M. Mendes Lopes, R. Pardini
We classify completely the surfaces of general type whose canonical map is 3-to-1 onto a surface of minimal degree in projective space. These surfaces fall into 5 distinct classes and we give explicit examples belonging to each of these classes. As far as we know, one of the examples thus constructed was unknown and it is a surface whose canonical system has two infinitely near base points.
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For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.
We construct a surface of general type with canonical map of degree 12 which factors as a triple cover and a bidouble cover of $\mathbb P^2$. We also show the existence of a smooth surface with $q=0,$ $χ=13$ and $K^2=9χ$ such that its canonical map is either of degree 3 onto a surface of general type or of degree 9 onto a rational surface.
In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 3$ for which the canonical map induces a triple cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ is again canonical and induces a triple cover. The extremal case when $Y$ is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. This is especially interesting as well, since it has n…
We show that the moduli spaces of non-triple cyclic coverings of genus 2 curves and that of even spin curves of genus 2 are birationally isomorphic via the Prym map. We describe the log canonical models of the moduli space of even spin curves of genus 2 and use this to extend the above birational map.
We show that Fay's trisecant identity follows from the A_{infinity}-constraint between certain triple Massey products in the derived category of coherent sheaves on a curve. We also deduce the matrix analogue of this identity that can be conveniently formulated using quasideterminants of matrices with non-commuting entries. On the other hand, looking at more special triple Massey products we derive a formula for the tangent line to a canonically embedded curve at a given point.
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…
We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.
We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.
In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.
We give a sufficient condition for the termination of flips. Then we discuss a semi-stable minimal model program for varieties with (numerically) trivial canonical divisor as an application. We also treat a slight refinement of dlt blow-ups.
On termination of minimal model program for log canonical pairs on complex analytic spaces
本文研究复解析空间上log canonical对的最简模型程序终止性，并探讨其与项目性log canonical对理论的关系。
Given a three-dimensional projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps.
We show that the minimal log discrepancy of any $\mathbb Q$-Gorenstein non-canonical threefold is $\leq\frac{12}{13}$, which is an optimal bound.
We investigate the stratified hyperbolicity properties of Birkar's moduli stack of log canonical (lc) stable minimal models. The main technical result is a construction of Viehweg-Zuo's system of Higgs sheaves associated with an admissible family of lc stable minimal models, using the theory of degenerations of Hodge structure and non-abelian Hodge theory.
We prove the normality of minimal log canonical centers on threefold pairs which residue fields are perfect of residue characteristics $p\neq 2,3 $ and $5$. We also show that the union of all log canonical centers on threefold pairs with standard coefficients are seminormal provided that the residue characteristic is large enough. We provide an example of a non-seminormal log canonical center on a threefold in characteristic $3$, and give sufficient conditions to construct similar examples.
In this paper, we prove the boundedness of foliated surfaces $(X,\mathscr{F})$ which are minimal partial du Val resolutions of canonical models $(X_c,\mathscr{F}_c)$ of general type. For applications, we show the boundedness of non-cusp singularities on canonical models of foliated surfaces of general type and the effective generation on the complement of the cusp singularities.
Minimal compactifications of the affine plane with nef canonical divisors
本文研究仿射平面的极小解析紧化，证明具有 nef 典范除子的此类紧化必含无理奇点，与先前对数典范情形形成对比。
We describe the foundation of the log minimal model program for log canonical pairs according to Ambro's idea. We generalize Kollár's vanishing and torsion-free theorems for embedded simple normal crossing pairs. Then we prove the cone and contraction theorems for quasi-log varieties, especially, for log canonical pairs.
Minimal model program for log canonical pairs on complex analytic spaces
本文研究复分析空间上log canonical对的最小模型程序，将Birkar和第二作者的结果推广到Fujino的设定中。
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.
We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $(X,Δ)/Z$ and establish the minimal model theory for the pair $(X,Δ)$ assuming the minimal model theory for all Kawamata log terminal pairs whose dimension is not greater than ${\rm dim}\,Z$. We also show the finite generation of log canonical rings for log canonical pairs of dimension five which are not of log general type.
The minimal model program for normal pairs along the log canonical locus in the complex analytic setting
本文在复解析情形下建立了沿对数典范轨迹的正规对的极小模型理论，推广了作者先前的结果。
On minimal model program and Zariski decomposition of potential triples
本文研究了由对与伪有效除子构成的势三元组的性质，证明若除子允许双有理Zariski分解则可关联广义对结构并运行广义MMP，同时证明了pklt对在特定分解条件下极小模型的存在性。
Minimal plus-one generated line arrangements with double and triple intersection points
该论文在弱组合学框架下，完整分类了复射影平面中具有二重和三重交点的极小加一生成线排列。
We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.
We show that there exists a positive real number $δ>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-δ$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.
Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Goren…
Existence of the minimal model program for log canonical generalized pairs
本文提出线性可分解通用对，结合改进的特殊终止和Kollár型粘合理论，证明了log canonical通用对的最简模型程序存在性，无需假设klt、NQC或$\mathbb Q$-因子性条件。
We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.
Log Canonical Minimal Model Program for corank one foliations on Threefolds
本文研究了三重形上秩为一的叶形的Log Canonical极小模型程序，证明了通过运行$(K_\mcF+\Delta)$-MMP可以得到极小模型或Mori纤维空间，并探讨了极小模型之间的连接及有限数量性质。
We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs. We also discuss log MMP for slc pairs.
We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.
We classify del Pezzo non-commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification of del Pezzo orders over projective surfaces given by Chan and Kulkarni in 2003
In this paper, we show that Shokurov's conjectures on the ACC for $a$-lc thresholds and the ACC for minimal log discrepancies are equivalent in the interval $[0,1)$. That is, the conjecture on ACC for $a$-lc thresholds holds for every $0\leq a<1$ if and only if the set of minimal log discrepancies for pairs with DCC coefficients do not have an accumulation point from below which belongs to $[0,1)$.
Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K^2_S=1$, so called by a minimal $(1,2)$-surface. Then we obtain that the global log canonical threshold of the surface $S$ via $K_S$ is greater than equal to $\frac{1}{2}$. As an application we have \[ {\rm{vol}}(X)\ge\frac{4}{3}p_g(X)-\frac{10}{3} \] for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. Jiang, "The Noether inequality for algebraic threefolds", arXiv:1803.05553] about Noether inequality for $X$ with $5\le p_g(X)\le 26$.
In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double then they are all fiber products of double covers. We construct examples to show that all the possibilities in the classification do exist. There are implications of this classification that include the existence of families with unbounded geometric genus and families with unbounded irregularity, in sharp contrast with the case of double and triple canonical covers. Together with the…
The main idea in this paper is merging two techniques that have been recently developed. On the one hand, we consider MCCGS, standing for Minimal Canonical Comprehensive Groebner Systems, a recently introduced computational tool yielding "good" bases for ideals of polynomials over a field depending on several parameters, that specialize "well", for instance, regarding the number of solutions for the given ideal, for different values of the parameters. The second ingredient concerns automatic theorem discovery in elementary geometry. Automatic discovery aims to obtain complementary hypotheses f…
For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.
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。
In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler…
Let $S$ be a nonsingular minimal complex projective surface of general type and the canonical map of $S$ is generically finite. Beauville showed that the geometric genus of the image of the canonical map is vanishing or equals the geometric genus of $S$ and discussed the canonical degrees for these two cases. We generalize his results to nonsingular minimal complex projective threefolds.
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 […
The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, a deformation of such $X$ could have a…
On the canonical degree of a Gorenstein minimal threefold of general type
本文证明一般型Gorenstein极小三维流形X的典范映射次数不超过72，当p_g(X)>243时，并刻画了等号成立时X的Albanese纤维的性质。
We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kähler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kähler manifolds, flopping contractions, and so on, in order to clarify our results.
In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.
The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.
On minimal varieties growing from quasismooth weighted hypersurfaces
本文研究小典范体积极小簇的构造，通过建立加权超曲面加权爆破的有效nef准则构造了多族极小3-折叠，并证明了一般型极小n-折叠典范体积的有效下界，证明了其在低维最优、高维近似最优。
This document contains notes from the lectures of Corti, Kollár, Lazarsfeld, and Mustaţă at the workshop ``Minimal and canonical models in algebraic geometry" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan.
Let $(X,Δ)$ be a projective log canonical pair such that $Δ\geq A$ where $A \geq 0$ is an ample $\mathbb{R}$-divisor. We prove that either $(X,Δ)$ has a good minimal model or a Mori fibre space. Moreover, if $X$ is $\mathbb{Q}$-factorial, then any Log Minimal Model Program on $K_X+Δ$ with scaling terminates. As an application we prove that a log Fano type variety $X$ with $\mathbb{Q}$-factorial log canonical singularities is a Mori dream space.
We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or "non-commutative algebraic varieties which are finit…
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…
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。
We give an explicit way of writing down a minimal set of generators for the canonical ideal of a non-degenerate curve, or of a more general smooth projective curve in a toric surface, in terms of its defining Laurent polynomial.
On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.
For any admissible subcategory of the bounded derived category of coherent sheaves on a smooth proper variety, we prove that sections of the canonical bundle impose a strong constraint on the supports of the objects of the subcategory or its semiorthogonal complement. We also show that admissible subcategories are rigid under the actions of topologically trivial autoequivalences. As applications of these results, we prove that the derived category of various minimal varieties in the sense of the minimal model program admit no non-trivial semiorthogonal decompositions, generalizing the result f…
Minimal compactifications of the affine plane with only star-shaped singularities
该研究针对复仿射平面的极小紧化问题，在已有对数典范奇点相关分类基础上，完成仅含星形奇点的仿射平面极小紧化的分类工作。
In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in arbitrary dimension and which expresses these invariants through lattice points on the Fine interior. From this formula we derive a formula for the volume of the canonical divisor of the toric hypersurface. We also study the pluricanonical mappings and show under some restrictions how to construct a canonical model of the toric hypersurface.
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