Yasuyuki Kachi, János Kollár
We prove that a smooth projective variety of dimension n is isomorphic to projective n-space iff the canonical class is -(n+1)-times an ample divisor. In characteristic zero this was proved by Kobayashi-Ochiai. We also extend the second adjunction theorem of Ionescu and Fujita to arbitrary characteristic.
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Let $\Bbbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\Bbbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\Bbbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\Bbbk} / G$ is rational for an arbitrary field $\Bbbk$ of characteristic zero.
We show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $χ^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.
We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.
In this paper we compute the number of rational curves with one node passing through a given number of points, lines and tangent to a given number of planes in $\mathbb{P}^3$.
This paper extends previous results of the authors, concerning the behaviour of the equimultiple locus of algebroid surfaces under blowing--up, to arbitrary characteristic.
Log Canonical Thresholds for Plane Curves in Arbitrary Characteristic
本文通过估值理论将复数域上平面曲线的log canonical阈值公式推广到任意特征情况。
The Bogomolov-Gieseker-Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
本文将Bogomolov-Gieseker不等式从特征零光滑射影曲面推广到任意特征、具典范奇点的射影曲面，并推广其若干经典应用。
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary characteristic. Based on this fact we will reduce many geometric properties of the singularities into the problem on jet schemes and try to avoid discussions which are distinctive for characteristic 0.
For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for such curves to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine the exact number of $\mathbb{F}_q$-rational points. In the remaining cases, an upper bound for the number of $\mathbb{F}_q$-rational points will follow from Stöhr-Voloch theory.
This paper has been withdrawn by the author due to the gaps in the proofs of Proposition 2.2 and Proposition 3.2
Arcs on rational double points in arbitrary characteristic
该研究证明纳什问题在所有特征下对二维有理二重点成立，核心方法是直接计算通过这些奇点的弧族。
Deformations of isolated cyclic quotient singularities in arbitrary characteristic
该研究证明了孤立循环商奇点在任意特征下均可变形，并验证了相关猜想。
A family of indecomposable rank-$n$ vector bundles on $\mathbb P^n\times\mathbb P^n$ in positive characteristics
本研究在正特征域上构造了$\boldsymbol{\rm P}^n\times\boldsymbol{\rm P}^n$上的一族不可分解秩$n$向量丛（$n\geq2$），证明其非任意射影空间映射的拉回丛，并详细刻画了$p=n=2$时由单子描述的具体情形。
We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. Then we classify parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţǎ-Nakamura.
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…
$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties
本文计算了低对称幂和分裂托里纤维品种的$\mathbb{A}^1$-欧拉特征数，并展示了其与Pajwani-Pál构造的幂结构预测的一致性。
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $Ø_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $Δ(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.
Over $\C$, Henry Laufer classified all taut surface singularities. We adapt and extent his transcendental methods to positive characteristic. With this we show that if a normal surface singularity is taut over $\C$, then the normal surface singularities with isomorphic dual graph over algebraically closed fields of characteristic exponent $p>1$ are taut for all but finitely many $p$. We conjecture that this is actually "if and only if".
We give an elementary construction of an arbitrary differentially closed field and of a universal differential extension of a differential field in terms of Nash function fields. We also give a characterization of any Archimedean ordered differentially closed field in terms of Nash functions.
The $\mathbb{A}^1$-Euler characteristic of symmetric powers
本文提出通过幂结构理论推测计算变种对称幂的$\mathbb{A}^1$-欧拉特征的方法，并探讨其证据、技术及应用。
A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2
本文研究特征2下ℙ²×ℙ²上的秩2不可分解向量丛E的零集，构造出双次数为(4,4)的非经典Enriques曲面族，并证明双线性正规的此类曲面均为此类零集。
Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.
Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$
本文研究了特征 $p$ 中平展 $\nathbb F_p$-层的六函子形式，证明上同调光滑态射仅由上同调平展态射构成，并刻画了后者在任意特征中实为平展态射（至普遍同胚）。
Fourier--Mukai partners of elliptic ruled surfaces over arbitrary characteristic fields
本文将复数域上椭圆直纹曲面Fourier--Mukai伙伴集合的显式描述推广至任意特征域，并在证明中获得Popa--Schnell猜想的部分证据。
A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number
针对维数不小于2的Q-可分典范弱Fano簇，证明当Q-Fano指数不小于3时其满足Kawamata–Miyaoka型不等式，并给出Gorenstein典范Fano三维簇的Q-Fano指数取值范围。
On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$
本文研究函数域$\mathbb{F}_q(x)$任意Kummer扩张中广义Weierstrass半群的结构、性质及极大元，给出极大元集合的明确描述并应用于特定极大曲线，推广统一了早期成果，为半群与函数域关系提供新视角。
Crepant resolution of $\mathbb{A}^4/A_4$ in characteristic 2
本文在特征 2 下构造了 $\mathbb{A}^4/A_4$ 的 crepant 消解，并通过计算其 Euler 数给出正特征 McKay 对应类似命题的新反例。
For quantum Hamiltonian reductions in arbitrary characteristics, it is known that derived localization holds if and only if the algebra of global sections has finite global dimension. In this paper we provide an alternative characterization of when derived localization holds: Derived localization holds if and only if it holds for an explicit finite set of (quantized) line bundles. As an application, we prove a new result that there are integral weights for which localization holds on in the positive characteristic hypertoric case for $p$ larger than an explicit bound. We also discuss how deriv…
Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.
We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then alternatively, F' can be obtained from F by at most two Galois extensions; if in addition P is zero-dimensional, then we only need one Galois extension. Certain rational places of rank 1 can be uniformized already on F. We introduce the notion of "relative uniformization" for arbitrary finitely generated extensions of valued fields. Our proofs are based solely on…
The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.
Given $\mathbb P^4_k$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset \mathbb P^4_k$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0\mathscr I_Y(s)\ne 0$ and $h^0\mathscr I_X(s)=0$ for some $s>0$. This determines a nonzero element $α\in H^1\mathscr I_X(s)$ such that $α\cdot H=0$ in $H^1\mathscr I_X(s)$. We find different upper bounds of $d$ in terms of $s$, $p$ and the order of $α$ and we show that these bounds are sharp. In particular, we see that $d\le s^2$ for $p<s$ and $d\le s^2-s+2$ for $p\ge s$.
In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi's original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is…
Using Moriwaki's calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g>2. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.
We consider characterizations of projective varieties in terms of their tangents. S. Mori established the characterization of projective spaces in arbitrary characteristic by ampleness of tangent bundles. J. Wahl characterized projective spaces in characteristic zero by cohomological condition of tangent bundles; in addition, he remarked that a counter-example in characteristic two is constructed from odd-dimensional hyperquadrics $Q_{2n-1}$ with $n > 1$. This is caused by existence of a common point in $\mathbb{P}^{2n}$ which every embedded tangent space to the quadric contains. In general, a…
Generalization of Gurjar's Hyperplane section theorem to arbitrary analytic varieties and A$\mathbb{m}$AC classes
本文将Gurjar超平面截面定理推广到交不孤立的任意解析簇，并在形式簇中引入代数m-进闭（AmAC）函数类以扩展定理适用范围。
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
Foliations with small singular set in arbitrary characteristic
研究任意特征下光滑代数簇上叶状结构几何性质，给出极小有理和德尔佩佐曲面上正则叶状结构分类，推广卡马乔 - 萨德指标，证明射影空间叶状结构相关结论，建立$p$-闭叶状结构的博特消没定理版本。
We investigate the vanishing of $H^1(X,\mathcal{O}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,Δ)$ over an algebraically closed field of positive characteristic $p$.
It is known that Chern characteristic numbers of compact complex manifolds cannot have arbitrary values. They satisfy certain divisability conditions. W. Ebeling and S. M. Gusein-Zade gave a definition of Chern characteristic numbers of singular compact complex analytic varieties. We prove that there exists a singular projective variety with an arbitrary set of characteristic numbers.
Let $k$ be a field of arbitrary characteristic. Let $S$ be a singular surface defined over $k$ with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation $\tilde{S}$ is finite dimensional. We give numerical conditions under which the Chow group of zero cycles of $S$ is finite dimensional.
We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.
We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of Borel-fixed ideals and our previous work classifying Hilbert schemes with unique Borel-fixed points.
We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties.
We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.
Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-covers in characteristic $2$
本文研究特征2中Z/2Z覆盖的Ekedahl-Oort类型，通过发展第二种增强微分理论，揭示了覆盖类型与底曲线性质之间的关系。
We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).
In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov's well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics…
In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.
We show that for a reductive group $G$ over a field $k$ the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck-Witt ring $\mathrm{GW}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle which allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.
Classification of objects in the singularity categories of rational double points in arbitrary characteristics
本文研究了任意特征代数闭域上的有理二重点，完全分类了其奇点范畴中的不可分解对象，对应于Auslander-Reiten箭图中的顶点，并给出了确定这些箭图配置的替代证明以及处理孤立超曲面奇点矩阵因子化同伦范畴的计算机代数方法。
Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers.
Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ …
Let $X$ be an integral affine or projective scheme of finite presentation over a perfect field. We prove that $X$ admits a resolution, that is, there exists a smooth scheme $\widetilde X$ and a projective birational morphism from $\widetilde X$ onto $X$.
We prove the equivalence between the category $\mathbf{RigDM}_{et}^{eff}(K,\mathbb{Q})$ of effective motives of rigid analytic varieties over a perfect complete non-archimedean field $K$ and the category $\mathbf{RigDM}_{Frobet}^{eff}(K,\mathbb{Q})$ which is obtained by localizing the category of motives without transfers $\mathbf{RigDA}_{et}^{eff}(K,\mathbb{Q})$ over purely inseparable maps. In particular, we obtain an equivalence between $\mathbf{RigDM}_{et}^{eff}(K,\mathbb{Q})$ and $\mathbf{RigDA}_{et}^{eff}(K,\mathbb{Q})$ in the characteristic $0$ case and an equivalence between $\mathbf{D…
Characteristic numbers of algebras
本文定义有限交换含单位元 C-代数的特征数，用其刻画 Gorenstein 与局部完全交代数，并在 Hilbert 概形相关轨迹上证明恒定性及显式界。
We show that for any integer n and any field k of characteristic different from 2 there are at most finitely many isomorphism classes of quadratic morphisms from the projective line over k to itself with a finite postcritical orbit of size n. As a consequence we prove that every postcritically finite quadratic morphism over a field of positive characteristic can be lifted to characteristic zero with the same combinatorial type of postcritical orbit. The associated profinite geometric monodromy group is therefore the same as in characteristic zero, where it can be described explicitly by genera…
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