Towards finite generation of the canonical ring without the MMP

MMP 对 Enriques 对和奇异 Enriques 变种的研究

MMP for Enriques pairs and singular Enriques varieties

本文研究了 Enriques 变种在 MMP 下的稳定性，并证明了 Enriques 流形的 MMP 终止性，同时探讨了其渐近理论。

关于三维簇上秩一叶状结构的极小模型计划（MMP）

On the MMP for rank one foliations on threefolds

本文证明 Q-因子射影 klt 三维簇上秩一对数典范叶状配对存在翻转，从而推广了三维秩一叶状结构极小模型的存在性结果。

Kähler 3折上广义对的MMP

MMP for Generalized Pairs on Kähler 3-folds

本文研究Kähler 3折的广义对MMP，证明了极值定理、翻转存在性、log终端模型等核心结果。

广义叶面三维流形秩为1的MMP

MMP for generalized foliated threefolds of rank one

本文针对秩为1的广义叶面三维流形建立MMP，并证明了相关基点自由定理和log canonical阈值的ACC

一个有趣的壁穿越：壁穿越/MMP 对应的失效

An interesting wall-crossing: Failure of the wall-crossing/MMP correspondence

本文发现 Bridgeland 稳定性条件空间中的壁穿越可诱导非 Q-因子奇点，导致其无法被 MMP 检测，并通过 $D^b(\mathbb{P}^3)$ 中的具体例子展示了小收缩与除子收缩的不对称性。

Finite generation and geography of models

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

Relations between two log minimal models of log canonical pairs

We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by a sequence of flops, and the two log minimal models share some properties. We also give examples of two log minimal models of an lc pair which have different properties.

三维簇上余秩一叶状结构的极小模型计划（MMP）

MMP for co-rank one foliations on threefolds

该研究针对Q-阶乘射影三维簇上余秩一的F-dlt叶状配对，证明了翻转存在、特殊终止、基点自由定理及对数一般型下极小模型存在，推导了相关修正与奇点性质，验证了数值平凡情形的丰饶性，推进了叶状结构极小模型计划研究。

反典范极小模型纲领的赋值方法

A valuative approach to the anticanonical minimal model program

该研究提出用拟单项式赋值计算潜在klt三元组的对数典范阈值，拓展了Xu关于klt配对的结果，还证明可对潜在klt三元组和配对运行相应MMP，为反典范极小模型纲领提供了更灵活的研究框架。

叶状极小模型与翻转

Foliated Minimal Models and Flops

本文研究叶状结构的极小模型与翻转，证明秩1叶状结构的$K_F$-MMP输出同构、三维簇余秩1叶状结构的$D$-翻转存在性，同时构造例子揭示秩1叶状结构的病态。

关于log canonical广义对极小模型存在性的注记

Remarks on the existence of minimal models of log canonical generalized pairs

本文研究了NQC log canonical广义对的极小模型存在性，证明了在较弱意义下存在极小模型或非pseudo-effective时带缩放的MMP可终止，并在边界含ample divisor等附加条件下证明了pseudo-effective广义对的极小模型存在性。

关于维数至多为4的某些对的锥猜想

On the cone conjecture for certain pairs of dimension at most 4

本文利用极小模型纲领和反正则纤维化，在维数至多为4且Iitaka维数满足特定下界条件下证明了klt Calabi-Yau对的Morrison-Kawamata锥猜想。

Finite Generation of Canonical Rings

We prove the finite generation of canonical rings of projective variety of general type defined over complex numbers.

On log canonical rings

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.

Canonical rings of Gorenstein stable Godeaux surfaces

Extending the description of canonical rings from \cite{reid78} we show that every Gorenstein stable Godeaux surface with torsion of order at least $3$ is smoothable.

The canonical ring of a stacky curve

Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Gröbner basis. We work in a general algebro-geometric context and treat log canonical and spin canonical rings as well. As an application, we give an explicit presentation for graded rings of modular forms arising from finite-area quotients of the upper half-plane by Fuchsian groups.

Canonical rings of surfaces whose canonical system has base points

In this paper we propose an approach to investigate the canonical rings of surfaces of general type whose canonical system has isolated base points and yields a birational map onto its image. We apply then the method in the concrete case of surfaces with $K^2=7$ and $p_g=4$, where we give three different descriptions of the canonical ring (under the presence of base points). The main result in the paper consists in showing an explicit deformation of the above canonical rings to the canonical rings of surfaces whose canonical system has no base points (the latter were already classified by the …

On the anti-canonical ring and varieties of Fano type

We study the anti-canonical ring of a projective variety and we characterise varieties of log Fano type depending on the singularities of these models.

Derived invariance of the Albanese relative canonical ring

We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.

On the log canonical ring of projective plt pairs with the Kodaira dimension two

The log canonical ring of a projective plt pair with the Kodaira dimension two is finitely generated.

Finite generation of the log canonical ring for 3-folds in char $p$

We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $κ=2$.

On the log canonical ring in Kodaira dimension two

We prove that the log canonical ring of a projective log canonical pair in Kodaira dimension two is finitely generated.

Birational geometry and the canonical ring of a family of determinantal 3-folds

Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.

Finite generation of a canonical ring

The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan whose method is based on the Minimal Model Program. A survey article for Current Development in Mathematics 2007.

Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers

In this paper, we prove the weak positivity theorem in positive characteristic when the canonical ring of the geometric generic fiber $F$ is finitely generated and the Frobenius stable canonical ring of $F$ is large enough. As its application, we show the subadditivity of Kodaira dimensions in some new cases.

On the canonical ring of curves and surfaces

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, ω_C) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with K_C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p_g(S)>0 and q(S)=0 the canonical ring R(S, K_S) is generated in degree \leq 3 if there exists a curve C in |K_S| numerically 3-connected and not hyperelliptic.

具有有限生成反典范代数的环中的超测试理想

Ultra-test ideals in rings with finitely generated anti-canonical algebras

本文在反典范环有限生成的条件下，利用超弗罗贝尼乌斯刻画伴随理想，并据此给出 Zhuang 关于 klt 型环的纯子环仍为 klt 型这一结果的替代证明。

具有有限生成典范环的klt广义对

A klt generalized pair with infinitely generated canonical ring

本文构造了一个复数域上的射影klt广义对，其广义典范环非有限生成，该结果由GPT-5.6-sol-ultra等生成式AI辅助取得。

二维亏格Gorenstein自旋曲线的半典范环

Half canonical rings of Gorenstein spin curves of genus two

该研究引入曲线上广义Gorenstein自旋结构概念，明确描述二维亏格丰富典范丛曲线的截面环，得到五种不同形式，为相关代数几何研究提供基础。

半dlt对的相对对数典范代数的有限生成

Finite generation of relative log canonical algebras of semi-dlt pairs

本文直接证明了半dlt对的相对对数典范代数的有限生成，避免Kollár粘合理论，并由此恢复并推广了相关MMP结果，为稳定对模空间固有性证明奠定基础。

Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is $F$-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases.

The canonical ring of a 3-connected curve

Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, ω_C)= \oplus_{k \geq 0} H^0(C, ω_C^k is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L)=\oplus_{k \geq 0} H^0(C,L^k)$ is generated in degree 1 if C is reduced with planar singularities and L is an invertible sheaf such that deg L_{|B} \geq 2p_a(B)+1 for every B \subseteq C.

On the Canonical Ring of Covers of Surfaces of Minimal Degree

Let f be a generically finite morphism from X to Y. The purpose of this paper is to show how the O_Y algebra structure on the push forward of O_X controls algebro-geometric aspects of X like the ring generation of graded rings associated to X and the very ampleness of line bundles on X. As the main application of this we prove some new results for certain regular surfaces X of general type. Precisely, we find the degrees of the generators of the canonical ring of X when the canonical morphism of X is a finite cover of a surface of minimal degree. These results complement results of Ciliberto […

Finite generation of the log canonical ring in dimension four

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundan…

The Beilinson complex and canonical rings of irregular surfaces

In the first part of the paper Beilinson's theorem on the bounded derived category of coherent sheaves on P^n is extended to weighted projective spaces in a rather explicit form. To this purpose the usual category of coherent sheaves is replaced by a suitable category of graded sheaves, and a more general theory of graded schemes is developed. In the second part of the paper the weighted version of Beilinson's theorem is applied to prove a structure theorem for certain canonical projections of surfaces of general type into a 3-dimensional weighted projective space. This result (which generaliz…

Singularities with G_m-action and the log minimal model program for $\bar{M}_g$

We give a precise formulation of the modularity principle for the log canonical models of $\bar{M}_g$. Assuming the modularity principle holds, we develop and compare two methods for determining the critical alpha-values at which a singularity or complete curve with G_m-action arises in the modular interpretations of log canonical models of $\bar{M}_g$. The first method involves a new invariant of curve singularities with G_m-action, constructed via the characters of the induced G_m-action on spaces of pluricanonical forms. The second method involves intersection theory on the variety of stabl…

What is missing in canonical models for proper normal algebraic surfaces?

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.

Note on asymptotic behaviour of the canonical ring

The theory of the Oukounkov body is a useful tool for studying the asymptotic behaviour of the canonical ring of a line bundle over a projective manifold. In this note, combined with the algebraic reduction, we study the asymptotic behaviour of the canonical ring of a line bundle over an arbitrary compact normal irreducible complex space.

Additional Explanatory Notes on the Analytic Proof of the Finite Generation of the Canonical Ring

This set of notes provides some additional explanatory material on the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type.

On the canonical ideals of one-dimensional Cohen-Macaulay local rings

In this paper we consider the problem of finding explicitly canonical ideals of one-dimensional Cohen-Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension two case, from a Hilbert-Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.

Canonical Maps to Twisted Rings

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest.

The Brauer group of a valuation ring

Let $V$ be a valuation ring and $K$ be its field of fraction. We show that the canonical map $\Br(V) \to \Br(K)$ is injective.

Canonical semi-rings of finite graphs and tropical curves

For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it is classically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, m K_C)$ is finitely generated in degree at most three. In this article, we study whether analogous statements hold for finite graphs and tropical curves. For any finite graph $G$, we show that the canonical semi-ring $R(G)$ is finitely generated but that the degree of generators are not bounded by a universal constant. For any hyperelliptic tropical curve $Γ$ with integer edge-length, we show that the canonical semi-ring $R(Γ)$ is not finitely ge…

The amalgamated duplication of a ring along a multiplicative-canonical ideal

After recalling briefly the main properties of the amalgamated duplication of a ring $R$ along an ideal $I$, denoted by $R\JoinI$, we restrict our attention to the study of the properties of $R\JoinI$, when $I$ is a multiplicative canonical ideal of $R$ \cite{hhp}. In particular, we study when every regular fractional ideal of $R\Join I$ is divisorial.

The divisor class groups and the graded canonical modules of multi-section rings

We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor due to Hashimoto.

Spin canonical rings of log stacky curves

Consider modular forms arising from a finite-area quotient of the upper-half plane by a Fuchsian group. By the classical results of Kodaira-Spencer, this ring of modular forms may be viewed as the log spin canonical ring of a stacky curve. In this paper, we tightly bound the degrees of minimal generators and relations of log spin canonical rings. As a consequence, we obtain a tight bound on the degrees of minimal generators and relations for rings of modular forms of arbitrary integral weight.

Triple canonical covers of varieties of minimal degree

In this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives. First we study in Section 2 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of $X$ also shows that, if the dimension of X is greater than or equal to 3, there does not exist a con…

Adjoint rings are finitely generated

This paper proves finite generation of the log canonical ring without Mori theory.

The degree of the generators of the canonical ring of surfaces with p_g=0

It is shown that the canonical ring of a minimal surface of general type with $p_g=0, K^2\geq 2$ is generated by its elements of degree lesser or equal to 5, provided $|2K|$ has no fixed components, and that this bound can be lowered to 4 under the assumption that $|2K|$ is base-point free.

The canonical module of a Cox ring

In this paper, we shall describe the graded canonical module of a Noetherian multi-section ring of a normal projective variety. In particular, in the case of the Cox ring, we prove that the graded canonical module is a graded free module of rank one with the shift of degree $K_X$. We shall give two kinds of proofs. The first one utilizes the equivariant twisted inverse functor developed by the first author. The second proof is down-to-earth, that avoids the twisted inverse functor.

通过Cox环的特殊典范嵌入证明Mukai猜想

The Mukai conjecture via Cox rings for smooth canonical ambient embeddings

本文通过Cox环理论研究特殊典范嵌入下的Fano簇，证明了Mukai猜想在局部因子Fano簇中的成立，基于Mori梦想空间的log版本。

Canonical threefold singularities with a torus action of complexity one and $k$-empty polytopes

We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold singularities. Our classification relies on lattice point emptiness of certain polytopal complexes with rational vertices. Scaling the polytopes by the least common multiple $k$ of the respective denominators, we investigate $k$-emptiness of polytopes with integer vertices. We show that two dimensional $k$-empty polytopes either are sporadic or come in series given by F…

The Chow ring of hyperkähler varieties of $K3^{[2]}$-type via Lefschetz actions

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift …

关于扭曲的 Kawamata 半正定性和广义 canonical 环的有限生成性

On twisted Kawamata's semi-positivity and finite generation of generalized canonical rings

本文提出了一种扭曲的 Kawamata 半正定性定理，并应用于广义对 canonical 环的有限生成性问题。