support is irreducible by condition (3); the relevant intersection number is then computed explicitly and forced nonnegative by condition (2). Theorem 1.3 and Theorem 3.2 use the same direct intersection-theoretic argument and the same effective-divisor bookkeeping. No parameter is fitted to a data subset and then renamed a prediction; the canonical volumes, plurigenera and Picard numbers in Tables 1-5 are computed from the stated weights, degrees and blow-up data via Proposition 2.12, Reid's formula, and Lemma 3.9. The only citations to the same research group's earlier paper [CJL24] are Theorem 1.1 as motivation and Lemmas 3.6 and 3.9 as auxiliary tools; both lemmas have stated assumptions independent of the target examples and have published proofs elsewhere, so they are independent support rather than a self-citation chain forcing the conclusions. The paper's genuine weakness is that most table rows are asserted to be manually verified without providing the verification or search code, for example: 'All examples in Table 1 and Table 2 have been manually verified' and 'All items in Table 3 and Table 4 can be manually verified similar to previous cases.' That is a verification-transparency gap for the 79-family enumeration, not a circular definition of the output. No equation or construction step reduces to its own input.

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

The paper introduces no new free parameters: the weights and degrees of the examples are explicit constructions, not fitted values. It relies on standard singularity theory and birational geometry results from the literature, and cites two lemmas from the same research group's earlier paper, which have independent published proofs. No new particles, forces, or exotic entities are introduced.

assumptions (7)

standard math Weighted blow-up canonical bundle formula and volume formula (Proposition 2.12)
Quoted from Andreatta; used in every example to compute K_Y and K_Y^3 after the weighted blow-up.
standard math Quasi-smoothness criterion for weighted complete intersections (Proposition 2.8)
From Pizzato-Sano-Tasin; used to certify that the constructed X are quasi-smooth.
domain assumption Stratum-by-stratum singularity description for weighted hypersurfaces and codimension-2 complete intersections (Propositions 2.10, 2.11)
From Iano-Fletcher; used to identify the non-canonical points to blow up and to compute Reid baskets.
standard math Reid's plurigenus formula (Lemma 2.5)
Used to compute P_2 and higher plurigenera of the resulting minimal 3-folds from the basket.
domain assumption Abundance for 3-folds with canonical singularities
Used to equate numerical Kodaira dimension with genuine Kodaira dimension for the constructed 3-folds.
standard math Existence of terminalization and crepant divisor count (Lemma 3.9)
Adopted from CJL24; used to replace canonical singularities by terminal ones and compute Picard numbers.
standard math Noether inequality and Noether line results for 3-folds (Theorems 4.4 and 4.5)
Used to identify examples as lying on or near the Noether lines and to justify sharpness claims.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Construction of minimal varieties from quasi-smooth weighted complete intersections." pith.science (2026). https://pith.science/paper/SCFNTVFD

@misc{pith2026250708826,
  author       = {Pith},
  title        = {Pith review of: Construction of minimal varieties from quasi-smooth weighted complete intersections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCFNTVFD}},
  note         = {Machine review of arXiv:2507.08826}
}

abstract

This paper is devoted to the generalization of the construction of minimal varieties from the previous work of Meng Chen, Chen Jiang and Binru Li. We first establish several effective nefness criterions for the canonical divisor of weighted blow-ups over a weighted complete intersection, we both consider the high codimensional case and the blowing up several points case, from which we construct plenty of new minimal $3$-folds including $79$ families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, $16$ families of minimal $3$-folds of general type on or near the Noether lines.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages

[1]

Andreatta, Lifting weighted blow-ups , Rev

M. Andreatta, Lifting weighted blow-ups , Rev. Mat. Iberoam. 34 (2018), no. 4, 1809--1820

work page 2018
[2]

Birkar, P

C. Birkar, P. Cascini, C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405--468

work page 2010
[3]

Brown, A

G. Brown, A. Kasprzyk, Four-dimensional projective orbifold hypersurfaces , Exp. Math. 25 (2016), no. 2, 176--193

work page 2016
[4]

Brown, A

G. Brown, A. Kasprzyk, et al. Graded Ring Database , website: http://grdb.co.uk/

work page
[5]

Brown, M

G. Brown, M. Kerber, M. Reid, Fano 3 -folds in codimension 4, Tom and Jerry. Part I , Compos. Math. 148 (2012), no. 4, 1171--1194

work page 2012
[6]

Brown, M

G. Brown, M. Reid, Diptych varieties, I , Proc. Lond. Math. Soc. (3) 107 (2013), no. 6, 1353--1394

work page 2013
[7]

Brown, M

G. Brown, M. Reid, Diptych varieties. II: Apolar varieties , In: Higher dimensional algebraic geometry--in honour of Professor Yujiro Kawamata's sixtieth birthday, pp. 41--72, Adv. Stud. Pure Math., 74, Math. Soc. Japan, Tokyo, 2017

work page 2017
[8]

J.A. Chen, M. Chen, C. Jiang, The Noether inequality for algebraic 3 -folds , Duke Math. J. 169 (2020), no. 9, 1603--1645

work page 2020

Show all 33 references

[9]

J.A. Chen, M. Chen, C. Jiang, Addendum to ``The Noether inequality for algebraic 3 -folds'' , Duke Math. J. 169 (2020), no. 11, 2199--2204

2020
[10]

Chen, A sharp lower bound for the canonical volume of 3 -folds of general type , Math

M. Chen, A sharp lower bound for the canonical volume of 3 -folds of general type , Math. Ann. 337 (2007), no. 4, 887--908

2007
[11]

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

arXiv
[12]

M. Chen, C. Jiang, B. Li, On minimal varieties growing from quasi-smooth weighted hypersurfaces, J. Differential Geom. 127 (2024), no. 1, 35-76

2024
[13]

Y. Chen, Y. Hu, On canonically polarized Gorenstein 3 -folds satisfying the Noether equality , Math. Res. Lett. 24 (2017), no. 2, 271--297

2017
[14]

Corti, M

A. Corti, M. Reid, Weighted Grassmannians , In: Algebraic geometry, pp. 141--163, de Gruyter, Berlin, 2002

2002
[15]

Dolgachev, Weighted projective varieties , In: Group actions and vector fields (Vancouver, B.C., 1981), pp

I. Dolgachev, Weighted projective varieties , In: Group actions and vector fields (Vancouver, B.C., 1981), pp. 34--71, Lecture Notes in Math., 956, Springer, Berlin, 1982

1981
[16]

Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, Berlin, New York, 1995

D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, Berlin, New York, 1995

1995
[17]

Esser, B

L. Esser, B. Totaro, C. Wang, Varieties of general type with doubly exponential asymptotics , arXiv:2109.13383v2

arXiv
[18]

Fujiki, On resolutions of cyclic quotient singularities , Publ

A. Fujiki, On resolutions of cyclic quotient singularities , Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 293--328

1974
[19]

Y. Hu, T. Zhang, Algebraic threefolds of general type with small volume, Math. Ann. 391 (2025), no. 1, 567-612

2025
[20]

Iano-Fletcher, Working with weighted complete intersections , In: Explicit birational geometry of 3 -folds, pp

A.R. Iano-Fletcher, Working with weighted complete intersections , In: Explicit birational geometry of 3 -folds, pp. 101--173, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000

2000
[21]

Ishii, Introduction to singularities , Second edition, Springer, Tokyo, 2018, x+236 pp

S. Ishii, Introduction to singularities , Second edition, Springer, Tokyo, 2018, x+236 pp

2018
[22]

Johnson, J

J.M. Johnson, J. Koll\'ar, Fano hypersurfaces in weighted projective 4 -spaces , Experiment. Math. 10 (2001), no. 1, 151--158

2001
[23]

Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces , J

Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces , J. Reine Angew. Math. 363 (1985), 1--46

1985
[24]

Kawamata, Abundance theorem for minimal threefolds , Invent

Y. Kawamata, Abundance theorem for minimal threefolds , Invent. Math. 108 (1992), no. 2, 229--246

1992
[25]

Kawamata, K

Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem , In: Algebraic geometry, Sendai, 1985, pp. 283--360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987

1985
[26]

Kobayashi, On Noether's inequality for threefolds , J

M. Kobayashi, On Noether's inequality for threefolds , J. Math. Soc. Japan 44 (1992), no. 1, 145--156

1992
[27]

Koll \'a r, S

J. Koll \'a r, S. Mori, Birational geometry of algebraic varieties , Cambridge Tracts in Mathematics 134 , Cambridge Univ. Press, 1998

1998
[28]

Miyaoka, Abundance conjecture for 3 -folds: case = 1 , Compos

Y. Miyaoka, Abundance conjecture for 3 -folds: case = 1 , Compos. Math. 68 (1988), no. 2, 203--220

1988
[29]

Pizzato, T

M. Pizzato, T. Sano, L. Tasin, Effective nonvanishing for Fano weighted complete intersections, Algebra Number Theory 11 (2017), no. 10, 2369-2395

2017
[30]

Reid, Canonical 3 -folds , In: Journ \'e es de G \'e om \'e trie Alg \'e brique d'Angers, juillet 1979/Algebraic Geometry, Angers, 1979, pp

M. Reid, Canonical 3 -folds , In: Journ \'e es de G \'e om \'e trie Alg \'e brique d'Angers, juillet 1979/Algebraic Geometry, Angers, 1979, pp. 273--310, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980

1979
[31]

Reid, Young person's guide to canonical singularities , In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp

M. Reid, Young person's guide to canonical singularities , In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp. 345--414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987

1985
[32]

Reid, Constructing algebraic varieties via commutative algebra , In: European Congress of Mathematics, pp

M. Reid, Constructing algebraic varieties via commutative algebra , In: European Congress of Mathematics, pp. 655--667, Eur. Math. Soc., Z\"urich, 2005

2005
[33]

Totaro, C

B. Totaro, C. Wang, Klt varieties of general type with small volume , arXiv:2104.12200v4

arXiv

Pith tools

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

[1] [1]

Andreatta, Lifting weighted blow-ups , Rev

M. Andreatta, Lifting weighted blow-ups , Rev. Mat. Iberoam. 34 (2018), no. 4, 1809--1820

work page 2018

[2] [2]

Birkar, P

C. Birkar, P. Cascini, C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405--468

work page 2010

[3] [3]

Brown, A

G. Brown, A. Kasprzyk, Four-dimensional projective orbifold hypersurfaces , Exp. Math. 25 (2016), no. 2, 176--193

work page 2016

[4] [4]

Brown, A

G. Brown, A. Kasprzyk, et al. Graded Ring Database , website: http://grdb.co.uk/

work page

[5] [5]

Brown, M

G. Brown, M. Kerber, M. Reid, Fano 3 -folds in codimension 4, Tom and Jerry. Part I , Compos. Math. 148 (2012), no. 4, 1171--1194

work page 2012

[6] [6]

Brown, M

G. Brown, M. Reid, Diptych varieties, I , Proc. Lond. Math. Soc. (3) 107 (2013), no. 6, 1353--1394

work page 2013

[7] [7]

Brown, M

G. Brown, M. Reid, Diptych varieties. II: Apolar varieties , In: Higher dimensional algebraic geometry--in honour of Professor Yujiro Kawamata's sixtieth birthday, pp. 41--72, Adv. Stud. Pure Math., 74, Math. Soc. Japan, Tokyo, 2017

work page 2017

[8] [8]

J.A. Chen, M. Chen, C. Jiang, The Noether inequality for algebraic 3 -folds , Duke Math. J. 169 (2020), no. 9, 1603--1645

work page 2020