Osamu Fujino
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.
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Minimal algebraic space filling curves on the product of projective lines
本文研究了射影线积上最小次数的光滑代数空间填充曲线，证明了存在大量例子并扩展了Homma和Kim的结果。
In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.
Let $X$ be an irreducible algebraic variety over $\mathbb{C}$, endowed with an algebraic foliation ${\cal{F}}$. In this paper, we introduce the notion of minimal invariant variety $V({\cal{F}},Y)$ with respect to $({\cal{F}},Y)$, where $Y$ is a subvariety of $X$. If $Y=\{x\}$ is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through $x$. First we prove that for very generic $x$, the varieties $V({\cal{F}},x)$ have the same dimension $p$. Second we generalize a result due to X. Gomez-Mont. More precisely, we prove t…
Let X be a smooth projective variety of dimension n on which a simple Lie group G acts regularly and non trivially. Then X is not minimal in the sense of the Minimal Model Program. In the paper we work out a classification of X via the Minimal Model Program under the assumption that the dimension of X is small with the respect to the dimension of G. More precisely we classify all such X with n smaller or equal to (r_G +1), where r_G is the minimum codimension of the maximal parabolic subgroup of G (for instance r_{SL(m)}= m-1). We consider also the case when G = SL(3) and X is a smooth 4-fold …
Let S be an algebraic space, A an S-abelian algebraic space, L an S-fiberwise numerically trivial invertible module on A, and L* the sheaf of regular sections of L considered as a G_m-torsor on A. We classify the S-minimal models of L* into two types.
We characterize K. Kato's log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.
Fix a very general hypersurface D in P^n of degree at least 2n + 1 and we show that the complement P^n - D does not contain any algebraic torus C^*.
ICM lecture on minimal models and moduli of varieties.
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.
We prove the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field.
Log algebraic hyperbolicity of $\overline{M}_{0,n}$
本文证明了带边界 Δ 的稳定 n 点有理曲线模空间 $\bar{M}_{0,n}$ 具有代数双曲性。
Algebraic Magnetism Invariants of Self-Actions of Diagonalizable Monoid Schemes
该研究提出一种计算可对角化幺半群概形自作用纯磁体的方法，通过锐幺半群极小生成元描述，且代数磁学能检测幺半群模可逆元后的锐性与极小生成元。
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 discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.
This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an orientable 3-manifold (this assumption can be weakened considerably). Then there is a fairly simple description on how the topology of real points changes under the minimal model program. The first application is to study the topology of real projective varieties which are birational to projective 3-space (Nash conjecture). The second application is a factoriz…
On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field
本文在特征大于5时去除了大性条件，将有限域代数闭包上三维对数典范射影对的基点自由定理推广至一般线丛，并讨论了此类域上的Mori dream空间。
The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero
本文在等特征零情况下，建立了代数和解析空间的相对极小模型程序与缩放，并证明了相关环的有限生成性及GAGA定理的应用。
We discuss the ideal-adic semi-continuity problem for minimal log discrepancies by Mustata. We study the purely log terminal case, and prove the semi-continuity of minimal log discrepancies when a Kawamata log terminal triple deforms in the ideal-adic topology.
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.
Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $X$. This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale and $p$-adic settings). In the specific case in which $k=\mathbb C$ and $X$ is the central fiber of a semistable degeneration over the complex disc, our …
We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log discrepancies for smooth varieties in positive characteristic.
We prove the ideal-adic semi-continuity of minimal log discrepancies on surfaces.
We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.
We describe the set of minimal log discrepancies of toric log varities, and study its accumulation points.
Minimal log discrepancies on a fixed threefold
本文证明任意固定三维簇上最小对数差异满足ACC，为三维簇奇点不变量的有界性提供结论。
We show the equivalence between two notions of log moduli stacks which appear in literatures. In particular, we generalize M.Olsson's theorem of representation of log algebraic stacks and answer a question posted by him (\cite{Ol4} 3.5.3). As an application, we obtain several fundamental results of algebraic log stacks which resemble to those in algebraic stacks.
Minimal log discrepancies on smooth threefolds
本文证明了光滑三维簇上极小对数差异的ACC，并推导出a-lc阈值ACC、一致m-adic半连续性及计算除子有界性等推论。
We completely decide which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the trucated witt rings of lengh 2.
In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for $n$-dimensional minimal log discrepancies of regularity one holds around zero. Furthermore, we show that there exists an upper bound for the minimal log discrepancy of any $n$-dimensional klt singularity of regularity one.
In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications …
We prove the existence of good log minimal models for dlt pairs of numerical log Kodaira dimension 0.
We prove $\mathbb{R}$-boundary divisor versions of results proved by Birkar or Hacon-Xu on special kinds of the relative log minimal model program.
We prove that the existence of log minimal models in dimension $d$ essentially implies the LMMP with scaling in dimension $d$. As a consequence we prove that a weak nonvanishing conjecture in dimension $d$ implies the minimal model conjecture in dimension $d$.
We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.
Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space.
We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.
This paper formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepacy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is $-/infty$.
Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X + D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min\{n-\barκ(V), n\}$ with $V := X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\barκ(V) = 0$ and $\bar{p}_g(V) = 1$. We (re)prove and generalize this classical result for all surfaces with $\barκ=0$ wit…
We construct a log algebraic version of the homotopy sequence for a quasi-projective normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.
Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $Λ\subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n / Λ$, thereby extending a result of Peterzil-Starchenko in the case when $Λ$ is cocompact. We also obtain a similar extension when $X\subset \mathbb{R}^n$ is definable in an o-minimal structure with no restrictions on $Λ$, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset \mathbb{C}^n$ (such as a complex algebraic variety) an…
Approximating Periodic Orbits with Algebraic Curves and Related Minimal Problems
本文提出用低次隐式代数曲线近似周期轨道，用于解决联络导航中的最优化问题，通过符号和数值方法计算解的个数，并设计了基于同伦持续的求解器。
Minimal model program for algebraically integrable foliations and generalized pairs
利用叶状结构理论，建立了一般lc广义对的锥定理与收缩定理，并发展了Q-因子叶状dlt代数可积叶状结构的最小模型纲领，从而完全得到了广义对的典范丛公式，并回答了Cascini与Spicer的一个问题。
The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.
On the density of strongly minimal algebraic vector fields
本文证明了两个定理，展示了任意阶几何平凡强极小自治微分方程的丰富性，并首次给出了在Painlevé意义下为新的且满足高阶自治微分方程的亚纯函数实例。
On the minimal algebraic complexity of the rank-one approximation problem for general inner products
本文研究了在一般内积下秩一近似问题的最小代数复杂度，通过验证弗罗贝尼乌斯内积为ED度数的局部最小值并探讨其全局最小性，结合奇点理论、莫尔斯理论和代数几何工具进行分析。
Minimal Problems for the Calibrated Trifocal Variety
本文通过数值代数几何和Bertini软件确定校准三焦点流形的最优化问题的代数次数。
We define a linear code $C_η(δ_T,δ_X)$ by evaluating polynomials of bidegree $(δ_T,δ_X)$ in the Cox ring on $\mathbb{F}_q$-rational points of the Hirzebruch surface of parameter $η$ on the finite field $\mathbb{F}_q$. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of $\mathbb{F}_q$-rational points of a non-filling curve on a Hirzebruch surface.
We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts, geometrical objects encoding the combinatorial structure of cylindrical cell decompositions in o-minimal geometry. Cylindical decompositions, refinements of such decompositions, and cylindrical parametrizations of such decomposition become morphisms in the category of forts. We formulate and prove the above results in the language of forts.
Every smooth minimal complex algebraic surface of general type, $X$, may be mapped into a moduli space, $\MM_{c_1^2(X), c_2(X)}$, of minimal surfaces of general type, all of which have the same Chern numbers. Using the braid group and braid monodromy,we construct infinitely many new examples of pairs of minimal surfaces of general type which have the same Chern numbers and non-isomorphic fundamental groups. Unlike previous examples, our results include $X$ for which $|π_1(X)|$ is arbitrarily large. Moreover, the surfaces are of positive signature. This supports our goal of using the braid grou…
The second largest accumulation point of the set of minimal log discrepancies of threefolds is $\frac{5}{6}$. In particular, the minimal log discrepancies of $\frac{5}{6}$-lc threefolds satisfy the ACC.
In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta-Iovita-Kim's article: obtaining in this way a complete algebraic criterion for good reduction for curves.
In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarith…
Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer-Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the…
We survey the known and expected properties of the minimal log discrepancy, the local invariant of a log variety.
We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period ma…
In this paper consisting of two parts, we study the integral of a logarithmic differential form on a compact semi-algebraic set in R^n or C^n. In Part I, we prove the convergence of the integral when the semi-algebraic set satisfies allowability (or admissibility), a condition on the dimension of the intersection of the set and the pole divisor of the differential form.
In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.
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