On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two

Ching-Jui Lai

On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two

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Pith's one-line read For a weak $\mathbb{Q}$-Fano threefold of Picard rank two, the anticanonical volume is at most 54 when both two-ray games end in Mori fiber spaces, and at most 72 otherwise unless one specified configuration occurs, with equality only for…

desk verdict A credible step on the rank-two weak Q-Fano volume problem; the one soft patch (Lemma 5.1) holds up, and the deferred exceptional subcase is honestly flagged. read the letter →

arxiv 2501.12555 v1 pith:3BCYXFER submitted 2025-01-22 math.AG

Ching-Jui Lai This is my paper · ORCID

classification math.AG MSC 14J3014J4514E30

keywords Fanovarietiesterminalthreefoldsminimalmodelprogramanticanonicalvolumetwo-raygamePicardranktwoweakQ-FanothreefoldMorifiberspace

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 an upper bound on the anticanonical volume of weak $\mathbb{Q}$-Fano threefolds---terminal, $\mathbb{Q}$-factorial threefolds whose anticanonical divisor is nef and big---when the Picard rank is two. Running two-ray games from the two extremal rays, the author shows that either both games end in Mori fiber spaces and the volume is at most 54, or one game produces a $K$-trivial divisorial contraction and the volume is at most 72, with a single exception where only the weaker bound 81 is obtained. Equality at 72 forces the variety to be the projective bundle $\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The result is a first step toward the conjecture that 72 is the optimal upper bound for all weak $\mathbb{Q}$-Fano threefolds.

What carries the argument

The engine of the proof is the two-ray game: starting from each of the two extremal rays of $X$, run the minimal model program until the ray becomes a Mori fiber space or a $K$-trivial divisorial contraction. The anticanonical class is then decomposed as $-K_X\equiv aA_l+bA_r$ in the double-Mori case, or $-K_X\equiv aF_X+bE$ (with $F_X$ replaced by $H_X$ when the left base is a surface) in the mixed case, and the coefficients are bounded by the length of extremal curves on del Pezzo surfaces and by a subadjunction estimate on the exceptional divisor. Key supporting identities are the negativity lemma for comparing divisors across birational models and the relation $-E\cdot\Gamma=-(K_X+E)\cdot\Gamma$ that translates intersection numbers on the exceptional divisor into coefficient bounds.

What would settle it

Take a $K$-trivial divisorial contraction to a point in the mixed case with $\dim Z_l=2$, resolve the pair $(E^\nu,\mathrm{Diff}(0))$ minimally, and check whether a moving curve $\Gamma$ with $-E\cdot\Gamma\le 3$ exists when the minimal resolution is a ruled surface over a positive-genus curve; if not, Lemma 5.1's patched classification fails and the $72$ bound in that case is unsupported.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is the Main Theorem: for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two, after running two-ray games on the two extremal rays, either both games terminate in Mori fiber spaces and $-K_X^3\leq 54$, or one game produces a $K$-trivial divisorial contraction and $-K_X^3\leq 72$, except when $\dim Z_l=1$ and $\dim\phi_r(E)=0$, where the estimate is only $-K_X^3\leq 81$. Equality $-K_X^3=72$ can occur only for $X\cong\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The proof splits the two-ray game into these two cases, writes $-K_X$ as a positive linear combination of a fibre class (or pullback of an ample class) from one side and the exceptional divisor from the other, and bounds the coefficients using extremal curve lengths on del Pezzo surfaces and a subadjunction estimate for the exceptional divisor contracted to a point.

Load-bearing premise

The load-bearing premise is that the patched surface classification in Lemma 5.1 is complete: after the footnote, every minimal surface without $(-1)$-curves is $\mathbb{P}^2$, a ruled surface of the form $F_n$, or a ruled surface over a positive-genus curve, and in each case a moving curve $\Gamma$ with $-E\cdot\Gamma\le 3$ exists; if this fails, the coefficient estimate $a\le 3b$ in Lemma 5.2, needed for the $72$ bound in the mixed case, does not follow.

Editorial extensions

If this is right

In case (I), where both two-ray games end in Mori fiber spaces, the stronger bound $-K_X^3\leq 54$ holds for every weak $\mathbb{Q}$-Fano threefold of Picard rank two.
In the mixed case, the bound $-K_X^3\leq 72$ holds whenever $\dim Z_l=2$ or $\dim\phi_r(E)=1$, and equality forces $X\cong\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$.
In the remaining exceptional configuration, the anticanonical volume is at most 81; the author expects the 72 bound to persist there as well.
Because terminalization and $\mathbb{Q}$-factorization preserve anticanonical volume, the same bounds apply to canonical weak Fano threefolds of Picard rank two.

Reading between the lines

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

Beyond the paper's cases, the coefficient-decomposition method suggests that a similar two-ray analysis could bound anticanonical volumes when the Picard rank is larger, provided the terminal models still admit a Mori fiber space on at least one ray.
The missing $\dim Z_l=1$, $\dim\phi_r(E)=0$ case is the natural testing ground: if any example there has $-K_X^3>72$, the conjecture of optimality 72 would fail, while a proof of 72 would complete the Picard-rank-two story.
The subadjunction bound on $E$ may generalize: analogous estimates for exceptional divisors of $K$-trivial contractions in higher dimensions could feed into volume bounds for canonical Fano varieties of dimension four and above.

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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

Dear colleague,

Short version: this paper proves a real new upper bound, -K^3 <= 72 for weak Q-Fano terminal threefolds of Picard rank two, with equality only for P_{P^2}(O + O(3)), except for one explicit subcase where it proves 81 and defers 72. That subcase is not a hidden gap; it is stated as open. The approach is a two-ray game decomposition plus intersection estimates from conic bundles. It is a genuine advance over known Gorenstein and rank-one bounds.

What's good: the reduction to two cases is clean, Lemma 4.2's volume inequality and the estimates in Section 4 are convincing, and the extremal example is correctly identified. The proof is honest about where it relies on standard MMP tools (BCHM, abundance, boundedness) rather than hiding assumptions. No circularity, no fitted parameters.

Soft spots, in order of seriousness. Lemma 5.1 states that if S contains no (-1)-curve, then S = P2 or F_n. That is false as written; minimal ruled surfaces over positive-genus curves also have no (-1)-curves. The footnote patches it by adding the ruled-surface case, and the estimate still goes through because -(K_S+Delta_S)·f <= 2 for a general fiber. But the main text should be corrected, not footnoted, since a classification claim is involved. Proposition 5.5's key inequality is compressed: the sign of the intersection term E''²·p_l^*F_l <= 0 is justified by pulling back to a general smooth surface, but the argument is terse and worth expanding for referees. Remark 5.3 honestly explains why a naive length-of-extremal-curves bound does not give a <= 4; I found that discussion useful rather than evasive.

The exceptional case (dim Z_l = 1, dim phi_r(E) = 0) leaves a genuine open subcase. Since the paper claims <=72 "except in one case" and proves <=81 there, the theorem as stated is accurate. The conjecture that 72 is optimal overall remains open, so readers should not expect a full resolution of the Main Question.

Who this is for: MMP people working on Fano threefolds and anticanonical volumes. It deserves serious peer review. The result is new and the proof is mostly solid; the two flagged issues are fixable in revision. I'd send it to a good journal's referee, not desk reject.

Recommendation: engage. Assign a referee who knows conic bundles and the two-ray game. Ask them to check Lemma 5.2's coefficient bound a <= 3b and Proposition 5.5's inequality line by line; those are the load-bearing numerical steps. If those survive, the paper is a solid contribution.

Referee Report

0 major / 5 minor

Summary. The paper studies weak Q-Fano terminal threefolds of Picard rank two and proves an upper bound for their anticanonical volume. The main theorem states that either -K_X^3 <= 54 when both two-ray games terminate in Mori fiber spaces, or, when a K-trivial divisorial contraction appears, -K_X^3 <= 72 except in the case where the base of the left Mori fiber space is a curve and the K-trivial contraction contracts its exceptional divisor to a point, in which case the bound -K_X^3 <= 81 is proved. The equality case -K_X^3 = 72 is characterized as X isomorphic to P_{P^2}(O_{P^2} + O_{P^2}(3)). The proof uses two-ray games, conic bundle geometry, subadjunction, and length estimates for extremal curves.

Significance. If the proof is correct, this is a substantial step toward the conjectural optimal bound -K_X^3 <= 72 for all weak Q-Fano threefolds, resolving the Picard rank two case up to a single explicitly isolated subcase. The paper gives a clear framework that could generalize to higher Picard rank, and it provides a concrete example attaining the bound. The exposition is honest about the unresolved subcase, which is deferred to a forthcoming paper. The argument relies entirely on established theorems (BCHM, abundance, boundedness, Prokhorov's bounds) and introduces no fitted parameters; the equality characterization is an explicit falsifiable statement.

minor comments (5)

[5, Lemma 5.1] The sentence 'If S contains no (-1)-curve, then S = P^2 or S = F_n' is false as written, since minimal ruled surfaces over curves of positive genus also contain no (-1)-curves. The footnote patches this, but the main text should incorporate the correction, for instance by explicitly noting that S is rational because E is rationally chain connected, so that the classical classification of minimal rational surfaces applies.
[5, proof of Lemma 5.2] The justification of the inequality F_X * Gamma >= 1 is compressed. Please expand the argument to show that a general member of the moving family Gamma from Lemma 5.1 is not contracted by the birational map to X_l and that its image in X_l has positive intersection with a general fiber F_l (or H_l when dim Z_l = 2).
[5, Proposition 5.5] The chain of inequalities in the proof of Proposition 5.5 would be easier to follow if the vanishing and sign of the exceptional terms E' and E'' were stated explicitly with the supporting projection-formula argument; currently several steps are compressed into a single line.
[Global] There are several typographical and grammatical errors: the title contains 'Q-F ano' instead of 'Q-Fano'; the abstract says 'can be served as' instead of 'can serve as'; and in the display 'S = P2 orS = Fn' there are missing spaces.
[6.1] In the proof of Proposition 6.1, the step where E_l -> P^2 is concluded to be an isomorphism uses the phrase 'bijective projective morphism'; for clarity, specify that E_l is a section of a smooth conic bundle, so it is isomorphic to P^2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Main Theorem is derived from independent external results (BCHM, MMP, Prokhorov's bounds) with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained with respect to circularity. The Main Theorem is established by running two-ray games, expressing -K_X as a linear combination of divisor classes pulled back from two Mori fiber spaces (or a Mori fiber space and a K-trivial exceptional divisor), and bounding the resulting intersections via Lemma 4.2, Lemma 4.3, Lemma 5.1, and Lemma 5.2. None of these lemmas defines its conclusion in terms of the target volume bound. Lemma 4.1 invokes BCHM and Mori dream space theory for independence of the two divisor classes; Lemma 4.3 uses the conic-bundle numerical formula from Proposition 2.3 (whose proof is included) and standard del Pezzo surface extremal lengths; Lemma 5.2 uses subadjunction, rational chain connectedness, and integrality of intersections with moving fibers. The equality case X = P_P2(O⊕O(3)) is obtained by tracing numerical equalities and identifying the projective bundle via a reflexive rank-two sheaf, not by assuming the conclusion. There is no fitted input called a prediction: the numerical coefficients a and b are computed from intersection products, not adjusted to match -K_X^3. There are no load-bearing self-citations: the bibliography cites external work (BCHM, Mori-Prokhorov, Prokhorov, Kawamata, etc.), and the single citation to work co-authored with an acknowledged colleague ([6]) concerns the unrelated lower bound 1/330. The unresolved exceptional subcase (dim Z_l = 1, dim phi_r(E) = 0) is openly stated with the weaker bound 81; this is an admitted incompleteness, not a circular derivation. The flagged Lemma 5.1 issue (the initial listing of P^2 and F_n omitting positive-genus ruled surfaces) is a correctness concern, patched in the footnote by allowing a P^1-bundle S -> C; even if the patch failed, that would be a gap in an auxiliary estimate rather than a reduction of the theorem to its own assumptions. No step in the paper equates a target quantity to an input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard MMP results (BCHM, abundance, termination of flips), Prokhorov's bounds for ρ=1 and Gorenstein cases, and classification results for singular threefolds. None are ad hoc.

assumptions (6)

standard math Existence of minimal models for klt pairs (BCHM), implying weak Q-Fano threefolds are Mori dream spaces.
Used in Lemma 4.1 to show divisors A_l and A_r are independent, and in Section 6 to show no nontrivial SQM of P(O⊕O(3)).
standard math Abundance and termination of flips in dimension three.
Used in Section 3 to ensure the two-ray game terminates and yields cases (I) or (II).
standard math Prokhorov's bounds: -K_X^3 ≤ 64 for smooth Fano threefolds, ≤72 for Gorenstein weak Q-Fano, and ≤125/2 for non-Gorenstein ρ=1 Q-Fano.
Used in case (D-) to conclude -K_X^3 ≤ 64 when a K-negative divisorial contraction appears.
standard math Mori-Prokhorov classification of Q-conic bundles: base Z has A-singularities, and -4K_Z ≡ f_*K_X^2 + Δ.
Essential for Lemma 4.3 (bound K^2·H ≤ 12) and Proposition 6.1.
standard math Kollár's subadjunction formula for the exceptional divisor E: (K_X+E)|_{E^ν} = K_{E^ν}+Diff(0).
Used in Lemma 5.1 to transfer the bound on -E·Γ to a surface statement.
domain assumption Classification of minimal surfaces without (-1)-curves: P^2, Hirzebruch surfaces F_n (n≠1), and ruled surfaces over positive-genus curves (the latter added by footnote).
Needed in Lemma 5.1 to find a covering curve Γ_S with -(K_S+Δ_S)·Γ_S ≤ 3; the initial text omits ruled surfaces over positive genus, patched by footnote.

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Pith. "Pith review of On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two." pith.science (2026). https://pith.science/paper/3BCYXFER

@misc{pith2026250112555,
  author       = {Pith},
  title        = {Pith review of: On anticanonical volumes of weak $\mathbbQ$-Fano terminal threefolds of Picard rank two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BCYXFER}},
  note         = {Machine review of arXiv:2501.12555}
}

abstract

We show that for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two ($\mathbb{Q}$-factorial with at worst terminal singularities), the anticanonical volume satisfies $-K_X^3\leq72$ except in one case, and the equality holds only if $X=\mathbb{P} (\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The approach in this article can serve as a general strategy to establish the optimal upper bound of $-K_X^3$ for any canonical Fano threefolds, where the described main result serves as the first step.

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Works this paper leans on

32 extracted references · 31 canonical work pages

[1]

Singularities of linear systems and boundedness of fano varieties, 2016

Caucher Birkar. Singularities of linear systems and boundedness of fano varieties, 2016. arXiv:1609.05543. (Cited on page 2.) 2From this and 12 = K 2 X · H, one can conclude K 2 X · E = KX · E2 = 0 and E3 = 9. ANTICANONICAL VOLUMES OF WEAK Q-F ANO THREEFOLDS 17

arXiv 2016
[2]

Anti-pluricanonical systems on Fano varieties

Caucher Birkar. Anti-pluricanonical systems on Fano varieties. Ann. of Math. (2) , 190(2):345–463, 2019. (Cited on page 2.)

work page 2019
[3]

Hacon, and James McKernan

Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. , 23(2):405–468,

work page
[4]

Algebraic surfaces

Lucian B˘ adescu. Algebraic surfaces . Universitext. Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Ma¸ sek and re vised by the author. (Cited on pages 10 and 15.)

work page 2001
[5]

Cone theorem via Deligne- Mumford stacks

Jiun-Cheng Chen and Hsian-Hua Tseng. Cone theorem via Deligne- Mumford stacks. Math. Ann. , 345(3):525–545, 2009. (Cited on page 14.)

work page 2009
[6]

Chen and Meng Chen

Jungkai A. Chen and Meng Chen. An optimal boundedness on wea k Q-Fano 3-folds. Adv. Math., 219(6):2086–2104, 2008. (Cited on page 2.)

work page 2008
[7]

On the anti-canonical geometry of Q-Fano threefolds

Meng Chen and Chen Jiang. On the anti-canonical geometry of Q-Fano threefolds. J. Diﬀerential Geom. , 104(1):59–109, 2016. (Cited on page 3.)

work page 2016
[8]

Hacon and James Mckernan

Christopher D. Hacon and James Mckernan. On Shokurov’s ratio nal connectedness conjecture. Duke Math. J. , 138(1):119–136, 2007. (Cited on pages 7 and 12.)

work page 2007

Show all 32 references

[9]

Hacon and James McKernan

Christopher D. Hacon and James McKernan. Existence of minimal models for varieties of log general type. II. J. Amer. Math. Soc. , 23(2):469–490, 2010. (Cited on page 1.)

2010
[10]

Algebraic geometry

Robin Hartshorne. Algebraic geometry . Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. (Cited on page 16.)

1977
[11]

Mori dream spaces and GIT

Yi Hu and Sean Keel. Mori dream spaces and GIT. Michigan Math. J. , 48:331–348,
[12]

A. R. Iano-Fletcher. Working with weighted complete intersect ions. In Explicit bira- tional geometry of 3-folds , volume 281 of London Math. Soc. Lecture Note Ser. , pages 101–173. Cambridge Univ. Press, Cambridge, 2000. (Cited on page 2.)

2000
[13]

V. A. Iskovskih. Anticanonical models of three-dimensional alg ebraic varieties. In Cur- rent problems in mathematics, Vol. 12 (Russian) , pages 59–157, 239 (loose errata). VINITI, Moscow, 1979. (Cited on pages 3 and 16.)

1979
[14]

The index of a threefold canonical singular ity

Masayuki Kawakita. The index of a threefold canonical singular ity. Amer. J. Math. , 137(1):271–280, 2015. (Cited on page 14.)

2015
[15]

Abundance theorem for minimal threefolds

Yujiro Kawamata. Abundance theorem for minimal threefolds. Invent. Math. , 108(2):229–246, 1992. (Cited on page 7.)

1992
[16]

Boundedness of Q-Fano threefolds

Yujiro Kawamata. Boundedness of Q-Fano threefolds. In Proceedings of the Interna- tional Conference on Algebra, Part 3 (Novosibirsk, 1989) , volume 131 of Contemp. Math., pages 439–445, Providence, RI, 1992. Amer. Math. Soc. (Cited on page 3.)

1989
[17]

Cambridge University Press, Cambridge, 2013

J´ anos Koll´ ar.Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2013. With a collab- oration of S´ andor Kov´ acs. (Cited on page12.)

2013
[18]

Boundedness of canonical Q-Fano 3-folds

J´ anos Koll´ ar, Yoichi Miyaoka, Shigefumi Mori, and Hiromichi Ta kagi. Boundedness of canonical Q-Fano 3-folds. Proc. Japan Acad. Ser. A Math. Sci. , 76(5):73–77, 2000. (Cited on page 2.)

2000
[19]

Birational geometry of algebraic varieties , volume 134 of Cambridge Tracts in Mathematics

J´ anos Koll´ ar and Shigefumi Mori. Birational geometry of algebraic varieties , volume 134 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge,
[20]

Abundance conjecture for 3-folds: case ν = 1

Yoichi Miyaoka. Abundance conjecture for 3-folds: case ν = 1. Compositio Math. , 68(2):203–220, 1988. (Cited on page 7.) 18 CHING-JUI LAI

1988
[21]

On the Kodaira dimension of minimal threefolds

Yoichi Miyaoka. On the Kodaira dimension of minimal threefolds. Math. Ann. , 281(2):325–332, 1988. (Cited on page 7.)

1988
[22]

Classiﬁcation of Fano 3-folds with B2 ≥ 2

Shigefumi Mori and Shigeru Mukai. Classiﬁcation of Fano 3-folds with B2 ≥ 2. I. In Algebraic and topological theories (Kinosaki, 1984) , pages 496–545. Kinokuniya, Tokyo,

1984
[23]

On Q-conic bundles

Shigefumi Mori and Yuri Prokhorov. On Q-conic bundles. Publ. Res. Inst. Math. Sci. , 44(2):315–369, 2008. (Cited on page 5.)

2008
[24]

Smoothing Fano 3-folds

Yoshinori Namikawa. Smoothing Fano 3-folds. J. Algebraic Geom. , 6(2):307–324, 1997. (Cited on page 2.)

1997
[25]

Yu. G. Prokhorov. The degree of Fano threefolds with canonic al Gorenstein singulari- ties. Mat. Sb. , 196(1):81–122, 2005. (Cited on pages 2 and 5.)

2005
[26]

Yu. G. Prokhorov. The degree of Q-Fano threefolds. Mat. Sb. , 198(11):153–174, 2007. (Cited on pages 2, 3, and 14.)

2007
[27]

On Fano indices of Q-Fano 3-folds

Kaori Suzuki. On Fano indices of Q-Fano 3-folds. Manuscripta Math. , 114(2):229–246,
[28]

Rational connectedness of log Q-Fano varieties

Qi Zhang. Rational connectedness of log Q-Fano varieties. J. Reine Angew. Math. , 590:131–142, 2006. (Cited on page 7.) Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan Email address : cjlai72@mail.ncku.edu.tw

2006
[1986]

(Cited on pages 5 and 6.)
[1998]

With the collaboration of C. H. Clemens and A. Corti, Translate d from the 1998 Japanese original. (Cited on pages 5 and 6.)

1998
[2000]

(Cited on page 9.)

Dedicated to William Fulton on the occasion of his 60th birthday. (Cited on page 9.)
[2010]

(Cited on pages 1, 9, and 16.)

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Singularities of linear systems and boundedness of fano varieties, 2016

Caucher Birkar. Singularities of linear systems and boundedness of fano varieties, 2016. arXiv:1609.05543. (Cited on page 2.) 2From this and 12 = K 2 X · H, one can conclude K 2 X · E = KX · E2 = 0 and E3 = 9. ANTICANONICAL VOLUMES OF WEAK Q-F ANO THREEFOLDS 17

arXiv 2016

[2] [2]

Anti-pluricanonical systems on Fano varieties

Caucher Birkar. Anti-pluricanonical systems on Fano varieties. Ann. of Math. (2) , 190(2):345–463, 2019. (Cited on page 2.)

work page 2019

[3] [3]

Hacon, and James McKernan

Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. , 23(2):405–468,

work page

[4] [4]

Algebraic surfaces

Lucian B˘ adescu. Algebraic surfaces . Universitext. Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Ma¸ sek and re vised by the author. (Cited on pages 10 and 15.)

work page 2001

[5] [5]

Cone theorem via Deligne- Mumford stacks

Jiun-Cheng Chen and Hsian-Hua Tseng. Cone theorem via Deligne- Mumford stacks. Math. Ann. , 345(3):525–545, 2009. (Cited on page 14.)

work page 2009

[6] [6]

Chen and Meng Chen

Jungkai A. Chen and Meng Chen. An optimal boundedness on wea k Q-Fano 3-folds. Adv. Math., 219(6):2086–2104, 2008. (Cited on page 2.)

work page 2008

[7] [7]

On the anti-canonical geometry of Q-Fano threefolds

Meng Chen and Chen Jiang. On the anti-canonical geometry of Q-Fano threefolds. J. Diﬀerential Geom. , 104(1):59–109, 2016. (Cited on page 3.)

work page 2016

[8] [8]

Hacon and James Mckernan

Christopher D. Hacon and James Mckernan. On Shokurov’s ratio nal connectedness conjecture. Duke Math. J. , 138(1):119–136, 2007. (Cited on pages 7 and 12.)

work page 2007